Alternative Loop Rings, Volume 184 (North-Holland Mathematics Studies)

Preface

T h e importance of group algebras has been clear since t h e work of T . Molien, G. Frobenius, I. Schur and H. Maschke in the early 1900s. Often studied because of their applications to group theory, they have also long since a t t r a c t e d attention as objects of interest in their own right. To t h e obvious interplay between group theory and ring theory which is offered through the theory of group rings, in this book, we incorporate a new element, lack of associativity. General quasigroup algebras were defined by R. H. Bruck in 1944, but it was not until the 1980s, with the discovery of a class of loop algebras satisfying important identities, the right and left alternative laws, t h a t the subject was seen to be substantive. For the past ten years, alternative loop rings have intrigued m a t h e m a t i cians from a wide cross-section of modern algebra, including loop theory, group theory and ring theory, both associative and nonassociative. As a consequence, the theory of alternative loop rings has grown tremendously. One of the main developments is the complete characterization of loops which have an alternative, but not associative, loop ring. Such loops are closely related to a class of 2-groups. As an application, a full characterization of the structure of alternative loop algebras has been obtained. In particular, one has complete descriptions of the loop representations in Zorn's vector matrix algebra. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups.

An-

other major topic of research has been the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group rings is extremely fascinating. This is the first survey of the theory of alternative loop rings and related issues. Because of the strong interaction between loop rings and certain ix

X

PREFACE

group rings, we have included many results on group rings, some of which are new and some of which are known though, in the latter case, we often have a new viewpoint and novel, elementary proofs. This work has been written with minimal expectations on the part of the reader. We have assumed that the reader has had the equivalent of a standard first-year graduate course and hence that he or she is familiar with basic ring-theoretic and group-theoretic concepts. We have made an earnest effort to present a work which is self-contained, in the hope that it will prove to be a useful reference to the student as well as the research mathematician. There is an extensive bibliography of references which are either directly relevant to the text or which offer supplementary material of interest. The first three chapters are intended to serve as "short courses" on the fundamental algebraic structures with which this book is primarily concerned—alternative rings, Moufang loops and loop rings. While much of what is presented is undoubtedly in print elsewhere, we are unaware of a single source. The pace is leisurely, all terminology is carefully defined, all theorems are proven in detail. This book is mainly concerned with loop rings which are not associative, but which are alternative and hence nearly associative. An alternative ring is a ring in which the subring generated by any two elements is associative. Associative rings are alternative, though much of the interest in this work derives from the study of alternative loop rings which are not associative. Chapters IV and V discuss the properties of and classify those loops which, in characteristic different from two, have alternative loop rings which are not associative. We call these loops RA ("ring alternative") loops. The remaining chapters are devoted, in the main, to an analysis of the loop rings themselves and to the group rings of certain groups which are closely related to RA loops. In Chapter VI, we describe the Jacobson and prime radicals of an alternative loop ring and establish a Maschke type theorem for alternative loop algebras: over many fields, alternative loop algebras are the direct sums of simple algebras. The structure of alternative loop algebras over a field is further investigated in Chapter VII. We fully describe the simple components of

PREFACE

xi

a (semisimple) alternative loop algebra over the rationals by establishing concrete isomorphisms with Zorn's vector matrix algebras. Furthermore, we give a concrete description of the primitive central idempotents which, as it turns out, are "group determined". This work requires knowledge of the primitive idempotents of the group algebra of a finite abelian group and so, consistent with our goal to write a self-contained work, there is a thorough treatment of this subject in this chapter. In Chapter VIII, we begin a study of the units in an integral alternative loop ring. For example, we determine by elementary methods when all the units are trivial and when all the torsion units are trivial and, as a consequence, also obtain new proofs of the well known theorems of G. Higman and S. D. Berman for group rings. As with group rings, the central units of finite order in the integral alternative loop ring of a finite loop are trivial. Moreover, most such loop rings contain a free subgroup. A finite RA loop L is determined by its integral loop ring; that is, if G is another RA loop and ZG = ZL, then G = L, Thus, the well known "isomorphism question" has a positive answer within the context of integral alternative loop rings (which are not associative), as is shown in Chapter IX. After establishing for composition algebras, an appropriate analogue, due to N. Jacobson, of the Skolem-Noether Theorem for simple associative rings, we also show in Chapter IX that every "normalized automorphism" of ZL, L a finite RA loop, is the composition of an automorphism of L and an inner automorphism of the rational loop algebra QL. We also prove variations of three conjectures which H . J . Zassenhaus has formulated for group rings; for instance, we show that every torsion unit in ZL is conjugate to a conjugate of an element of ±L. We next turn our attention to the loop algebras of RA loops over arbitrary fields where our primary interest is the isomorphism problem. Since abelian groups play such an important role in the structure of RA loops, we first study this problem for group algebras of finite abelian groups and devote an entire chapter. Chapter X, to this topic. In Chapter XI, we apply our results to loop algebras. The structure of the unit loop and, indeed, finding all the units of an integral alternative loop ring, both in the associative and not associative cases, are hard and exciting areas of current research. Even for many loops

xii

PREFACE

of small order, the structure of the unit loop is not known. As a compromise, one aims for the structure of a subloop of finite index in the unit loop or, sometimes, just for a finite set of generators for such a subloop. These are the themes of Chapter XII. Under what conditions does each element of a unit loop possess only finitely many "conjugates" (in the usual associative sense of the word)? This is the central theme of our final chapter, for alternative loop algebras over fields. In the process, we also describe conditions under which all idempotents of a loop algebra are central and determine when aU the nilpotent elements are trivial. At the end of a preface, it is customary to thank family members, colleagues, typists and proofreaders for various things. Certainly, the three of us owe enormous debts of gratitude to our wives, Linda, Griet and Sandra, who have displayed remarkable patience and understanding during our years of work on this project. Many people have read various parts of our manuscript and made helpful comments, including Luiz de Barros, Guilherme Leal, Yuanlin Li, Daniel Robinson, Yiqiang Zhou and Yongxin Zhou. We would, however, like especially to identify and thank Michael Parmenter, our long time friend and colleague, for his extraordinary contributions. As for typists and proofreaders, we have only ourselves to thank, and to blame. The typesetting and proofreading were shared by the three of us, who assume the entire responsibility for errors.

Introduction

The formula (a^ + b'^)(c'^ + d'^) = (ac - bdf + {be + adf, which shows that the product of two sums of two squares is again the sum of two squares, is very familiar and most elegantly established with reference to the field of complex numbers. The first noncommutative "field" (more commonly called "division algebra") was discovered in 1843 by Sir William Rowan Hamilton and is denoted H(R) in Hamilton's honour. This algebra is the four-dimensional vector space over the real numbers, R, with basis {1, i,i, k}. Its elements, called quaternions^ are thus formal sums of the form (1)

q — a-\- bi + ej + dk

where a, b^e^d £ R. Two such sums are added "coordinatewise" (ai + bii + eij + d^k) + (a2 + 62^ + C2J + d2k) = {ai + a2) + (61 + b2)i + (cj + 02)7 + (di + d2)k and multiplied using distributivity and Table 1, which shows how to multiply basis elements. As in the complex numbers, there is in H(R) a notion of TABLE

1. Multiplication table for l , i , j . A: in the quaternions.

I1

~T1nr

i

i i i -1 J 3 -k k \k 3

3

k

k 3 k -3 i -1 —i - 1

2

INTRODUCTION

norm; for q = a-\- bi + cj + dk £ H(R),

\q\ = a' + b' + c' + d\ This norm is multiplicative: for any ^1,^2 € H(R), we have \qiq2\ = This fact corresponds to the formula

\QI\\Q2\'

(aj + al + al + al)(bl + bl + bl + bj) = (ai6i - 0262 - cisbs - 0464)^ + (ai62 + ^^2^1 + ^364 - 0463)^ + («1^3 + ^3^1 - Ct2&4 + ^^4^2)^ + (^164 + ^^461 + a263 - ^^362)^, which shows that the product of two sums of four squares is another sum of four squares. This particular formula plays an important role in number theory; for example, it shows that to prove that every natural number is the sum of four squares, it is sufficient to establish the result just for primes. For what integers n are there identities of the form n

n

n

t=l

t=l

t=l

with each Ci = X)j jt=i ^ijkO^ibj-, Zij^ G C, a biUnear form in the variables a^, bjl The answer, which was given by A. Hurwitz in 1898, is n = 1,2,4,8 [Hur98]. We refer the reader to a beautiful article on the history and solution of the n-squares problem by Charles Curtis [Cur63]. That there exist n-squares formulas for n = 1, 2, 4 and 8 follows from the existence of the real numbers, complex numbers, quaternions and a certain nonassociative algebra called the Cayley numbers. That these are the only integers for which formulas exist is a consequence of the fact that the reals, the complex numbers, Hamilton's quaternions and the Cayley numbers are the only alternative division algebras of finite dimension over the real numbers, a theorem which we prove in Chapter I. Alternative rings arose out of the work of Ruth Moufang in the 1930s [Mou33, Mou35]. Given a projective plane, one can label the points and the lines with elements from a set R and then define the addition and multiplication of elements of R in terms of incidence relations in the plane. (See [Hal59, Chapter 20] for an introduction to projective planes and their coordinatization.) One can then relate various geometrical properties of

INTRODUCTION

3

the plane to algebraic properties of the "ring", (i?, +, •). Two of the nicest theorems in this regard concern planes in which certain theorems due to Desargues and Pappus hold. A plane is desarguesian (pappian) if and only if it can be coordinatized by a planar alternative division ring (field). Since a field is a particular kind of alternative division ring (one which is commutative), a pappian plane is always desarguesian. Since a finite alternative division ring must be a field, a finite desarguesian plane is necessarily pappian. Much of Moufang's attention was directed to the m^ultiplicative structure of an alternative division ring. Just as the nonzero elements of a field form a group under multiplication, so the nonzero elements of an alternative division ring form a Moufang loop under multiplication. Much of this book concerns Moufang loops, which are introduced in Chapter II. Group rings were implicitly introduced in a paper by Arthur Cayley in 1854 [Cay54]. Given Hamilton's definition of a quaternion several years earlier, it was natural to consider more general algebraic expressions c^i^i

+ 0262 +

h

ajiCn,

where the a^ belong to some field F (originally the real or complex numbers) and { e i , . . . ,6^} is a basis for a vector space over F. Multiplication of two elements with the above form could be defined quite naturally by first showing how the basis elements multiply, as with the quaternions, and then extending this multiplication to linear combinations of basis elements by means of the left and right distributive laws. As a specific example, Cayley considered the case where the €{ were the six elements of the symmetric group 53, thereby giving the first instance of the now familiar group algebra 0^3. The importance of group algebras became clear in the early 1900s after the work of T. Molien, G. Frobenius, I. Schur, H. Maschke, and later R. Brauer and E. Noether, on group representation theory. Since then, group algebras have taken on a life of their own. The appearance of two large books on the subject [Seh78, Pas77] at almost the same time made it clear how the subject had grown by the late 1970s to an important field within its own right. The idea of relaxing the requirement of associativity and considering general loop rings or algebras is due to R. H. Bruck, who introduced the idea of a quasigroup algebra in a paper in 1944 [Bru44]. In this work.

4

INTRODUCTION

Bruck proved that over a nonmodular field, the loop algebra of a finite loop is the direct sum of simple algebras; thus, the well known theorem of H. Maschke for group algebras [CR88, Theorems 10.8 and 15.6] holds in the more general nonassociative context. Two years later, Bruck determined the centre of a loop algebra [Bru46], but the subject of loop algebras then seems to have laid dormant until 1955. In that year, Lowell Paige proved that if a commutative loop algebra over a field of characteristic diflFerent from 2 satisfies even the very weak identity, x'^x'^ = x^x, then the loop is an abelian group [Pai55]. In other words, there are no "interesting" nonassociative commutative loop algebras which are not already group algebras. This result strongly suggested that it was fruitless to expect that a loop algebra could satisfy any important identity without, in fact, being a group algebra. Nevertheless, in 1983, E. G. Goodaire showed that there do exist alternative loop algebras (which are not group algebras) [Goo83]. It is this paper which gave birth to a subject which has grown rapidly, as we hope to illustrate with this monograph.

Chapter I

Alternative Rings

1. Fundamentals A (nonassociative) ring is a triple (i?,-h,-)? where (i2,+) is an abehan group, (i?, •) is a groupoid (that is, • is a binary operation) and both distributive laws hold: a(b + c) = ab + ac^ (a + b)c = ac+ bc^ for all a^b^c ^ R. If, in addition, (i2,+) is a module over a commutative, associative ring $ such that a{ab) = {aa)b = a{ab) for all a G $ and all a, 6 G R^ then (i2, +, •) is said to be a (nonassociative) algebra. Except in Chapter VI, we assume that any algebra or ring R is unital; that is, we assume that R possesses an element 1 / 0 , called a unity, with the property that al = la = a for all a € ^ and, in the case of an algebra over a ring $, that J? is a unital ^-module. Unless we indicate specifically to the contrary, we also assume that all fields have characteristic different from 2. The term "nonassociative" means "not necessarily associative". If we intend that a ring (or some other structure) be not associative, we say so. Also, we usually suppress the operations when specifying a ring, referring to the ring "J?", instead of the technically correct "(i?, +,•)"• The concepts of subring, subalgebra, homomorphism and ideal are the same in general as they are in associative algebra and the basic isomorphism theorems hold as well. Nevertheless, certain well known "facts" in associative ring theory are not true in general. For instance, in a ring which is not associative, the square of an ideal / (the linear span of all products of the form xt/, x,y G / ) is not necessarily an ideal. Two important functions in nonassociative ring theory are the commutator and associator which, for elements a^b^c in a ring, are defined 5

6

I. ALTERNATIVE RINGS

respectively by (1)

[a, 6] = ab — ba

and (2)

[a,b,c] = (ab)c-

a(bc).

Each of these is a linear function of each of its arguments. For example, if ai, a2, 6 and c are elements of a ring R, we have [ai + a2,6,c] = [ai,6,c]+ [a2,&,c]; furthermore, if R is an algebra over a ring $ and a ^ ^, then [aa, 6, c] = a[a^ 6, c] for any a, 6, c in R. If X ^Y and Z are subsets of a ring, we often find it convenient to denote by XY the set of products xy, x £ X^ y ^ Y and to denote by [X, Y] and [X^Y^Z]^ respectively, the sets of all commutators [x^y] and all associators [x,y,z], X e X,y eY, z e Z, In any ring, there holds the Teichmuller identity: (3) g{w,x,y,z)

=

[wx, t/, z] - [w, xy, z] + [w, X, yz] - w[x, y, z] - [w, x, y]z = 0, which can be readily verified. 1.1 Definition. The nucleus of a ring R is J\f{R) = {a e R \ [a^ x, y] = [x, a, y] = [x, y, a] = 0 for all x, y G R} and the centre of i? is Z{R) = {ae N{R) I [a, x] = 0 for aU x € i?}. Note that in nonassociative algebra, elements of the centre are required also to associate with all pairs of elements. 1.2 Proposition. In any ring R, the nucleus and centre are subrings.

1. FUNDAMENTALS

7

Since the commutator and associator are linear functions of their arguments, both the nucleus and centre are additive subgroups of (i2,+). li w^x € M{R)^ it follows from the Teichmuller identity that [wx^y^z] = 0 for dX[ y^z ^ i?, so M{R) is also closed under multiplication and hence a subring. Finally, let w^x E Z{R), Then w, x and wx are all in Af{R) and hence in the centre since the calculation PROOF.

{wx)y = w{xy) = w{yx) = {wy)x = {yw)x = y{wx) shows that wx commutes with any y ^ R.

D

Just as in associative ring theory, a nonassociative ring R is simple if the only ideals of R are 0 and R itself. It is not hard to see that the centre of a simple ring with 1 is necessarily a field. 1.3 Definition. A ring R is alternative

if it satisfies

[x,x,y] = 0,

the left alternative

[2/,x,x] = 0,

the right alternative

identity,

and identity.

An alternative ring R with 1 is a division ring if every nonzero element of R is invertible; that is, given any x e R^ there exist y and z in R such that xy = I and zx = I. Any associative ring is alternative, but the converse is not true. In Sections 2 and 3 we shall meet some important examples of alternative rings which are not associative. Linearization is a fundamental and extremely useful tool in nonassociative algebra. By this, we mean the replacement of a repeated variable in an identity by the sum of two variables in order to obtain another identity. For example, linearizing the identity [x,x,y] = 0 involves replacing x hy x + z^ thereby obtaining 0=[x + z,x + z,y] = [x,x,y] + [z,x,y]+

[x,z,y]

+[z,z,y],

from which we conclude [z,x,t/]-|-[x,^,y] = 0 because [x^x^y] = [z^z^y] = 0. Thus, in an alternative ring, [x,z,y] = —[z,x,y] is also an identity. Similarly, linearization of [x^y^y] = 0 gives [x,y,z] = —[x,z,y]. In particular.

8

I. ALTERNATIVE RINGS

alternative rings satisfy (4)

[x, y, x] = 0,

since [x,y,x] = —[x^x^y] = 0. If i2 is a ring, a function f:R^-^Ris permutation o" of 1,2,.. .n,

the flexible identity,

said to be alternating if, for any

f(^^',y] are identities. Since any permutation of {1,2,3} is a product of the transpositions (12) and (23), it follows that [Xa(l)^Xa(2),^a(3)]

= S g n ( a ) [ x i , 2:2, X3]

for any a:i,X2,X3. In expressing a nonassociative product, it is common to employ dots instead of or in addition to parentheses to indicate the order of multiplication, juxtaposition always taking precedence over dot. So, for example, we might write (x • yz)x instead of {x{yz))x. The flexible identity says that xy • x = x ' yx; in other words, the expression xyx is unambiguous in an alternative ring.

1.4 Definition. Given an alternative ring i2, the Kleinfeld function is the function f \ R!^ -^ R defined by f{w, X, t/, z) = [wx, y, z] - x[w, y, z] - [x, y, z]w. 1.5 Proposition (Kleinfeld). The Kleinfeld function is an alternating function of its arguments.

1. FUNDAMENTALS

9

P R O O F . In this proof, as will always be the case, we use g to denote the TeichmuUer identity. For any z,w,x,y ^ R, we have

f{z,w,x,y)

=

f{z,w,x,y)-

g{w,x,y,z)

=

[zw, X, y] - w[z, X, y] - [w, x, y]z - [wx, t/, z] + [w, xy, z] - [w, X, yz] + w[x, y, z] + [w, x, y]z

=

[zw, X, y] - [wx, t/, z] + [w, xy, z] - [w, x, yz]

=

[zw, X, y] - [wx, y, z] + [xy, z, w] - [yz, w, x]

and so f(w, X, y, z) = [wx, y, z] - [xy, z, w] + [yz, w, x] - [zw, x, y] =

-f{z,w,x,y).

Linearizing the identity f{w, X, y, y) = [wx, y, y] - x[w, y, y] - [x, y, y]w = 0 {y ^-^ y -\- z) gives f(w,x,y,z) = —f(w,x,z,y) and hence the result, since the cycles (1234) and (34) generate the symmetric group on {1,2,3,4}. D 1.6 Corollary. In any alternative ring, the following identities are valid, (5)

[x^ y, z] = x[x, y, z] + [x, y, z]x

(6)

[xy,x,z] == [y,x,z]x

(7)

[yx,x,z] = x[y,x,z]

(8)

{{xy)x)z = x{y{xz))

the left Moufang identity

(9)

{{xy)z)y = x{y{zy))

the right Moufang identity

(10)

{xy){zx) = (x(yz))x

the middle Moufang identity

The first three identities are immediate consequences of Proposition 1.5. For (8), PROOF.

{{xy)x)z - x{y{xz)) = ((xy)x)z - {xy){xz) + {xy){xz) -

x{y{xz))

= [xy,x,z] + [x,y,xz] = [y, x,z]x + [xz, X, y] = [y, x, z]x + [z, x, y]x = 0.

10

I. ALTERNATIVE RINGS

Similarly, the right and middle Moufang identities follow from

{{^y)z)y - ^{{yz)y) = {{xy)^)y - {^y){zy) + {xy){zy) - x{y{zy)) = [xy.z.y]^

[^.y.zy]

= -[xy, y, z] - [zy, y, x] = -y[x, y, z] - y[z, y,x] = 0 and {xy){zx) - {x(yz))x = {xy){zx) - ({xy)z)x + {(xy)z)x = -[xy,z,x]

{x{yz))x

+ [x,y,z]x

= [xy, X, z] + [x, y, z]x = [y, x, z]x + [x, y, z]x = 0.

D

1.7 Proposition (Kleinfeld [Kle63]). Let R be an alternative ring with nucleus M. Then 1. [x,7i][x,y,^] = 0 jor all n ^ M and for all x^y^z £ R, and, 2. if [a, 6, x] = 0 for all x e R, then [a, b] £ Af and [a, ab] G AfIn what follows, / denotes the Kleinfeld function, g denotes the Teichmuller identity and we make liberal use of identities (6) and (7). Let x^y^z £ R and n G A/*. Then PROOF.

/ ( x , y, z, n) = [xy, z, n] - y[x, z, n] - [y, z, n]x = 0, so 0 = / ( n , X, y, z) = [nx, y, z] - x[n, y, z] - [x, y, z]n, which gives [nx^y^z] = [x,y,2^]n. Similarly, [xn,y,2r] = n[x,y,z]. But 0=

g{x,n,y,z)

= [xn, y, z] - [x, ny, z] + [x, n, yz] - x[n, y, z] - [x, n, y]z = [ x n , y , z ] - [x,ny,z], so [xn, y, z] = [x, ny, z]. Since [x, ny, z] = [ny, z, x] = [y, z, x]n = [x, y, zjn = [nx,y,z], we obtain [[ri,x],y,>2r] = 0; thus [x,n] G AT. A direct calculation shows that [yx,n] = y[a:,n] + [y,n]x, so 0 = [[yx,n],x,2r] = [y[x,n],x,z] + [[y,n]x,x,z] = [x,n][y,x,z] + [x,x,z][y,n], implying [x,n][y,x,z] = 0 and hence the first statement.

1. FUNDAMENTALS

11

Next, if [a, b,x] = 0 for all x G iZ, then / ( x , t/, a, 6) = [xy, a, 6] - y[a:, a, 6] - [y, a, b]x = 0 and so 0 = g{a, 6, X, y) + f(b, x, y, a) - / ( 6 , a, x, y) = [[a, 6], x, y], which shows that [a, b] G Af. But [a, 6, x] = 0 for all x implies [a, a6, x] = — [a6, a, x] = —[6, a, x]a = 0 for all x, so, as just shown, [a, ab] G JV. D One of the most useful properties of alternative rings is given by the "Generalized Theorem of Artin", which says that if a, b and c are three elements of an alternative ring which associate, then the subring which they generate is associative. This theorem is a consequence of a more general theorem due to R. H. Bruck and E. Kleinfeld [BK51, Appendix I]. 1.8 T h e o r e m . If A, B and C are subsets of an alternative ring R such that [A,A,R] = [B,B,R] = [C,C,R] = [A,B,C] = 0, then the subset D = AL} B U C is contained in an associative subring. always, we denote by / the Kleinfeld function on R and use the fact that this function, and the associator, are alternating and zero if two of their arguments are equal. For subsets X, y, Z, W of denote by / ( X , F, Z, W) the set of elements of the form / ( x , ?/, z, t/;), x£X^y£Y^z^Z^w^ W. Our proof is that originally offered by and Kleinfeld. First we identify certain subsets of R:

P R O O F . AS

freely hence jR, we where Bruck

K = {ke

R\ [k, D, D] = f{k, D, D, D) = 0}

M = {me

K \ mK C K and [m, D, K] = 0}

S = {se M \ [s,M,K]

= 0}.

By assumption, [D^D^D] = 0. Also, for ai,a2 G A and any x,y G i2, we have /(x,t/,ai,a2) = [xy,ai,a2] - y[x,ai,a2] - [y,ai,a2]x = 0; thus f{A, A, R,R) = 0 and, similarly, f{B, B, R, R) = f{C, C, R, R) = 0. It follows that D C K. Next, for di, c/2, ds e D and k G K^ [did2,d3, k] = d2[di,d3, k] + [^2, ds, k]di + / ( d i , ^2, ds, k) = 0

12

I. ALTERNATIVE RINGS

and so, for any k £ K^ d ^ D and x G DD, we have [x, d, A;] = 0. For such k and x, and for any ai,a2 € A, it follows that [aiA;,a2,x] = k[ai^a2jx] + [k^a2^x]ai + / ( a i , fc, a2,x) = 0. Also, for ai^a2 E A^ d £ D and fc G /iT, [aik,a2,d] = k[ai,a2,d] + [/;;, a2,c?]ai + f{ai,k,a2,d)

=0

so, ioT b £ B and c e C^ 0 = [6c, ai,a2A:] = c[6, ai, a2/;:] + [c, ai, a2A;]6 + f(b, c, ai, a2A;) = /(6, c, ai, a2A;). By symmetry, 7(^1,^2,^3, d/j) == 0 for any d, ^1,^2,0^3 G i?. Since also [dk, di, ^2] = ^[d, di, ^2] + [A:, di^d2]d + f(d^ A:, ^1,^2) = 0, we obtain Z^A^ C K; hence D C M. It follows that D C 5 C M C A, so 5 is an associative subset of R. We prove it is the desired subring. Since it is clearly an additive group, it remains only to prove that S is closed under multiplication. For m G M, k G K and ^1,52 G 5 , f{m,k,

81,82) = [mk,8i,82] - k[m, 81,82] - [k,8i,82]m - 0,

since each associator on the right is in [5, M, A'], and so \8x82,m,k\ - 52[5i,m, fc]+ \82,m,k\8x + / ( ^ i , 52, m. A:) = 0. Since 5 C M C A' and MK C K, we have SK C A' and [^1,52,/;:] = 0, so {^8\82)k — 8\{^82k) G A\ Also [5162,m,A;] = 0 implies \8\82',d,k\ — 0, so 51^2 G M. Thus 5 5 C 5 as required. D 1.9 Corollary (Generalized Theorem of Artin). /fa, 6 andc are elements of an alternative ring which associate in some order, then the subring generated by a, 6, c is associative. Since an alternative ring is one in which {yx)x = yx'^ and x{xy) = x'^y are identities, the preceding corollary implies the following "Theorem of Artin".

2. THE REAL QUATERNIONS AND THE CAYLEY NUMBERS

13

1.10 Corollary (Artin). A ring R is alternative if and only if the subring generated by any two elements of R is associative. Without a doubt, Artin's Theorem describes the most fundamental property of alternative rings. As one small application, we derive an identity which will be needed later. 1.11 Proposition. Let Xj y and z be three elements of an alternative ring. Then {xy)[x,y,z\ = y{x[x,y,z\). PROOF.

(11)

First note that

{xy)[x, y, z] = {xy)(xy • z) - {xy){x • yz) = {xyfz

- (xy){x • yz).

Replacing y hy y + w in the left Moufang identity—y{x • yz) = {yx • y)z—we obtain y{x ' wz) + w{x • yz) — (yx • w)z + (wx • y)z and, replacing w by xy, we have y{x{xy'

z)} + {xy){x • yz) = {yx •xy}z+

{{xy • x)y} z = {yx'^y)z +

{xyfz,

by Artin's Theorem. Substituting for {xy){x • yz) in (11), we get {xy)[x,y, z] = {xyfz

+ y{x{xy • z)} - {yx'^y)z -

[xyfz

= y{x{xy'

z)} - {yx'^y)z.

By Artin's Theorem and the left Moufang identity, (yx'^y)z = {yx'^ • y)z = y{x'^ -yz) = y {x{x • yz)} , so the result follows.

D

2. T h e real q u a t e r n i o n s and t h e C a y l e y n u m b e r s The real quaternion algebra is the four-dimensional vector space over R with basis {l,i,j,A:} and multiplication given by first defining multiplication of basis elements according to the rules z^ = p z=: k'^ = ijk = —I and then extending to real linear combinations of basis elements by the distributive laws. This (associative) algebra is denoted H(R) in honour of Sir

14

I. ALTERNATIVE RINGS

William Rowan Hamilton who discovered it in 1843. As we noted in the introduction, the quaternions are historically significant because they provided the first example of a real associative algebra in which every nonzero element is invertible, but which is not commutative. Indeed, the birth of noncommutative ring theory is often credited to their discovery. If q = ai + a2i + a^j + a^k^ ai G R, is a quaternion, the conjugate

of q

is the element (12)

q = ai — a2i — a^j — a^k.

Direct calculation shows t h a t (13)

qq = a\-\- a\-\- a\ + a\

and t h a t (14)

qiq2 = Q2qi

for any qi,q2 ^ H(R). 2.1 D e f i n i t i o n . An involution

of an algebra ^ is a linear m a p a \-^ a oi A

which satisfies ab = ba

and

a = a

for all a,6 G i4. Thus equations (12) and (14) together show t h a t conjugation in the real quaternion algebra is an involution. If ^ is a quaternion, we define the norm of q by n{q) = qq. Notice t h a t n is multiplicative]

t h a t is, n{qiq2) = ^(^1)^(^2) for any ^1,^2 €

H(R), because (15)

n{qiq2) = qiq2qTq2 = ^ 1 ^ 2 ^ W = q\n{q2)qi

= ^(^1)^(^2),

n[q2) being real and hence central. It is this multiplicativity of the norm, together with (13), which shows t h a t the product of two sums of four squares is the sum of four squares, an idea we also mentioned in the introduction. T h e first example of an alternative ring which is not associative is the ring of Cayley numbers which was discovered by J. T. Graves in 1843 and appeared for the first time in print two years later [ C a y 4 5 ] . Like the real

3. GENERALIZED QUATERNION AND CAYLEY-DICKSON ALGEBRAS

15

quaternions, the Cayley numbers form a division ring in which multiplication is not commutative. Unlike the quaternions, however, multiplication of Cayley numbers is not associative. The Cayley numbers, which we denote C, can be constructed from the real quaternions in very much the way that the complex numbers are usually constructed from the real numbers. To wit, a Cayley number is an element of the form a + b£^ where a and b are real quaternions and i is an indeterminate. Elements of C are added in the obvious way (a + b£) + {c+ di) = {a+c) + (b + d)i and multiplied in a way which mimics multiplication of complex numbers, (a + b£){c + di) = (ac - db) + (da + bc)i, where a, 6, c and d are quaternions. The map a + bi y-^ a — bi is an involution in C and n{w) = ww for w £ C defines a multiplicative norm. Since n{w) is the sum of eight squares of real numbers, the existence of the Cayley numbers gives rise to an 8-squares formula. The Cayley numbers form an eight-dimensional algebra over R with basis (16)

{l,iJ,k}u{e,iiJi,M}

where {l,i,j,A;} denotes the usual basis for H(R). The real and complex numbers, the algebra of real quaternions and the Cayley numbers often arise in the same context. For example, these four algebras are the only alternative division algebras of finite dimension over the field of real numbers. It was G. Frobenius who, in 1877, proved that R, C and H(R) are the only finite dimensional associative division algebras over R. The extension of this theorem to alternative algebras is a celebrated result of R. H. Bruck and E. Kleinfeld [BK51]. Although we will later prove this theorem ourselves (Corollary 4.13), we also draw attention to the highly readable and self-contained article by E. Kleinfeld [Kle63].

3. Generalized quaternion and Cayley-Dickson algebras The real quaternions and the Cayley numbers represent special cases of more general algebras which we describe in this section. Our main sources

16

I. ALTERNATIVE RINGS

for the material we present here and in the next section are the books of R. D. Schafer [Sch66, Section III.4] and K. A. Zhevlakov, A.M. Slin'ko, I. P. Shestakov and A. I. Shirshov [ZSSS82, Chapter 2]. Let F be a field of characteristic different frona 2. (Remember that we make this assumption, often implicitly, throughout this monograph.) Let A be a nonassociative algebra of finite dimension n over F with an identity element 1. Suppose that A has an involution a H^ a such that a+a G Fl and aa = aa ^ Fl for all a € A. Given any nonzero a £ F and an indeterminate ^, we construct an algebra 5 , denoted (A, a ) , of dimension 2n over F and with properties similar to ^ , by the Cayley-Dickson process. The elements of B are formal expressions of the form a + 6^, a, 6 G A. Addition and multiplication are given by (a + bt) + (c + di) = (a + c) + (6 + d)i and (a + hi){c + dt) = {ac + adh) + {da + bc)L Defining

a-\-bi = a-be, it is straightforward to show that the map x »-> x defines an involution on B such that x + x and x'x are in F l for any x ^ B. li x = a + bi and y = c + df^^ we have (17)

[x,x,y]

=

{[a, a, c] — a[c, 6,6] — a[a, d, 6]} + {[d, a, a] + [6, c, a] + a[6,6, d]} i. Here we have used freely the centrality of a + a and bb and the fact that 6 and b commute (which is the case because 6 + 6 is central). Furthermore, the centrality oi b + b implies that [c,6,6] = [c,(6 + b) — 6,6] = —[c,6,6]. Using this idea, we can rewrite (17) as (18) [x,x,y]

=

{[a, a, c] + a[c, 6,6] - a[a, d, 6]} + {[of, a, a] + [6, c, a] - a[6,6, d]}^. Now suppose that B is an alternative algebra. Then so is A and 0 = [x, x, y] = —a[a, d, 6] + [6, c, a]^

3. GENERALIZED QUATERNION AND CAYLEY-DICKSON ALGEBRAS

17

for all a^b^c^d. It follows t h a t A is associative. On the other hand, associativity of ^ clearly implies t h a t [x^x^y] = 0 and, similarly, t h a t [x^y^y] = 0. Thus we obtain the following proposition. 3.1 P r o p o s i t i o n . / / A is an algebra with 1 over a field F and a ^ a / 0, then {A^a)

is alternative

if and only if A is

F,

associative.

Iterating the construction of ( ^ , a ) , we can produce algebras of dimensions 4n, 8n, and so forth although, as we shall see, few of these algebras are alternative. Let AQ = F and, for a G AQ, define a = a. The m a p a »-> a is certainly an involution so, if 0 7^ a G F , we can, by the Cayley-Dickson process, construct the two-dimensional algebra Ai = (Ao^a).

For 0 7^ /? G F , the

four-dimensional algebra A2 = (>li,/3), which we denote ( F , a,/3), is called a generalized

quaternion

algebra. Hamilton's real quaternion algebra H(R) is

(R, — 1, — 1) . More generally, for any field F , we write H ( F ) for ( F , — 1, — 1) and call this the quaternion

algebra over F .

For nonzero 7 G F , the eight-dimensional algebra A3 — (^12,7) (F^a^fi^j)

is called a Cayley-Dickson

algebra.

=

Note t h a t the algebra of

Cayley numbers is (R, - 1 , — 1, - 1 ) . Clearly AQ is associative and it can be verified t h a t Ai and A2 are associative as well, whereas A3 is alternative but not associative. Thus, the algebras which we might define by continuing the Cayley-Dickson process beyond A3 are not alternative. We note in passing t h a t a four-dimensional algebra A over a field

F

is generalized quaternion if and only if it has a basis { l , i , j , i j } satisfying i'^ — a ^ F . , p

— (3^F.,a(i^Q

and ji

= —ij.

In this case, it is

routine to check t h a t A = A2 = ( F , a,/3) where Ai = ( F , a ) = F + Fi and A2 = Ai +

Aij.

The algebra AQ is certainly simple and, if a is not a square in F , so is Ai = (F^a).

In fact, if a is not a square, Ai is a field since it is easy to

check t h a t ii a + bi ^ 0^ the element (a — bl)/(a'^ — ab^) is a (two-sided) inverse. On the other hand, if a = C"^ for some ( E F^ then | ( 1 + C~^^) ^^^ 1(1 — C~^i) are orthogonal idempotents with sum 1 and (F^a) By contrast, the algebras A2 and A3 are always simple.

= F Q F.

18

I. ALTERNATIVE RINGS

3.2 P r o p o s i t i o n . / / F is a field of characteristic generalized

quaternion

algebra ( F , a, /?, 7) is PROOF.

different

from

algebra ( F , a,/3) is simple and any Cay

2,

any

ley-Dickson

simple.

P. M. Cohn gives a very elegant proof of the simplicity of a

generalized quaternion algebra in [ C o h 7 7 , §8.6,p.

293]. We a d a p t this

proof to obtain the same property of Cayley-Dickson algebras and omit the proof for quaternion algebras since what follows should make Cohn's proof quite clear. Let ^1,^2,^3 be the elements i used to construct ( F , a ) from F , ( F , a,/3) from ( F , a ) and ( F , a , / J , 7 ) from (F^a^fi)^

respectively, by the Cayley-

Dickson process. Let 1^1,1^2?... , ^^7 be the elements of the set {^1,^2,4} U {^1^2,^14,^24} U { 4 4 • 4 } In the definition of B Q Bi which is part of the Cayley-Dickson process, notice t h a t ib — bi for b ^ B and I = —i. It follows t h a t UiU^ + u^Ui = 0 if i / j and also t h a t each u\ is a nonzero element of F, Now let A be any nonzero homomorphic image of ( F , a , / 3 , 7 ) .

Since

A is alternative, the subalgebra generated by any two elements of A is associative, by the Theorem of Artin. Suppose t h a t Vi is the image in A of Ui. Since vl / 0 and ViV^ + v^Vi = 0 for i 7^ j , no Vj is a scalar. Suppose (19)

aol + a^vi -f a2^2 +

h ^7^7 = 0

for scalars a o , . . . ,a7. Multiplying on the left by v~^ and on the right by Vj changes the term aiVi to —aiVi if i ^ j , while leaving ajVj unchanged. Adding the resulting equation to (19) gives 2aol + 2ajVj = 0. Since Vj is not a scalar and char F / 2, we get aj = ao = 0. Thus all the coefficients in (19) are 0, so {1, t;i, t>2, • • • ,^7} is a linearly independent set in an algebra of dimension at most eight. It follows t h a t dim A = 8 and A = Since (F^a^(3^j)

{F^a^l3^^).

has no proper nonzero homomorphic images, it is simple.

D 3.3

Split q u a t e r n i o n a n d C a y l e y - D i c k s o n a l g e b r a s . Let A be any

unital algebra with involution a -^ a such t h a t aa G i^l for every a ^ A. li A is a division algebra, it certainly has no (nonzero) divisors of 0. Conversely, if A has no (nonzero) divisors of 0 and a e Ais nonzero, then aa is a nonzero

3. GENERALIZED QUATERNION AND CAYLEY-DICKSON ALGEBRAS

scalar, so a is invertible with inverse a(aa)~^.

19

In particular, if an algebra

constructed by the Cayley-Dickson process is not a division algebra, then it has nonzero divisors of zero. In the latter case, the algebra is called split and, as we shall soon see (Theorem 4.16), it is unique up to isomorphism. The algebra ( F , 1) is split because it is easy to see t h a t ^ ^ and ^ ^ are orthogonal idempotents, hence ( F , 1) = F^^

©F ^

= F ® F. In fact.

a + bi i-^ (a + b^ a — b) is an isomorphism (F^l)

^

F ® F and the involution a + bi ^-^ a — bi on

( F , 1) corresponds to the exchange involution (a, b) y-^ (6, a) in F © F . The algebra ( F , 1,—1) is split because it is isomorphic to the full ring M 2 ( F ) of 2 X 2 matrices over F . Indeed, one can check t h a t

(a, 6)+ {c,d)i\

a -d

is an isomorphism ( F , 1, — 1) -^ M 2 ( F ) . With respect to this isomorphism, the involution in ( F , 1, — 1) becomes the familiar adjoint: -b a On the other hand, as shown in the next result, it is more complicated to determine whether or not H{F) — ( F , — 1 , - 1 ) is split. The first part of this theorem is classical (see, for example, [ C o h 7 7 , §8.6, p. 294]); its extension to Cayley-Dickson algebras is due to L. G. X. de Barros [ d B 9 3 b ] . 3.4 T h e o r e m . A generalized

quaternion

algebra (F^a^f3)

only if the equation ax^ -\-(3y^ — z^ has nontrivial

solutions

is split if and in F. A

Cayley-

Dickson algebra ( F , a , / 3 , 7 ) is split if and only if the equation x^ — ay'^ — Pz'^ + aj3w^ = jt^ has nontrivial the existence of nontrivial

solutions

in F, and this condition

is equivalent

to

solutions in F to either x^ — ay^ — /Sz^ -\- a/Sw^ = 7

or ax^ + ^y^ — z^. PROOF.

Analysis of the Cayley-Dickson process shows t h a t one can find

F-bases {6o == 1,€1,62,63} of ( F , a , / 3 ) and { e o , e i , e 2 , . . . ,€7} of ( F , a , / 3 , 7 ) whose elements multiply according to Table 1. Suppose t h a t ( F , a , / 3 , 7 ) is a division algebra and t h a t a, 6, c, d, / are elements of F , not all 0, such t h a t a} — ab^—^(P'-\-aii^

— 7/-^. Then, for q — aeo + bei+ce2 + des + fe4^ we have

20

I. ALTERNATIVE RINGS

T A B L E 1. A multiplication table for a basis of ( F , a , / 3 , 7 )

1

ei

e2

es

64

^5

ee

er

1

ei

e2

es

64

ee

e?

ei

ei

a

es

ae2

€5

es a64

-^7

-c^ee

e2

^2

-es

/?

-/?ei

^6

e?

/5e4

/^e5

1 1I es

es

-ae2

fie.

e4

64

-^5

-e6

es

^5

—ae4

ee

^6

e?

e? 1 e?

aee

-ej

-Pe4 -Pes

-a/3

,

e7

Q;66

-/3e5

—a/364

-€?

1

-761

-7^2

-7^3

-aee

761

—a7

a762 1

/Jes

7^2

-7^3 -a762

7^3 -/37

a/3e4

7^3

-he\ 1

/?76i

a/37

g = aeo — bei —662 — ^/63-/64 and ^^ = a^— ab^ —/3c^ + a/3d^ —jf^ there exist nonzero divisors of zero in (F,a^/3^j),

= 0. Thus

so this algebra is not a

division algebra, a contradiction. Thus the equation a:'^ — a?/^—/3z'^+a/3ii;^ = 7^^ has no nontrivial solutions in F, Similarly, one shows t h a t if (/^, a, /3) is a division algebra, then the equation ax^ + f3y^ = z^ has no nontrivial solutions. For the converses, we first show t h a t if the equation ax^ + fiy^ — z^ has no nontrivial solutions, then (F^a^P)

is a division algebra.

let 0 7^ ^ = C060 + 6161 + 6262 + C363 e {F,a,(3),

For this,

60,61,62,63 G F.

Write

q = r + 862 with r — 6060 + 6161 and .s = 6260 + 6361. If s — 0, then the assumption implies t h a t

6Q

— ac\ ^ 0 and thus q = r =

CQCO

+ C\ei

has inverse {c^ — a6i)~^(6o6o — 6161). On the other hand, if 5 7^ 0, then s is invertible in ( F , a,/3) and we can write s~^q — rfo^o -\- d^e, + 62, with do^di G F. Again, by the assumption, C/Q — Oid\ — /3 7^ 0, and thus s~^q has inverse (C/Q - Oidl - /3)"^(c/o6o - rfi6i - c/2). So q is invertible in (F^a^/3).

Hence, {F^a^/3) is a division algebra.

Next, we consider the Cayley-Dickson algebra ( F , a , / 3 , 7 ) .

Clearly, if

the equation x^ — ay^ — /Sz'^ + afiw^ = jt^ has no nontrivial solution, then neither x"^ — ay^ — Pz'^ + a(iw'^ = 7 nor ax^ + Py^ = z"^ has a nontrivial solution. Thus (F^a^P)

is a division algebra and the equation x^ — ay^ —

3. GENERALIZED QUATERNION AND CAYLEY-DICKSON ALGEBRAS

fiz'^ + afiw^

= 7 has no nontrivial solutions.

conditions imply t h a t (F^a^f3^j)

21

We show t h a t these two

is a division algebra.

Observe t h a t the

nonzero elements in ( F , a , / ? , 7) of the form gi = aoeo + a i e i + a2€2 + a3€3 + e4 and q2 = 60^0 + ^1^1 + 62^2 + ^363 are invertible, with -1 Qi

m 1 \ a o e o - aiei - a2e2 - aaea - 64)

and Q2 ^ = ^ 2 ^ 6 0 ^ 0 -hei

- 62^2 - ^a^a),

where mi =: a Q - a a i - / ? a 2 + a / J a 3 - 7 / 0 and 7722 = bl-abj-Pbl

+ aPbl ^ 0

(because (F^a^/S) is a division algebra). Let g' = Y^J=o^i^i 7^ 0 be in (F^a^j3^j)

and write q = r + 864 with

r = aoeo + ai6i + 0262 + ^363 and s = a^e^ + a^ei + 0562 + 0763. If 5 = 0, then gr == r 7^ 0 is invertible in (/^, a,/3). If 5 7^ 0, then 5 is invertible in (F, a,/3) and we can write s~^q — 60^0 -\-b\e\-\- 62^2 + ^3^3 + ^4 with 60, ^ 1 , 62 and 63 in F . Thus s~^q is invertible in (F^a^fi^j)^

For example, in case F = Q or R, H{F)

so g' is invertible in

= ( F , —1,—1) is a division

algebra because —x^ — y'^ = z^ has no nontrivial solutions. The Cayley numbers (R,—1,—1,—1) form a division algebra because x'^ + y'^ + z"^ + w'^ = — t^ has no nontrivial real solutions, whereas (R, 1 , - 1 , - 1 ) is a split CayleyDickson algebra over R. We now show t h a t this algebra has a particularly nice realization. 3.5

Zorn's v e c t o r m a t r i x a l g e b r a .

For any commutative and associa-

tive ring R (with unity), let R^ denote the set of ordered triples over R and consider the set of 2 X 2 matrices of the form a

X

J \ where a^b ^ R and x,y E /?^. Add such matrices entrywise. but multiply them according to the following modification of the usual rule: «i

xi

a2

X2 _

yi

^ij |_y2 &2J

aia2+xi-y2

[ct2yi + &iy2+ xi X X2

^^1X2 + 62X1 - yi X y21

frit2

+ yi-x2

where • and x denote the dot and cross products respectively in i2^. By this construction, we obtain an alternative algebra which we denote 3 ( ^ )

I. ALTERNATIVE RINGS

22

in honour of Max Zorn who used such an algebra to represent the Cayley numbers (with R = R, oi course). In the case t h a t i2 is a field F , Zorn's algebra is precisely the split Cayley-Dickson algebra ( F , — 1 , - 1 , 1 ) , as we proceed to show. Replacing ^ by i in the construction of ( F , — 1), the elements of ( F , — 1) are of the form ai + ^2^, where a i , a 2 € F.

Replacing £ by j in the

construction of ( F , —1,—1), the elements of ( F , —1,—1) are of the form (ai + ci2i) + (^3 + o.4i)j' Thus the elements of ( F , —1, —1) are of the form a = a + X where a £ F and x = Xii + X2J + x^k^ Xi € F . Let p + qi e ( F , - 1 , - 1 , 1 ) , p,q e ( F , - 1 , - 1 ) . Write p = a + x and q = b + y where a^b £ F , x is as above and y = yii + y2J + ysk, yi G F . It is helpful to think of x and y both as quaternions and as vectors in F ^ , by identifying 2, j and k with the usual unit vectors i, j , k. It is routine to show t h a t a -\- b y — X

P + qi

y+ X a— b

is an isomorphism ( F , — 1 , - 1 , 1 ) ^ 3 ( F ) . Furthermore, the involution p + qi \-^ p — qi on ( F , — 1, — 1, 1) corresponds to the map b

(20)

-X

-y

in 3 ( F ) , and the norm x

a

XX on ( F , —1, —1, 1) corresponds to the m a p ab — X' y 0

0 ab — X

In particular. (21)

det:

H^ a6 — X • y

is a multiplicative map which, as its label suggests, we call the determinant. The Cayley-Dickson algebra ( F , 1,—1,—1) is also split and isomorphic to 3 ( F ) . Representing ( F , 1, —1) as M 2 ( F ) , as in §3.3, it can be shown t h a t

\a b c d

+

u w

V X

i^

a (c, —x,t;)

(b^u^w) d

4. COMPOSITION ALGEBRAS

23

is an isomorphism, and that the canonical involution and norm on the Cayley-Dickson algebra M2(F) ® M2{F)£ correspond respectively to the involution (20) and determinant (21).

4. C o m p o s i t i o n algebras Let $ be a commutative and associative ring with unity and of characteristic different from 2. Suppose F is a unital (left) module over $. While we shall assume that $ is a field in much of this section, we find it convenient to record some definitions in the more general setting. 4.1 Definition. A bilinear form on K is a function f: V X V -^ ^ which satisfies 1. f{u + v,w) = f(u,w) + f{v,w), 2. f{u, V + w) = f{Uy v) + f{u, w), and 3. f{au,v) - f{u,av) af(u,v) for all u^v ^ V and a G ^ . The form is symmetric if f{u^v) = f{v^u) for aM u^v ^ V and nondegenerate if f(v^x) ~ 0 for all x G V and some v ^V implies t; = 0. If / is a bilinear form on a ^-module V and if f/ is a submodule of K, the submodule l]-^ (pronounced "f/ perp") is defined by \J^ ^{v^V

\ f{v, u) = 0 for all u G U}.

Notice that nondegeneracy of / on K is just the assertion that V-^ = {0}. Suppose V is free of finite rank with ^-basis S = {ei, 6 2 , . . . , e^}. For u = Y^ aiCi and v = ^ biCi in V, bilinearity of / implies n

f{u,v)

= ^

aibjf{ei,ej)

= u^Av

where u^ and v^ are the row vectors ( a i , a 2 , . . . ^an) and (61,62?-•• -i^n) respectively, and A is the matrix [/(e^, e^)]. This matrix is called the matrix of f relative to £. Suppose that !F is another basis for V and that u^ and v^ are the row vectors of coordinates of u and v^ respectively, with respect to ^ . Then

24

I. ALTERNATIVE RINGS

u = Fui and v = P v i for some invertible matrix P. Hence

f{u,v) = u'(p'^APn. showing t h a t the matrix of / relative to !F is P^AP.

It follows t h a t the

matrix of / relative to some basis of V is invertible if and only if t h e m a t r i x of / relative to any basis of V is invertible. The following proposition is basic. 4.2 P r o p o s i t i o n . A bilinear form f on a finite V is nondegenerate of V is

if and only if the matrix

dimensional

vector

space

of f with respect to any basis

invertible.

PROOF.

Let ^ = { e i , 6 2 , . . .,67^} be a b a s i s for

V^.

Bilinearity shows t h a t

f(x^ v) = 0 for all V E V if and only if f{x, Cj) = 0 for all ej G ^ . Writing X = J2 ^i^i ^^^ denoting by x-^ the row vector ( x i , X 2 , . . . , Xn)-, we see t h a t X G V-^ if and only if x^A

= 0, where A is the matrix of / relative to C,

Since this equation has a nonzero solution if and only if A is not invertible, the result follows.

D

Let / be a bilinear form on a ^ - m o d u l e V, An element v E V \s isotropic if / ( t / , u) = 0 and a submodule U of V is totally isotropic if every element of U is isotropic. 4 . 3 C o r o l l a r y . Let f be a nondegenerate vector space V of finite dimension from 2.

Then any totally isotropic

symmetric

bilinear form

over a field F of characteristic subspace of V has dimension

on a

different at most

|dimV. PROOF.

Let e i , e 2 , . . .,6^; be a basis for [/, extend this to a basis E of

V and let A be the matrix of / relative to 8.

Since f{u^ tz) = 0 for all

u E U and since char F / 2, it follows t h a t f{u^v) In particular, f^Ci^Cj)

= 0 for all u^v E U.

— 0 for all i , j G { 1 , 2 , . . . , A:} so t h a t A h a s di K /\ K

block of zeros in its upper left hand corner. Since det ^ / 0, A; < ^ dim K, as required.

D

Recall t h a t for submodules U and W of 3b ^-module F , the sum of U and W is the subspace U + W defined by

u + w = {u + w\ue u,we w}.

4. COMPOSITION ALGEBRAS

25

The sum is said to be direct^ and we write U ® W^ if every element ofU + W can be written uniquely diS u + w] t h a t is, if ui + wi = U2 + W2', Ui^U2 ^ U^ w\^W2 G W^ implies u\ = U2 and w\ = W2. Clearly U -{-W \s direct if and only if C/n W = {0}. The dual space of a ^-module V is the ^ - m o d u l e y* = { r | T : V - > $, TUnear} of linear functionals on F . If / : V X F —> $ is a bilinear form on V and V ^ V^ the m a p (^y'. V —^ ^ defined by (^v{x) = f{v^x)

for x G V^ is an

element of V* (because / is linear in its second argument) and t h e m a p (f: V -^ V

defined by (p{v) = (fy is linear (because / is linear in its first

argument). 4 . 4 P r o p o s i t i o n . Let f:VxV—^Fbea a finite dimensional

nondegenerate

vector space V over a field F,

(i) The linear map (f:V—^

bilinear form on

Then

V*, defined above, is an

isomorphism.

(ii) / / U is a subspace of V, then dim V = dim U + dim f/-^. (iii) / / U is a subspace of V such that U nU^ PROOF,

= {0}, then V = U ® U-^.

(i) Let { e i , e 2 , . . . , e ^ } be a basis of V.

For i = 1 , 2 , . . . , n ,

define T^ G V* by

{

1

if i = J

0

otherwise.

The set { T i , . . . ,Tn} is linearly independent since if ^aiTi

— 0, evaluation

at Cj gives aj — 0. It spans V* because if T G K*, then T = ^ c^zT'z, where ai = T(ei),

Hence dim V = dim K*.

Since / is nondegenerate, we have kev(f = V-^ = {0}. Since dim K = dim y * < 00 and (f is one-to-one, it is necessarily surjective. (ii) For v eV,

define V^: V ^ (7* by {i^{v)){u)

= f{v,u).

T h e kernel of

V^ is precisely f/-^ so, by a theorem of elementary linear algebra, dim V = dim [/-'- -h dimim^/?. Since dim [/* = dim ?7, t h e proof will be complete if we establish i m ' 0 := U*. For this, suppose T G f/*. Thus T is a linear m a p U -^ F which can be extended to a linear m a p T': V -^ F^ t h a t is, to an element of V*.

By part (i), T ' = (fi(v) for some 1; G V , so T , being the

26

I. ALTERNATIVE RINGS

restriction of T' to C/, is the restriction of (p{v) to f/, which is ip(v). So ^^ is surjective, as required. (iii) For any subspaces U^ W oi V, we have dim(f/ + W) = dimU + dim Pr - dim{U D W). Since [/ n t/^ = {0}, dim(U + [/^) = dim [/ + dim U^. Thus f/ + C/-^ = F by part (ii).

D

4.5 Corollary. / / / is a nondegenerate bilinear form on a finite dimensional vector space V and if U is any subspace ofV^ then (U-^)^ = U. Clearly U C (U^)-^ and these subspaces have the same dimension since dim U + dim [/-'- = dim V = dim f/-^ + diin{U^)^. D PROOF.

4.6 Corollary. Let f be a nondegenerate bilinear form on a finite dimensional vector space V and let U be a subspace of V, Then the following statements are equivalent. (i) The restriction of f to U is nondegenerate.

(ii) t/nf/^ = {0}. (iii) The restriction of f to U-^ is nondegenerate. If / is nondegenerate on a subspace U and u ^ U f) U-^^ then f{u^v) = 0 for all t; G f/, so ^ = 0 and (i) implies (ii). Conversely, assume (ii), let u e U and suppose f{u^v) — 0 for all v ^ U. By Proposition 4.4, V = U + f/-"-, so /(ii, v) = 0 for all v £ V. Since / is nondegenerate, u = 0. Thus (i) and (ii) are equivalent. Noting that (ii) is also the statement f/-^ r\{U-^)^ = {0}, we see also that (ii) and (iii) are equivalent by the same reasoning which established the equivalence of (i) and (ii). D PROOF.

4.7 Definition. A function q: V -^ ^ is di quadratic form on the 4>-module V if 1. q{av) = a^q{v) for all a € $ and all v ^ V., and 2. the function / : (u^v) \-^ q{u + v) — q{u) — q(v) is bilinear on V. We refer to / as the bilinear form associated with q and caD q nondegenerate if the associated bilinear form is nondegenerate.

4. COMPOSITION ALGEBRAS

27

4 . 8 D e f i n i t i o n . A unital algebra A over a ring $ is quadratic

if each

element of A satisfies an equation of the form (22)

x^ -t{x)x

where t{x)^n(x)

+ n{x)l

= 0

G $ . The elements t{x) and n{x) are known, respectively,

as the trace and norm of x. The trace and norm of an element x ^ $ 1 are clearly unique.

Upon

defining (23)

t{al)

= 2a

and

n ( a l ) = a^

for Q; G ^ , we are assured t h a t t{x) and n ( x ) are uniquely defined for any X e A. Any algebra which arises from the Cayley-Dickson process is quadratic, with t(x) = X +'x and n{x) = x'x. 4 . 9 D e f i n i t i o n . An algebra A with unity over a field 7^ is a

composition

algebra if there is a nondegenerate quadratic form q on A such t h a t q{ab) — q{a)q{b) for all a^b £ A. A form with this multiplicative property is said to permit

composition.

The complex numbers form a composition algebra with respect to the form q defined by q{a + bi) = a'^ + 6^. As we showed in Section 2, the real quaternion algebra and the Cayley numbers are composition algebras. These algebras are division algebras whereas Zorn's vector matrix algebra is a composition algebra with respective to the determinant function (21), which is not a division algebra. Quadratic alternative algebras are almost composition algebras, as the following theorem shows. 4 . 1 0 T h e o r e m . Let A be a quadratic algebra over a field F of tic different from 2. Then the trace t is linear and the norm n is If in addition, PROOF.

A is alternative,

then n{xy) = n(x)n{y)

+ n{ax)l

Multiplying (22) by a^ gives a^x^ - a'^t{x)x + a ^ n ( x ) l = 0, so — n(ax)l

= a^t{x)x

—

quadratic.

for all x^y £ A.

Replacing x by ax in (22) gives a^x'^ — at(ax)x

at{ax)x

characteris-

a^n{x)l.

= 0.

28

I. ALTERNATIVE RINGS

Thus, if 1 and x are linearly independent and a / 0, we get t{ax)

=

at(x)

and n(ax)

= a^n(x).

=

at{x)

and n(ax)

= a^n(x)

Our assumptions in (23) show t h a t t{ax)

if x G F l , so these equations hold for all a G F and

X E A. To prove t h a t t is linear, it remains to show t h a t (24)

t{x + y) = t{x) + t{y)

for all x^y ^ A. This is clearly the case if both x and y are in Fl.

Suppose

X ^ F l . Let a,/3 G F , then, in (22), replace x hy ax + (31 and then x^ by t(x)x

— n(x)l.

a'^{t{x)x

We obtain

- n(x)l)

+ 2af3x + /J^l - t(ax + pi){ax

+ pi) + n{ax + /?1)1 = 0.

Since x is not in F l , the coefficient of x on the left of this equation must be 0; t h a t is, a'^t{x) + 2a(3 - at{ax

+ (31) = 0.

If a ^ 0, we obtain t{ax -\- (31) = at{x) + 2(3, Since this equation is also true if Q; = 0, it holds for all a ^ F. Thus it suffices to estabfish (24) in the case x ^ Fl^ y ^ Fl.

Moreover, since what we have just shown impfies t h a t

(24) holds if, for example, y = ax -\- (3^ we may also assume t h a t , whenever we have an equation of the form a x + /?i/ + 7 I = 0, necessarily a = /3 = 0. Replacing x by a: + 2/ in (22) gives x'^ + xy + yx -\- y'^ - t{x + y){x + y) + n{x + y)l - 0, and replacing x hy x — y \n (22) gives x'^ - xy - yx -\- y'^ - t{x - y){x - y) + n{x - y)l = 0. Adding these equations and replacing x"^ by t{x)x — n(x)l n{y)l

and y'^ by t{y)y —

gives

2{t{x)x

- n{x)l)

+ 2{t{y)y -

- t(x + y)(x + y)-

n{y)l)

t{x - y){x - y) + n{x + y)l + n{x - y)\ - 0.

Using our assumption to set to 0 the coefficients of x and y, we obtain 2t{x) - t{x + y)-t{x

- y) = 0

4. COMPOSITION ALGEBRAS

29

and 2t{y) - t{x + y) + t{x - y) - 0. Addition and division by 2 gives the desired result. To prove that n is quadratic, it remains to prove that the function / defined by / ( x , y) = n{x + y)-

n{x) - n{y)

is bilinear. This follows immediately from {x + y^ — t(x + y){x -\-y) + n{x + y)l = 0 and linearity of /, which shows that (25)

/ ( x , y) = t{x)y + t{y)x - {xy + yx).

Now suppose that A is alternative (and quadratic). We use Artin's Theorem (Corollary 1.10) freely, constantly employing the fact that the subalgebra generated by any two elements of A is associative. Multiplying (25) on the right by xy gives / ( x , y)xy = t{x)yxy + t{y)x'^y - (xyf

- yx'^y

= t{x)yxy + t{y){t{x)x - n{x)l)y - {xyf - y{t{x)x - n{x)l)y = t{x)yxy + t{x)t{y)xy - t{y)n{x)y - {xy^ - t{x)yxy +n{x){t{y)y - n{y)l) = t(x)t(y)xy - (xy)'^ - n(x)n{y)l and hence {xyf

+ / ( x , y)xy - t{x)t(y)xy

+ n{x)n{y)l

= 0.

Comparison with {xyY — t{xy)xy + n{xy)\ = 0 gives n{xy) = n{x)n{y) unless xy E Fl. To establish that n is also multiplicative in this case, we may assume that x (and hence also y) are not in F l , since (23) gives the result for elements of F l . So let xy = a l , a £ F^ with x ^ Fl and y ^ Fl, Multiplying x'^—t{x)x+ n{x)l = 0 on the right by y^ and noting that xy^ — ay and {xy^" = a-^l, we have a^l — at{x)y + n[x)y^ = 0.

30

I. ALTERNATIVE RINGS

If n{x) = 0, then a = 0 too; otherwise, a l — t{x)y = 0, so t(x) ^ 0 and y G Fl, Thus, when n{x) = 0, we have n(xy) = n{al) = 0 = n{x)n{y)^ as desired. In the case n{x) / 0, we have a^ ^ n[x)

at(x) n[x)

9

which, by comparison with y^ — t{y)y + n(y)l = 0, gives n{y) = and hence n{x)n{y) — o? — n ( a l ) = n{xy).

a'^/n(x) O

4.11 Theorem (Hurwitz). Let A be a composition algebra over a field F of characteristic different from 2. Then A is an alternative algebra, specifically, an algebra of one of the four types—F, (F^a), (F, a,/3), (F, a,/3,7) — which arise from the Cayley-Dickson process. In particular, any composition algebra which is not associative is a Cay ley-Dickson algebra. Conversely, any of the algebras F, (F^a), {F^a^/3), (F, a,/?,7) is a composition algebra. Let / be the bilinear form associated with the given quadratic form q on A; that is, PROOF.

/ ( x , y) = q{x + y)~ q{x) - q{y) for x^y E A. Note that / is symmetric. Since f(x^x) — 2q(x) and q is multiplicative, we have f{xy^xy) — 2q{xy^xy) = 2q{x)q{y) = f{x^x)q(y). Linearizing (x ^-^ x + z) gives f{xy, zy) + f{zy, xy) = f{x, z)q{y) + f{z, x)q{y) for all a:, y and z m A and hence (26)

f{xy,zy)

=

f{x,z)q(y)

since / is symmetric and char F 7^ 2. Setting y = 1, we see that g(l) = 1; otherwise f(x^z) — 0 for all x and z in ^ , contradicting nondegeneracy. Linearizing (26) {y y~^ y + w) gives f{xy, zy) + f{xy, zw) + f{xw, zy) + f{xw,

xw)

= / ( x , z)q{y +w) = f{x, z)[q{y) + q{w) + / ( y , w)] and thus (27)

f{xw, zy) + f{xy, zw) = f{x, z)f(y,

w),

4. COMPOSITION ALGEBRAS

31

SO, setting z = x, we get (28)

f{xy,

for all x,y,w

e A.

0 for all u eU}, (29)

xw) = f{y,

w)q(x)

Now let U = Fl

and U-^ = {a e A \ f{a,u)

=

With z = 1 and x e U-^ in (27), we get f{xw,y)

+ f{xy,w)

= 0

for an y,w e A and x G U-^. Since [/ n f/-^ = {0} (for example, / ( 1 , 1 ) = 2 ^ 0), we have A = U ® JJ-^ by Proposition 4.4. Thus any a G A can be written uniquely as a = a l + a' with a' G U-^. Then / ( a y , ti;) = / ( a y , ii;) + f{a'y,

w)

= f{ay,w)-f{a'w,y) = f{otw,y)-

by (29) f{a'w,y)

= /((al -

a)w,y).

Defining (30)

a =

a\-a'

we obtain (31)

fio^y-,^)

= f(G^'w,y)

for all a^y^w ^ A. Similarly, setting w = I and y G U-^ in (27), we get f(xy,z)

+ f(x,zy)

= 0

for all x^z ^ A and y G f/"*" and hence, again writing a = al + a', f{xa, z) - f{oLX, z) + f{xa', = f{x,az)

- f{x,za')

z) = f{x,z{al

Therefore (32) for all a^XyZ^

f(xa,z)

=

f{za,x)

A. So, for any a^y^w £ A^ f{ay,w)

= f{aw,y)

by (31)

= f{yw,a)

by (32)

= /(I/^,^)

by (31).

- a')).

32

I. ALTERNATIVE RINGS

In particular, with a = 1, we get f{y^w) fiWy^w)

— f{ay^w)

= f(ya,w).

= /(VyW) for any y^w ^ A and so

By nondegeneracy, we obtain ay = ya

for any a^y ^ A; thus, a i-^ a is an involution on A, Then aa = aa and (30) show t h a t aa € Fl for all a G ^ . Thus, for a £ A^ 2q{a)l = / ( a , a ) l = / ( l a , a ) l = / ( a a , 1)1

by (32)

= a a / ( l , 1) = 2aa and so (33)

^(^)1 = oM-

Similarly, ^ ( a ) l = aa^ so a commutes with a. Now (31), (28) and the fact that q(x) G F imply t h a t for any x^y^w f{x{xw),y)

= f{xy,xw)

£ A^

= f{y,w)q{x)

= f{y^q{x)w)

=

f{y,{xx)w).

Using again the nondegeneracy of / , we obtain (34)

x{xw)

=

{'xx)w.

Writing ^ = Ic + x — x and noting t h a t 'x + x G Fl^ we have also x{xw)

=

x^w.

Similarly, we can show t h a t {wx)x = wx^. Thus A is an alternative algebra. Now let B be any finite dimensional subalgebra of A which contains U = Fl and on which the restriction of / is nondegenerate (for example, B = U). Note t h a t B is closed under the involution since 6 = (6 + 6 ) — 6 and b + be Fl C B. Suppose B y^ A. By Proposition 4.4, we have A = If f(x^x)

= 0 for all x G B^^ linearization would give f(x^y)

B®B-^.

= 0 for all

x^y £ B^ and hence / ( x , a ) — Q for all x G B^ and all a G ^4, contradicting nondegeneracy of / . t£

Thus / ( x , x ) ^ 0 for some x G 5"*", so there exists

B^ with q{t) r= - a / 0. Now 1 G 5 implies t h a t / ( 1 , £ ) = 0, so £ G C/-^,

Z = - £ and ^2 =. -a

= -q(i)l

= al.

Suppose bi = 0. Then f{bi, ai) = 0 for all a G A, so / ( 6 , a)q(i) = 0 for all a, by (26). Since ^(£) = —a / 0, we get f(b^a)

= 0 for all a and, since

/ is nondegenerate, 6 = 0. Thus 6 i-^ 6^ is a one-to-one surjective linear transformation from B to 5 £ , so Bi and B are isomorphic as vector spaces

4. COMPOSITION ALGEBRAS

over F. Recalling equation (31), t h a t £ G B^

33

and t h a t B is closed under

the involution, we have f{xi,y) for any x^y £ B.

= f{xy,i)

= 0

Thus Bi C B^ so t h a t the sum B + Bi is direct. We

proceed to show t h a t multiplication m B ® Bi is given by (35)

(a + bi){c + di) = (ac + adb) + {da + bc)i

so t h a t B®Bi

is the algebra ( 5 , a) obtained by applying the Cayley-Dickson

process to B. (See Section 3.) Towards this end, we begin by remembering t h a t xlc = q{x)\ for any x, so t h a t {x + y){x -\- y) — q(x + y)l and hence / ( x , y)l = {q{x + y)-

q{x) - q(y))l

= xy + yx

for all X and y. In particular, with x = b ^ B and y — i ^ B^ ^ we see t h a t bl+ib^G,

Therefore

(36)

ib = -bl = bi.

Since, by(34), x{'xy) = {x'x)y — q{x)y for any x , y G ^4, linearization (x \-^ X + z) gives x{'zy) + z{xy) = for any x^y^z

f{x,z)y

^ A. With x^y ^ B and z = £, we get

(37)

-a:(£y) + £(xy) - 0

and so x(yi)

— x(iy)

= i{'xy)

= i{'yx) = {yx)L

Applying the involution to

(37), we obtain {yi)x - {yx)t — 0, so {yi)x = {y'x)(. for all x,y ^ B. Finally, for any x^y ^ B^ {xi)(yi)

= {tx)[ytj

flexible) identities, so [xi)[yi)

— i{'xy)i

by the middle Moufang (and

= 'xyt^ — ayx.

Thus (35) has been verified.

Note t h a t the involution on B ® Bi takes the form a-\-bi Since Bi

= a + ib = a-ib

=

a-bi.

C B^ ^ for any a^b^c^d G 5 , we have f{a^di)

— f{bi^c)

= 0.

Therefore / ( a + bi,c+

di) = f{a, c) + f{bi,

di)

= / ( a , c) + / ( 6 , d)q{i) = / ( a , c) - a / ( 6 , d)

34

I. ALTERNATIVE RINGS

for any a + bi^c + di G 5 © Bi,

Because a 7^ 0, nondegeneracy of / on

B implies nondegeneracy of / on B Q Bi.

IncidentaUy, this shows t h a t if

B Q Bi IS an alternative algebra and 5 is a composition algebra, then so is B ® Bi^ justifying the final statement of our theorem. Now apply the above process to the algebra B = Fl.

If A ^ B^ then A

contains Bi = F ® Ft which, as we have just shown, is the algebra li A ^ B\^ then A contains B2 — B\ ® B\i\

— ( F , a,/?) for some nonzero

(3 e F. li A j^ B2, then A contains 5 3 = ^ 2 ® ^2^2 = (F,a,(3,j) nonzero 7 G F .

{F^a).

for some

Since £ • £^£2 = h^ ' ^2 = ^h • ^2 = -^^1 • ^2 /

^h ' ^2,

the algebra B3 is not associative, so the next step in this process yields an algebra which, by Proposition 3.1, is not alternative, contradicting the fact t h a t A is alternative. It follows t h a t A is the Cayley-Dickson algebra (F,a,/?,7).

•

4 . 1 2 C o r o l l a r y . Over a field of characteristic quadratic alternative PROOF.

algebra is a composition

different from 2, any

simple

algebra.

Let A be an alternative algebra with linear trace t and qua-

dratic norm n.

(See Theorem 4.10.)

Since n is multiplicative, we have

only to prove t h a t the bilinear form / associated with n is nondegenerate; t h a t is, we must prove A-^ = 0.

For this, and because

/(l,l)=:2/0

shows t h a t / is not identically 0, it suffices to show t h a t A-^ is an ideal of A, Proceeding exactly as in the proof of Theorem 4.11, we reach equation (31)—f{ay^w)

= f(aw^y)—which

equation (32)—f{xa^z)

shows t h a t A-^ is a left ideal of A, and

— f(za^x)—which

shows t h a t A-^ is a right ideal of

A.

n In 1958, R. Bott and J. Milnor proved t h a t a finite dimensional real

division algebra must have dimension 1, 2, 4 or 8. The only such algebras which are alternative are listed in the next corollary. 4 . 1 3 C o r o l l a r y . The only finite dimensional over the real numbers PROOF.

are R, C, H(R)

alternative

division

algebras

andC.

Let /I be a finite dimensional division algebra over R. Then

every a ^ A satisfies a polynomial with coefficients in R. Such a polynomial of lowest degree is necessarily irreducible and hence of degree 1 or 2.

4. COMPOSITION ALGEBRAS

35

It follows t h a t A is quadratic. Since A is simple, it is a composition algeb r a by Corollary 4.12 and hence one of the four algebras which arise from the Cayley-Dickson process starting with R. Reviewing the proof of Theorem 4.11, it is evident t h a t the scalar a which defines the algebra B ® Bi must be negative, for otherwise {\-\-{y/a)~^t){\

— {y/a)~^t)

= 0, which can-

not occur in a division algebra. But then q{i) = —a > 0, so, replacing £ by {^yq(J))~^i•,

we may assume t h a t P = —1. Thus the algebras Bi obtained

at the end of the proof are, respectively, R, C, H(R) and C.

D For what integers n is there a formula of the type (38)

{al + al + --- + a^ibl

+ b] + • • • + bl) = c] + 4 + • • • + cl

where the Ci have the form Q = Y^ijk^jk^j^^

^^^ ^he coefficients aijk are

real numbers independent of the ai and 6^? This so-called n-squares

problem

was first considered by A. Hurwitz in 1898. Trivially there is a 1-squares formula, and we have previously noted t h a t there are n-squares formulae for n = 2 , 4 , 8 which can be obtained using the fact t h a t the norm functions on C, H{R) and C are multiplicative. At this point, it is not hard for us to show that there are n-squares formulas only for n— 1 , 2 , 4 , 8 . 4 . 1 4 C o r o l l a r y ( H u r w i t z ) . An n-squares

formula

exists

if and only if

n = 1, 2, 4 or 8. P R O O F . AS

noted, we need prove only necessity. So suppose there exists

an n-squares formula. Let A be the vector space R^ and, for a = ( a i , . . . , a^) and b = {b\^.. .,&n) in A-, define ab = c, where c = ( c | , . . . , Cn) is the n-tuple giving the solution to (38). This product is well defined because the aijk are independent of a and b. For a = ( a i , . . . ^an) G A^ define q[a) = a^ + ' - • + a^. Note t h a t q{ab) = q(ci)q{b) for any a, 6 G ^ and t h a t q{a) = 0 if and only if a = 0. Without loss of generality, we may assume t h a t A has a unity. To see this, choose f ^ A with q{f) = 1. Then q{fa) = q{(if) = q(ci) for all a £ A. Thus the linear transformations L{f) (39)

aL{f)

= fa,

and i 2 ( / ) , defined by aR{f)

= af

for a £ A, are one-to-one and hence bijections of the finite dimensional vector space A. Equations (39) say t h a t q(aL(f))

= q{a) = q{aR{f))

for

36

I. ALTERNATIVE RINGS

all a G ^ , and hence also q{aL(f)~^)

= q(a) = q(aR{f)~^)

for all a. Now

define a new product • in A by setting

a*b =

{aR(fr')(bL{f)-').

Then, for any a G ^ ,

f*a

= [{f)Rifr'][aL{fr']

and, similarly, aicp

= f[aL(fr']

=a

= a. Thus f^ is a unity for the algebra (A, + , ^ ) , which

is therefore a composition algebra since q(ai^b) = q{a)q{b) for all a, 6. Thus, we may now assume t h a t (A, + , •) is a composition algebra with respect to q and hence alternative. If ab — 0, then q{ab) — q{a)q(b) = 0, so q(a) = 0 or q{b) — 0 and hence a = 0 or 6 = 0. Thus A has no nonzero zero divisors, so A is an alternative division ring. The result now follows from Corollary 4.13.

D

For further discussion of the n-squares problem, we refer the reader to "The Four and Eight Square Problem and Division Algebras", by Charles W. Curtis [Cur63]. In Section 3, we defined the notion of split Cayley-Dickson algebra. More generally, a split composition

algebra

is a composition algebra which

is not a division algebra. Split composition algebras can be characterized in several ways. 4 . 1 5 P r o p o s i t i o n . Let A be a composition

algebra with respect to the qua-

dratic form q over a field of characteristic

different from 2. Then the fol-

lowing conditions

are

(i) A has (nonzero)

equivalent, zero

divisors.

(ii) A is split. (iii) There exists a nonzero a ^ A with q{a) — 0. (iv) There exists an idempotent Furthermore,

e 6 ^, e / 0,1.

if A '^ F ® F, then each of the above conditions

is

equivalent

to (v) A has nonzero nilpotent PROOF.

elements.

Certainly (i) implies (ii). Next note t h a t if q{a) ^ 0 for some

a G ^ , then a is invertible, with inverse a ( g ' ( a ) ) ~ \ since q{a) — aa by (33). Thus (ii) implies (iii). Since e^ = e implies e(l — e) = 0, (iv) implies (i). We

4. COMPOSITION ALGEBRAS

37

now show t h a t (iii) implies (iv). To establish (iv), it is sufficient to show t h a t there exists x ^ A with xx = 0, x + x == a l / 0, since then, a~^x

is

the desired idempotent. If no such x exists, then whenever q(x) — 0, so also X + x" = 0. Now assume (iii). Then there exists a ^ A with q{a) = 0 but a / 0. Then q{ax) = 0 for all x, so ax + ox = 0 for all x; thus, ax +^a

= 0

for all X. Replacing x by x, we obtain ax + xa = 0 for all x, but / ( a , x) = q{a + x) - q{a) -

q{x)

= (a + x)(a + x) — aa

xx = ax + xa

so t h a t / ( a , x) = 0 for all x. By nondegeneracy of / , a == 0, a contradiction. Thus the equivalence of the first four conditions has been established. Since (v) always implies (i), to complete the proof, it suffices to show t h a t (iv) implies (v) unless A '^ F® F, Recall that any composition algebra except F ® F \s simple (see Proposition 3.2 and preceding remarks). e /

0, 1 is an idempotent in A and eA{l

— e) = (1 — e)Ae

If

= 0, then

(1 — e)ae — ea[\ — e) — ^ and hence ea — ae for any a ^ A.

(In these

calculations, we use Artin's Theorem freely, thus regularly assuming that any subalgebra generated by two elements is associative.) Thus e is a central idempotent and Ae is a proper ideal, contradicting simplicity. It follows that either eA{l — e) / 0 or (1 ~e)Ae

^ 0. In the former case, there exists a such

t h a t /i = ea( 1 — e) 7^ 0. Notice that // is nilpotent. In the case (1 — e)Ae ^ 0, there exists a ^ A such t h a t (1 — e)ae is a nonzero nilpotent element, so (v)

follows.

n

We now establish a very important and rather striking result. 4 . 1 6 T h e o r e m . Any two split composition over a field of characteristic PROOF.

algebras of the same

different from 2 are

dimension

isomorphic.

Let ^ be a split composition algebra with respect to the qua-

dratic form q over a field F of characteristic different from 2. We again freely use the fact t h a t any subalgebra of A generated by two elements is associative (since A is alternative). By Proposition 4.15, A contains an idempotent e 7^ 0, 1. Note that q(e) = ee = 0 because e is not invertible and t h a t the scalar e + 6 is 1 because (e + e)e = e. It follows easily t h a t the restriction of q to the subalgebra of A generated by 1 and e is nondegenerate. Now let B be any subalgebra of A which contains 1 and e and on which the restriction

38

I. ALTERNATIVE RINGS

of q is nondegenerate. As in the proof of Theorem 4.11, there exists t ^ A such t h a t Bi C B^ and (? = - g ( ^ ) l = a l 7^ 0. Since et = ie by (36), using the middle Moufang identity we get {ety

= {ie){ei)

setting ii = i + a-^{l-

C B-^ and

ij = f + a-\l

a)ei, we have Bii - a)(iei

= f + a-^{l-a)(e

+ + e)f

= (i • €e)i = 0 and,

ef) = a l + a - ^ ( l - a ) a l = 1.

With reference to the final paragraph of the proof of Theorem 4.11, but starting with the subalgebra B generated by 1 and e/\i

B — A, then A =

F e F = ( F , 1). li B ^ A, then A contains B ® Bii

with e\ = 1. This

algebra is ( F , 1,1). If yl is not ( F , 1,1), then yl = ( F , 1,1,1). The theorem follows.

D

4 . 1 7 C o r o l l a r y . Any split generalized quaternion characteristic

different from 2 is isomorphic

Dickson algebra over a field F of characteristic

algebra over a field F of

to M2{F),

Any split

Cayley-

different from 2 is

isomor-

phic to Zorn's vector matrix algebra 3 ( ^ ) PROOF.

In Section 3, we saw t h a t M2{F) and 3 ( ^ ) are split composition

algebras, so the result follows from Theorem 4.16.

D

5. T e n s o r p r o d u c t s In this section, we study an important construction t h a t will be a useful tool throughout this book, the tensor product. This notion can be defined in the very general setting of modules over arbitrary rings; however, since we are interested only in algebras over commutative associative rings with unity (often fields) we shall work within the setting of unital modules over commutative associative rings. 5.1 D e f i n i t i o n . Let $ be a commutative associative ring with unity and let M , A^ and T be ^-modules. A m a p u)\ M x N ^ T \s bilinear if 1. cj(mi + m 2 , n ) = u;(7ni,n) + Ct;(m2,n), 2. u ; ( m , n i + 712) = a ; ( m , n i ) + c j ( m , n 2 ) , 3. a ; ( a m , n ) = Q;a;(m, n) = u{ni^an) for all m , m i , m 2 G M , all n, 711,712 G N and all a G $ .

5. TENSOR PRODUCTS

39

5.2 D e f i n i t i o n . A tensor product of ^-modules M and TV is a ^ - m o d u l e M ®^ N together with a bilinear m a p u: M X N -^ M ®^ N with the property t h a t any other bilinear map from M x A^ to a ^-module T factors uniquely through cj; t h a t is, given any bilinear m a p f:MxN-^T^

there

exists a unique homomorphism of ^-modules (f: M ®^ N ^^ T such t h a t the following diagram commutes.

M XN

^ M ®^ N I

I

I T After such a definition, it is natural to ask whether tensor products actually exist. As we shall see, they do, and they are essentially unique. We begin by giving a construction of one particular tensor product. Given two ^-modules M and A^, we first consider the set

[m,n)^My.N

where all sums here are understood to be

finite.

We can turn S into a

module over $ by defining addition and scalar multiplication by

(m,n)eMxN

(m,n)eMxN Yl

(^(m,n) + y(m,n)){m,

u).

(m,n)6Mx7V and

(m,n)eMxN

(m,n)eMxN

Let H be the submodule of S generated by all elements of the form (mi + 1712,n) - i'rni.n)

- (m2,n),

( m , n i + 712) - ( m , n i ) - ( m , n 2 ) (aTn, n) — a ( m , n )

and

(m^an)

— a{m^n)

40

I. ALTERNATIVE RINGS

where m , m i , m 2 G M , n , n i , n 2 € N and a G # . quotient module S/H.

Let M ®^ N be the

Clearly there exists a natural embedding M X N -^

S. If we denote by m ® n the image of (m^n)

in the quotient S/H^

then it

is clear from the definition of H t h a t (mi + m2) ®n = ( m i ® n) + ( ^ 2 ® n) m ® (ni + ^2) = ( m ® n i ) + ( m ® 712) and ( a m ) ® n = a{m ® n) =^ m®

(an)

for any m , m i , m 2 € M , any 7i,ni,n2 € A^ and any a G ^ . It is also clear t h a t the m a p u: M X N -^ M ®$ N defined by ( m , n) H-» m ® n is bilinear. 5.3 P r o p o s i t i o n . With the notation tensor product

of M and N.

established

Furthermore,

above^ (M 0^ N^uj) is a

if (T^f)

is any other

product of M and N^ then there exists an ^-isomorphism

tensor

(f: M ®4> N -^ T

such that f = (f 0 (jj. P R O O F . TO

prove t h a t ( M ®^ N.uj) is a tensor product of M and A^,

let U be another ^-module and let ^ : M x iV ^ f/ be a bilinear m a p . Since the elements of the form ( m , n ) ^ M X N form a basis of 5 , we can define a module homomorphism ip^: S —^ U hy ^'i

Yl

^im,n){m, n))=

(rn,n)eMxN

^

X(^^^^)g{m, n).

(7n,n)eMxN

Since g is bilinear, it is easy to see t h a t (p^ maps the generators of H to 0, so (f^ induces a module homomorphism (f: M ®^ N ^ ip{m ® n) = g{m^n)

U such t h a t

for all m G M and n E N; thus g = (f o uj.

The

uniqueness of (f follows easily. To prove uniqueness of the tensor product up to isomorphisms, assume that ( T , / ) is another tensor product of M and N,

Since

f:MxN^T

is bilinear, the first part of the proof shows t h a t there exists a module homomorphism (p: M ® N -^ T such t h a t f = (p o uj. On the other hand, since u: M x N ^^ M ®<j) A^ is also bilinear and (T^f)

is a tensor product,

there exists a module homomorphism ip: T -^ M ® ^ N such t h a t uj = tpo f. Notice t h a t u = ip o ip o uj.

Since we also have u = I 0 u {I denoting

the identity homomorphism on M ®<^ iV), it follows from uniqueness t h a t

5. TENSOR PRODUCTS

ip 0 (f = L

41

In a similar way, we obtain t h a t ip o i/j = I and the claim

follows.

D

Henceforth, we shall use the symbol M ®a> N to denote the tensor product of ^-modules M and N.

Sometimes, if there is no obvious reason to

emphasize t h a t the tensor product is taken over t h e ring $ , we drop the subscript and write simply M ® N. T h e universal property of tensor products turns out to be the most useful tool in establishing facts about them. The next examples and propositions will illustrate this very clearly. 5.4 P r o p o s i t i o n . Let M be a module over a (commutative,

associative)

ring $ . Then ^ 0^ M = M. P R O O F . Let g: ^ x M ^

am.

M he the bilinear m a p defined by ( a , m ) H-^

Since g is bilinear, there exists a homomorphism ip: $®<|)M -^ M such

t h a t g = (fou.

Now consider the module homomorphism ip: M -^ ^ ®^ M

defined by m ^-» 1 (g) m . Clearly '0 is also linear and direct calculations show t h a t (i[) o (f)(a ® m) — a ® m,. Since elements of the form a ® m generate $ ®4) M , we have ip o (f — I, An even easier calculation shows t h a t also (f o ip = I; hence, the result follows.

D

Using t h e universal property, it is routine to verify the following basic properties of tensor products. 5.5 P r o p o s i t i o n . Let Mi, M2, M3 be modules over a ring $ .

Then

(i) Ml® M2 = M2® Mi; (ii) Ml ® (M2 ® M3) = {Ml ® M2) ® M 3 ; (iii) Ml ® (M2 e M3) = {Ml ® M2) © {Ml ® Ms). 5.6 C o r o l l a r y . Let M and N be free ^-modules and {uj I j G J7} respectively. {nii ®nj I i G 2 , J G J}.

with ^-bases

{mi \ i E 1}

Then M ® N is a free ^-module

with basis

Moreover, every element in M ® N can be written

uniquely as a finite sum of the form ^ m , - ®yi,

Vi G N,

i

and also as a finite sum of the form \

Xj ® Uj.)

Xj G M.

42

I. ALTERNATIVE RINGS PROOF.

By Proposition 5.4, a ® m y-^ am defines an isomorphism $ ®

M -^ M. As a free ^-module, M is the direct sum of copies of $ , so

M®^N

is free with the indicated basis because m i ® (m2 + ma) i-^ ( m i ® 1712) + (mi ® ma) is an isomorphism Mi ® (M2 ® M3) ^ ( M i ® M2) © ( M i ® M3). To prove the last statement, let a G M ®^ N. can be written uniquely in the form a = ^ -

We have shown t h a t a

aijrrii ® Uj for some a^j G ^ .

Thus

z,j

with yi — V

t

j

z"

a^j^j 6 A^". To show t h a t this representation is unique, it is

sufficient to show t h a t if 53^-m^ ® 2/t = 0 with yi G A , then each y^- = 0. So suppose X^- mi ® yi = Q with y^- = Ylj ^ij'^j 0 = ^

m,- ® ?/,• = ^

^ ^ - Then

m,- ® ( ^ <^zj^j) = ^

and, because the elements m.i®nj

ctijrrii ® n^

form a <]^-basis for M®^ A , each a^j = 0.

Thus y^ = 0. Similarly, one shows t h a t each element of M ®^ N can be written uniquely in the form ^

Xj ® n^, with Xj £ M.

D

It is often the case that the ^-modules under consideration are also algebras over $ .

In this case, we show that the tensor product of these

modules also admits the structure of an algebra over $ . 5.7 P r o p o s i t i o n . Let A and B be algebras over a commutative Then A®B

is also an algebra over 4> with multiplication

(40)

(Xlt ttz ® &t)(Ej ^3 ® ^3) = E z j (^if^3 ® ^^^J••

/ / furthermore,

A (respectively

a ^-^ a ® 1 (respectively B) into A® PROOF.

6 i-^ \®b)

B) is free as a ^-module,

ring $ .

given by

then the

is an algebra embedding of A

mapping

(respectively

B. Note t h a t we cannot define the product m A ® B directly by

(40) because this might not be well defined. Instead, we shall introduce a multiplication in ^ ® 5 in several steps. First, for fixed elements a ^ A and b £ B^ consider the bilinear map fab-' Ax

B -^ A® B given by fab{x,y)

=

xa®yb

5. TENSOR PRODUCTS

f o r ( x , i / ) £ AxB.

43

Since/at is bilinear, there exists a module homomorphism

(fab'- A ® B -^ A ® B such t h a t fab = ^ab o ^- For an arbitrary element t = J2i Q'i ®b,e

A®

B,

Vab^t) = X)i Vabidi ® bi) = ^- fab{cii, bi) = X)i ^ i ^ ® bib. Let End(A 0 B) be the ^-module of endomorphisms of A ® B. The m a p "tp: A X B —» End(yl ® 5 ) given by tp{a,b)

= fab for {a,b)

bilinear, so there exists a homomorphism ^ : A ® j5 ^

e A X B is

End(>l ® 5 ) such

t h a t ip = "^ ouj. So we have "^{ai ® bi) = fa^btFinally, given arbitrary elements t = ^^- a^- ® 6^ and ^' = ^ ^ a^ ® 6^ in yl ® 5 , we define the product of these elements by

This product is clearly well defined and it is only routine to verify t h a t A® B^ with this product, is an algebra over $ . To see that it coincides with the product of our statement, we compute tt' = ^{t){t')

= * ( E , a, ® b,){Zj

= E , * ( a t ® b,)iZ,

ar ® 6;.)

«; ® br) = E , / a . 6 . ( E , «• ® b',)

= Eij /a.6.(« • ® ''•) = J:^J «.«;• ® ^t^-;Furthermore, if /I is a free ^-module, then it follows from Propositions 5.4 and 5.5 that the module homomorphism a »^ a ® 1 is an algebra embedding.

D 5.8 R e m a r k . As might be expected, in considerations involving the tensor product of algebras, there is a stronger version of the universal lifting property previously described for modules. and gs'

Specifically, suppose fj[' A -^ T

B -^ T are algebra homomorphisms into a ^-algebra T .

the map f: A X B ^

T defined by f{a^b)

= fA{o.)fB{b)

Then

is bilinear and

so there exists a bilinear module homomorphism ip: A ® B —^ T such t h a t (^{a ® b) — fA{^)fB{b)

for all a G ^ and b E B.

It is straightforward to

show, and important to note, t h a t (f is, in fact, an algebra homomorphism. 5.9 E x a m p l e . Let $ be a commutative ring. Then M ^ ( $ ) ®^ Mn{^) Mmn{^)

2^s ^-algebras.

=

44

I. ALTERNATIVE RINGS

To see this, define / : Mm{^) Given A = [aij] e Mm(^)

X M n ( $ ) -> Mmn{^)

in the following way.

and B G A/n($), set aiiB

CI12B

...

CLirnB

a2iB

a22B

...

CL2mB

a^alB

am2B

...

ClmmB

/(>1,5) =

Then f(A,B)

= f{A^Im)f{In',

matrices in Mn{^)

B)^ where /^ and / ^ denote the identity

and Mmi^)-, respectively. Clearly the mapping M ^ ( $ ) —>

M ^ n ( ^ ) defined hy A ^

f{A,Im)

and the mapping M ^ ( $ ) -> M ^ n ( ^ )

defined by 5 i-^ /(^n? ^ ) are algebra homomorphisms. Since / is bilinear, by Remark 5.8, there exists an algebra homomorphism (^: Mm{^)®^Mn{^) Mmn{^)

-^

such t h a t g — (^ o (jj. To see t h a t (/? is an isomorphism, we shall

show t h a t it maps a $-basis oi Mm{^)®^

Mn{^)

onto a ^-basis oi

Mmni^)-

Denote by {eij} the set of elementary matrices of Mmi^)-) which is a basis for this free module, and, by {e^^}, the set of elementary matrices of

Mn{^).

From the definition of (^^ we see t h a t (f{eij ®^'}^^) is an elementary matrix in ^mn{^)

and that Lp is actually a one-to-one mapping of the set {cij ® e^^}

onto the set of all elementary matrices of

Mrani^)-

We wish to provide more examples which will be useful in later chapters. For this, we need the following lemma. 5.10 L e m m a . Let F C E be a field extension over F.

and let V be a vector space

Then E ®p V is a vector space over E, where the product

scalar A G E and an element (41)

of a

^ ^ Xi ® yi ^ E ®p V is given by

^J2^^^ ® Vz = I^^{>^X^)®y^.

Moreover,

if {vi | i G X} is a basis of V over F, then {\ ® v^ | i G X} is a

basis of E ®p' V over PROOF.

by fxi^^y)

E.

For each element A G £^, we define a m a p f\:

E xV

-^ E®p V

= Ax 0 y, for (a:,y) £ E X V. Since f\ is bilinear, there exists

a homomorphism (f\: E ®f? V —^ E ®/? V such t h a t fx = (fx o cj. a £ E ®F y and X £ E^ define Xa = f\{a).

For

With this scalar product, it is

routine to check t h a t E (g)/? 1/ is a vector space over E. Given an arbitrary element t — ^

Xj®yj

G E®pV^

we can write yj in the form y^ = Y^ikijVi^

5. TENSOR PRODUCTS

45

kij G F. Then

= E j Et(^ij-^j) ® ^z = E t ( E j ^zj2^^)(i ® ^z)Since ^

^2i2:j G £" for every index j , it follows t h a t {1 ® t;^ | i € X} is a

set of generators. The fact t h a t this set is also linearly independent follows from Corollary 5.6.

D

Let F be a field and let a be an element in F , the algebraic closure of F. We denote by F(a) a.

The field F{a)

the smallest subfield of F containing both F and

is an algebraic extension of F; t h a t is, every element

of F{a) is the root of a polynomial with coefficients in F.

Let /o be the

monic polynomial in F[X] of least degree which has a as a root. It is well known that F{a) is isomorphic to the quotient field F[X]/(fo).

If F = C,

the field of complex numbers, then /o = n i - i ( ^ ~ ^i) ^^ ^^e product of r polynomials X — a^ in C[A^], where r = d e g / o and the a^, 1 < i < r^ are distinct elements of C. We refer the reader to Chapter 5 of [ C o h 7 7 ] for more details. 5.11 E x a m p l e . Let F be a field, let /o G F[X] and let F be a field containing F . Then

as algebras. In particular, if F is a subfield of C and F{a) is an algebraic extension of F , then C ® F F{a) is isomorphic to the direct sum of r copies of C, where r is the degree of a polynomial of least degree which has a as a root. To prove the first part of this statement, consider the map

'•"''

(/o)

(/o)

given by (A, / + (/o)) i-^ A / + (/o)- This map is clearly bilinear, so there exists a hnear m a p ^ F[X] E[X] cp: F ® ——- -> ——-

46

I. ALTERNATIVE RINGS

such t h a t g — (^ o uj. Denoting the degree of /o by r, the set {X^ + (/o) | 0 < i < r — l } i s a basis of F [ X ] / ( / o ) over F , so, according to L e m m a 5.10, {1 ® {X' + (/o)) I 0 < i < r - 1} is a basis of E t h a t if [I® {X' + (/o)) = X' + (/o) G map of a basis of E

®F F [ X ] / ( / O ) . NOW

JE;[X]/(/O);

thus (/? is a one-to-one

onto a basis of E[X]l{fo)

®F F [ X ] / ( / O )

notice

and hence a

bijection. Since it is easy to see t h a t (p is also an algebra homomorphism, the result follows. For the second p a r t , suppose F C C, t h a t a is algebraic over F and that /o is a polynomial of least degree over F which has a as a root. Then /o = n i = i ( ^ ~ ^i) ^^^ ^^^ ^i-) 1 ^ ^ ^ ^? ^re distinct elements of C. Hence, by the first part and the Chinese Remainder Theorem (see [ C R 8 8 ] ) , we have

/;[x]^c[x]^ ® (/o)

"

(/o) ~

^

C[X]

qx] {{X-a,){X-a2)'"{X-ar)) C[X]

- (^-ai)

C[X]

(^-c.2)

"^

{X-ar)

= c®c® •••ec. ^

V

'

r summands

5.12 E x a m p l e . For any commutative ring /Z, we find it convenient to denote by H(/?) the algebra R+ Ri + Rj + Rk^ where P — p — k^ — ijk

=

— 1. Let F be a field and let A' and E be field extensions of F. Then (42)

E®FWO

=

H(F®FA')

as algebras. In particular, \i E — C and K = E{a) is an algebraic extension of F of degree r, then C®FH(A')^H(C)ffi..-®H(C). ^

V

'

r summands

To prove (42), we define g: E X H(A') -> H{E ®F K) by g{\,a) ( A ® a i ) + (A®a2)i + (A®a3)j-|-(A(8)a4)fc, for A G £" and a = a\ +a2i-\-a^j

= +

a^k G H(A'). This map is bilinear, so there exists an algebra homomorphism (y9: E ®F H(A') -^ H(E ®F A ) such t h a t g = (f o LJ. If {vi \ i ^ 1} is a basis of A' over F and if A' = { l , 2 , j . A:}, then the set B = {viX | i G I , x G A'} is a basis of H(A^) over A"; thus I ® B = {1 ® ViX | i G I , x G A'} is a basis of E ®F H ( A ) over E. In a similar way, it is easy to see t h a t the set

5. TENSOR PRODUCTS

B' = {{l®Vi)x

I i € I , a: G X} is a basis of

47

over E. Finally, since

H{E®FK)

(/? is a one-to-one m a p oi I® B onto B\ it follows t h a t (f is an isomorphism. The last statement follows from (42) and Example 5.11 because C ®F H{K)

^ H{C ® A^) ^ H(C e . • • ® C) ^ H(C) 0 • • • 0 H(C). sy

^

'

^

V

r summands

'

r summands

As we have seen in Proposition 5.7, the tensor product of two algebras is again an algebra. It is easy to check t h a t if the given algebras are associative, then their tensor product is also associative. In general, the tensor product of alternative algebras is not alternative. (This can be shown as a consequence of Corollary IV. 1.4 and Lemma VII.0.1.) We do, however, note the following. 5.13 P r o p o s i t i o n . Let A be an alternative pose B is a commutative A ®7? B is

associative

algebra over a field F and sup-

algebra over F. Then the tensor

product

alternative.

P R O O F . Let x = ^ a^ ® bj and y = J2^i

® ^j ^ ^ elements of A ® F B.

To verify the left alternative identity, we must show t h a t x'^y — x{xy) Now x^y — x{xy)

= 0.

is a linear combination of terms of the form

{(a, ® bj){ak ® 6 ^ ) } « ® &!) + {{a^ ® 6^)(a, ® 6 ^ ) } « ® 6^) - (a^• ® bj){{ak ® 6^)(a; ® t^)} - (a/, ® 6^){(a, ® bj){a^ ® t ' J } which can be rewritten as {a^ak)a'^ ® {bjbi)^^ -f {akai)a'^ ® ( M j ) ' ^ ! - a,{aka[) ® bj{b(^b'^) - ak{a^a[) ® b^{bjb'^). Since B is associative and since, for fixed b ^ B^ the map a \-^ a®b\s

linear,

this expression can in turn be rewritten in terms of associators, [a,-, a/,, a^] ® bjb^b^^ + [a/,, a,-, a^] ® b^bjb'^ and then, since B is commutative, also as {[a,, a/,, a^] + [ak, a^-, a'^]] ® bjbf>b'^, which is 0 because the associator is an alternating function on A. establishes the left alternative identity in A ®F B.

This

Similarly, the right

alternative identity holds as well, so A ®F B is alternative, as claimed.

D

Chapter II

A n Introduction to Loop Theory and to Moufang Loops

1. What is a loop? The theory of loops has its origins in geometry, combinatorics and nonassociative algebra. In geometry, the coordinatization of a projective plane leads to various loop structures on the set of labels from which coordinates are chosen. Any Latin square with first row and column in standard position is the multiplication table of a loop. In rings with 1 for which there is a well defined notion of inverse, it is often the case that subsets closed under products and inverses are loops. R. H. Bruck's "Survey of Binary Systems" [Bru58] and G. Pickert's "Projektive Ebenen" [Pic75] were the first comprehensive treatises on loops and, for many years, these books were the standard references on the subject. More recently, H. 0 . Pflugfelder wrote an introduction to loop theory intended to be useful as a textbook for a graduate level course [Pfl90]. This book was soon accompanied by what was first intended to be a sequel, "Quasigroups and Loops, Theory and Applications" [CPe90], but which became, in fact, an extensive collection of papers amply illustrating the scope and diversity of the field today. We recommend all these books to the reader. The brief introduction to loop theory and, in particular, to the theory of Moufang loops which we present in this chapter, is based primarily upon the books by Bruck and Pflugfelder cited above. 1.1 Definition. A quasigroup is a pair (L,-) where L is a nonempty set and (a, 6) — I > a • 6 is a closed binary operation on L with the property that 49

50

II. LOOP THEORY AND MOUFANG LOOPS

the equation a -b = c determines a unique element b £ L for given a^c ^ L^ and a unique element a ^ L for given b^c £ L. We call a • b the product of a and 6. A /oop is a quasigroup with a two-sided identity element 1. Both cancellation laws hold in a quasigroup (L^-). For example, the equation a\'b — a2'b implies ai = a2 by uniqueness of the element x in the equation x • b = c. In any quasigroup (Z/, •), a critical role is played by the right and left translation maps R(x) and L{x)^ x G L, which are defined by (1)

R{x):a\-^a'X

and

L(x):a\-^X'a

for a ^ L, For example, the solution to a-x = 6, guaranteed by the definition of quasigroup, is x = bL{a)~^ ^ while the solution to y-a = b\s y = bR[a)~^, (In loop theory, it is common to write maps to the right of their arguments and we shall adhere to this practice in this work whenever translation maps are involved.) The definition of quasigroup is equivalent to the assertion that R(x) and L{x) are bijections of L, for any x ^ L. The right translation R{x)^ for instance, is one-to-one since a\R{x) = a2R{x) implies a\ - x = a2 • x so that ai = a2, by cancellation. The map R{x) is onto because, if 6 G //, the definition of quasigroup assures the existence of a G // such that a - x = b. Thus, the set of left and right translation maps generates a group M(L), called the multiplication group of L, which is a subgroup of the symmetric group on the set L; thus, MiL)

= {L(x),Rix)\xe

L).

If (L, •) is a loop, the equation a - x — \ has a unique solution a^ called the right inverse of a, and the equation x - a = I has a unique solution a^ called the left inverse of a. In general, the left and right inverses of an element are different. When they are equal, however, we use the notation a~^ to denote the unique (two-sided) inverse of a. This is the case, for example, in any associative loop, which is just a group. As with rings, so with quasigroups and loops, we shall often omit reference to the binary operation, referring simply to "the quasigroup L" or to "the loop L". Similarly, we routinely write ab rather than a • b to denote the product of elements a and b.

1. WHAT IS A LOOP?

51

A subloop of a loop (i/, •) is a subset of L which, under the inherited binary operation, is also a loop. Since, for h ^ H^ the equation hx = h has a unique solution in L, it follows t h a t the identity elements of H and of L are necessarily the same. 1.2 P r o p o s i t i o n . Let L be a loop and let H be a nonempty The following

are

subset of L.

equivalent,

1. H is a subloop of L, 2. If x^y G HJ then xy^ xR(y)~^ 3. If x^y^z

£ L, xy = z and two of x^y^z

these elements PROOF.

and xL(y)~^

are all in H.

are in H, then the third of

is in H too.

If Zf is a subloop of L and x,y

£ H^ then xy e H because H

is closed under the binary operation of L. T h e element z = xR(y)~^ satisfies zR{y)

G L

= x. Thus zy = x. Since ^ is a loop and ay = x has a unique

solution a both in H and in L, necessarily z ^ H. Similarly, xL(y)~^

G H.

Thus (1) implies (2). Now assume (2) and consider the equation xy = z. Because of (2), if x^y ^ H ^ then so is z; if a;,z G H ^ then so is y = zL{x)~^; so is X = zR(y)~^.

if y, z G H ^ then

This gives (3).

Finally assume (3) and let x^y e H. There exists a unique z = xy £ L, By (3), z ^ H. Similarly, there exists a unique z £ L such t h a t xz — y and, by (3), z £ H. Also, there exists a unique z £ L such t h a t zx = y and, by (3), z £ H.

All this shows t h a t ^ is a quasigroup. For x G H^ we have

x l = X, so 1 G ^ by (3); hence, /^ is a loop.

1.3

D

C o s e t d e c o m p o s i t i o n s . In general, Lagrange's Theorem does not

hold for loops. (For example, it is easy to construct a loop of order five in which every element has order two.) On the other hand, we can often still be assured t h a t the order of a finite loop is divisible by the order of certain subloops. Let ^ be a subloop of a finite loop L. Since the right and left cancellation laws hold in L, it is certainly the case t h a t \xH\ = \IIx\

= \H\ for every

X G I/. T h u s , if L is the disjoint union of cosets of IT, it is clear t h a t the order of L will be divisible by | ^ | . A necessary and sufficient condition for

52

II. LOOP THEORY AND M O U F A N G LOOPS

L to be the disjoint union of right cosets is (2)

Hx n Hy 7^ 0, a:, ^ G i/ if and only if Hx = Hy,

and this condition is equivalent to (3)

H{hx)

= Hx

for all x G L and all he

H.

To see the equivalence of (2) and (3), first assume (3) and suppose t h a t hix = h2y e Hx n Hy.

Then Hx = H{hix)

= H(h2y)

= Hy.

hand, assuming (2), for any x E L and any h e H ^we have hx 6 so H{hx)

=

On other H(hx)r\Hx,

Hx.

Summarizing, when condition (3) holds, there exists a subset T of L, called a transversal

of ^ in L, such t h a t

L = [j

Hx

with HxDHy

= (/} for x,y

eT

and, when this is the case, \L\ is divisible by \H\. A loop L is diassociative

if, for any x, t/ G L^ the subloop (x, y) generated

by X and y is associative (and hence a group). If L is diassociative, a e L and H = (a) is the cyclic group generated by a, then condition (3) clearly holds, so, if L is finite, \L\ is divisible by \H\. Thus the order of any element in a finite diassociative loop L divides the order of L. T h e concepts of commutator, associator, nucleus and centre, previously defined for rings, have natural analogues in loop theory. 1.4 D e f i n i t i o n s . Let a, b and c be three elements of a loop L. T h e (loop) commutator

of a and b is the unique element (a, 6) of L which satisfies ab = (6a)(a,6)

and the (loop) associator

of a, b and c is the unique element (a, 6,c) of L

which satisfies (ab)c = {a(6c)}(a,6, c). If X , y and Z are subsets of a loop L, we denote by (X^Y)

and ( X , y , Z ) ,

respectively, the set of all commutators of the form (x, y) and all associators of the form (x^ y^ z)^ where x e X, y EY,

z e Z.

1. WHAT IS A LOOP?

The commutator-associator

53

subloop of a loop L is the subloop V gen-

erated by the set of all commutators and associators. Thus

L' =

{{L,L,LUL,L)).

The left nucleus of L is the set Mx{L) = {a e L\ {a,x,y)

= 1, for all a:,y G L},

the right nucleus is the set Afp(L) = {a ^ L \ (x^y^a)

= 1^ for all x, y 6 L}^

the middle nucleus is Af^{L) = {a e L \ (x^a^y)

= 1, for all a:,y G L}

and the nucleus of L is A/'(L) =

Mx{L)r\Mp{L)r\N^{L).

The centre of L is Z ( L ) = {x G A/'(L) I (a, x) = 1 for all a G I } . As with rings, in expressing the nonassociative product of more t h a n two elements in a loop, we often employ dots instead of, or in addition to, parentheses to indicate the order of multiplication, juxtaposition always taking precedence over dot. 1.5 P r o p o s i t i o n . Each nucleus hence a group. PROOF.

of a loop is an associative

The centre of a loop is an abelian

subloop

and

group.

Let L be a loop and consider first the middle nucleus A^^ of L.

Clearly 1 G N^ and, if x, y G N^ and a^b ^ L, then (a • xy)b = (ax • y)b = (ax){yb)

since y e N^

= a{x ' yb)

since x G N^

= a{xy ' b)

since y G N^

which shows t h a t xy G N,j,. x^l

since x G N^

Let x G N^.

— Ix^ implies t h a t x^x = 1 = x^x.

unique two-sided inverse, x~^.

Then

{X^X)XP

= x^{xx^)

=

Thus x^ = x^^ so t h a t x has a

For a £ L^ {ax)x~^

= a(xx~^)

= a implies

54

II. LOOP THEORY AND M O U F A N G LOOPS

R{x)R{x~^) = / , the identity map on L, and so R{x)~^ = R{x~^), Similarly, x~^{xa) — a implies that L{x)~^ — L{x~^). Now let x ^ N^ and a,6 G L. By the definition of loop, we can write a = tx for some t £ L, Then (ax"^)6 = (tx ' x-^)b =: tR{x)R{x)~^

- b = tb

and a{x~^b) = {tx){x~'^b) = t{x • x"^6) == t • 6L(x~^)L(x) = tb; thus {ax~^)b = a(x~^6), so that x~^ £ N^. Since we have already shown that N^ is closed under multiplication, we now have that, for any x^y £ 7V^, the elements xR{y)~^ = xR{y~^) = xy~^ and xL{y)~^ = xL{y~^) = y~^x are both in L, so, by Proposition 1.2, N^ is a subloop of L, It is obvious that N^ is associative. Turning our attention to the left nucleus, we note first that 1 G A^A sind then, that if x, y G Nx^ also xy G A^A because {xy • a)b — {x • ya)b

since x G A^A

= x{ya ' b)

since x £ N\

= x(y • ab)

since y G A^A

= {xy){ab)

since x G A'A-

Let y G A^A- Then {yy^)y — y{y^y) = yl = ly shows that yy^ = 1 and hence that y^ = y^. As before, we see that y has a unique two-sided inverse y~^. Since y G A'A, for any a e L^ we have y(y~^a) = a, so L{y)~^ — L{y~^), Let a^b £ L. Then y{y~^ ' ab) = {yy~^){ab) = ab and t/(t/"^a . 6) = (y • y"^a)6 = aL{y~^)L{y) 'b = ab. Thus y{y~^ • a6) = y{y~^a • 6) and, after cancelling y, we obtain ^/"^ • ctb = y~^a ' 6, so that y"^ G A'^A- Since A^'A is closed under multiplication, it now follows that, if x,y G A^A, then xL{y)~^ - xL{y~^) = y~^x G A^A- The computation xR{y)~^ • y = xR{y)~^R(y) — x = x(y~^y) — {xy~^)y shows, after cancelling y^ that xR{y)~^ = xy~^ G A^A ^is well. Now Proposition 1.2 says that A^A is a subloop of L. Clearly this subloop is associative.

1. WHAT IS A LOOP?

55

W i t h any loop (i/,-)^ there is an associated opposite loop fined by i/^PP = L and x ir y = y - x for x^y Since the left nucleus of

J\/A(Z/°PP).

L^PP

G L,

(L^PP,*)

de-

Evidently, Afp{L)

=

is associative, as we have just seen,

the right nucleus of L is also associative. As the intersection of associative subloops of L, the nucleus of L is also an associative subloop of L, subset of L, To see t h a t Z{L)

Clearly Z{L)

is a commutative, associative

is a subloop, let x,t/ G Z{L),

Af(L)^ we have already established t h a t xy^ xR(y)~^

Since x^y E

and xL(y)~^

are also in

Af{L) and, moreover, t h a t y has a two-sided inverse y~^ such t h a t R(y)~^ R(y~^)

and L(y)~^

= L{y~^).

=

The usual group theory argument now works

in the present context to show t h a t ay~^ = y~^a for all a G i>, hence t h a t y~^ G 2(L).

Similarly, it is routine to verify t h a t xy G 2{L),

follows immediately t h a t xR(y)~^

= xy~^ and xL(y)~^

from which it

= y~^x are also in

a

z{L),

While as noted in §1.3, Lagrange's Theorem fails for loops, as we have already observed in §1.3, it is interesting to observe t h a t the order of any nucleus of L will divide |L|, since condition (3) clearly holds for H = JVA, AT^ or A^, and a condition corresponding to (3) for left cosets holds for A/"p. More generally, the order of any subloop of any nucleus of L divides |L|. The definition of "normality" in loop theory is a generalization of the definition for groups. 1.6 D e f i n i t i o n . A subloop H oi di loop L is normal^

and we write H <\ L^

if, for all x^y e L^ Hx = xH^ {Hx)y

= H{xy)^

{xH)y

— x(Hy)

and y(xH)

=

(yx)H.

1.7 R e m a r k . It is easy to show t h a t , if Hx = xH for all x, then any two of the last three equations here imply the third, and we usually make use of this fact whenever we want to show t h a t a certain subloop of a loop is normal. For example, Hx = xH, (Hx)y all x , y imply {xH)y {xH)y

= x{Hy)

= {Hx)y

= H(xy)

and y{xH)

= {yx)H

for

for all x , y since

= H{xy)

= {xy)H

= x{yH)

=

x{Hy).

The centre of a loop is clearly a normal subloop. Less obvious is the fact t h a t the commutator-associator subloop is also normal.

56

II. LOOP THEORY AND M O U F A N G LOOPS

1.8 Proposition. The commutator-associator normal subloop of L.

subloop V of a loop L is a

Let a, 6 € L. Since ah = (ba)(a^ b) and ab - c = (a • bc)(a^ 6, c), for any a, 6, c, we have ab G L^ if and only if ba G L^ and PROOF.

(4)

ab ' c ^ L' \i and only if a • 6c G L'.

Thus (5)

ab ^ c ^ L' \i and only if 6c • a G L'.

Let k = (a, 6) and suppose that ab • c = (ba • k)c is in V. Iterating (5), we have (c • ba)k G i/'; hence c-ba £ V and 6a • c G L\ Since this argument can be reversed, we have (6)

ab ' c ^ L' \i and only if 6a • c G L'.

Since (231) and (12) generate the permutation group on {1,2,3}, equations (5) and (6) imply that (7)

a6 • c G i/' if and only \i xy - z ^ L'

whenever {x^y^z^ — {a,6,c}. Let X £ L. Since L is a loop, for any a £ L^ there exists 6 G Z/ such that xa = bx. Using (4) and (7), we see that a e L^ <=)> x^x -ae L' ^^=^ x^ - xa e L' <=^ x^ - bx e L' <^=> x^b'X e l! <=> x^x 'b e L^ <==> 6 G L\ Thus xV = L'x. Let x,2/ G L, By (4), x{y • {xyY) G L' since {xy){xyy = I ^ L'. Since // is a loop, if a G //, there exists b ^ L such that ax - y = b - xy. Then a G //' <==^ a{x • 2/(2^2/)^} G L' <^=^ ax{y(a:2/)^} G L' ^=:> (ax ' y){xyy

e L' <^=> [b - xy){xyY

^ L'

^=^ b{xy ' [xyY] e L' ^=> b e L\ Thus {L'x)y = L'{xy). Similarly, one proves that y{xL') = {yx)V and so, by Remark 1.7, it follows that L' <\ L, completing the proof. D Just as in group theory, a useful characterization of the concept of a normal subloop makes use of an appropriate definition of "inner map".

1. WHAT IS A LOOP?

57

1.9 Definition. Let x and y be elements of a loop L. Define bijections T(x), R{x,y) and L{x,y) by

(8)

T(x)

=

R{x)L{x)-^

i?(x,s/)

=

R{x)R{y)R{xy)-^

L{x,y)

=

L{x)L{y)L{yx)-\

(Evidently, each of these maps belongs to the multiplication group M.{L) of L.) The inner mapping group of L is the subgroup Inn(i/) of A4{L) generated by all maps of the form T{x), R{x^y), L{x^y)^ Inn(L) = {T{x),R{x,y),L{x,y)

\x,ye

L),

and elements of Inn(Z/) are called inner maps. Noting that Hx = xH if and only \i HT{x) = H, that {Hx)y = H{xy) if and only if HR{x^y) = H and that y{xH) = (yx)H if and only if ^ L ( x , y) = if, we see that /T is a normal subloop of L if and only if HO = H for all e e Inn(L). If H is Si normal subloop of a loop L, if a;,i/,xi,yi G i/, and if ^ x = Hxi and //^i/ = Hyi^ then //(xy) = (/ra:)y = {Hxi)y = {xiH)y = xi(Hy)

= xi{Hyi)

= xi{yiH)

= {xiyi)H

=

H{xiyi).

Let H he di normal subloop of a loop L. The observations in §1.3 show that the cosets of H in L are disjoint. Thus, as in group theory, if L is finite, the order of H divides the order of L. Moreover, when H <\ L, the set of cosets of H forms a loop L/H = {Hx \x e L} in the expected way, called the quotient loop of L by H. This construction allows one to see that normal subloops of L are precisely the kernels of homomorphisms with domain L. Moreover, the three basic isomorphism theorems of group theory carry over, verbatim^ to loop theory. 1.10 Definition. A (right) pseudo-automorphism of a loop L is a bijection 6 of L with the property that, for some fixed c £ L^ (9)

{xe){y0 . c) = {xy)e • c

58

II. LOOP THEORY A N D M O U F A N G LOOPS

for all x, J/ G L. T h e element c is called a companion

of 6.

Just as in any algebraic structure, an automorphism bijection 0 of L which satisfies {xy)9 — (x9)(y9)

of a loop L is a

for all x , y G L. T h u s an

automorphism is also a pseudo-automorphism, with companion 1. As our wording implies, a pseudo-automorphism generally has more t h a n one associated companion. In fact, if c is a companion of a pseudo-automorphism ^ of a loop i/, then so is en for any n G Afp{L)^ since (xO)(yO ' en) = (xe){(ye

- e)n] = {{xe){y0

. e)}n = {{xy)9 • e}n =

{(xy)e}(en).

Since a pseudo-automorphism of a group is an automorphism (given associativity, the e on each side of (9) can be cancelled), the notion of pseudo-automorphism does not arise in group theory, but it is an i m p o r t a n t idea in loop theory and closely allied with the concept of "autotopism". 1.11 D e f i n i t i o n . An autotopism

of a loop L is a triple ( a , / 3 , 7 ) of bijec-

tions of L with the property t h a t (10)

{xa){yp)

=

{xy)j

for all X, J/ G L. Evidently, a bijection 0 of a loop L is an automorphism of L if and only if (0^0^ 9) is an autotopism; more generally, ^ is a pseudo-automorphism of L with companion c if and only if (^, 0R{e)^ 0R{e)) is an autotopism of L. If ( a , / ? , 7 ) is an autotopism of a loop L, then, replacing x by xa~^

and

y by yf3~^ in (10), we obtain xy — {xa~^ • y/5~^)7, hence

for all x^y.

Thus (a~^,/J~^,7~^) is an autotopism of L. Furthermore, if

( a i , / 3 i , 7 i ) and (a2,/32,72) sire autotopisms of L, then, for any x^y £ L, we have (xaia2)(yl3i/32)

= {{xai)(yl3i)}j2

since (a2,/?2,72) is an autotopism

= (a:y)7i72

since

(Q:I,/3I,7I)

Thus (aia2,/3i/32 5 7i72) is also an autotopism of L.

is an autotopism. It follows t h a t the

set of autotopisms of a loop forms a group with identity (I^I^I)

under

1. WHAT IS A LOOP?

59

coordinatewise composition, ( t t l , ^ l , 7 l ) ( « 2 , / ? 2 , 7 2 ) = (0:1^2,/3l^2, 7172).

W i t h o u t the notion of autotopism, many results in loop theory would be difficult to find and not particularly enlightening when presented. The s t a n d a r d proof of the theorem which follows illustrates the usefulness and elegance of arguments by autotopism. 1.12 T h e o r e m , The set of (right) pseudo-automorphisms a group under PROOF.

of a loop

forms

composition.

Since the set of right pseudo-automorphisms of a loop L is a

subset of the symmetric group on the set L and since the identity m a p on Z/ is a pseudo-automorphism, it is sufficient to show t h a t the set in question is closed under products and inverses. Suppose t h a t 0i is a pseudo-automorphism of a loop L with companion ci and t h a t 62 is a pseudo-automorphism with companion C2. Then (^1, 0ii?(ci), OiR(ci))

and (^2? ^2-^(^2), ^2^(c2)) ^re autotopisms of L, hence

so is their product, (11)

(^1^2,a,a),

where a = 6\R[ci)92R{c2).

Now

xa = {(X^i • C1)02}C2 = (a;^1^2){(ci^2) • C2}

for x G L, because 62 is a pseudo-automorphism with companion C2. Thus a = 6\02R{c\62'C2)

and (11) simply expresses the fact t h a t ^1^2 is a pseudo-

automorphism of L, with companion c i ^ • C2. Next, let 0 be a pseudo-automorphism of L with companion c. Then {9^6R{c)^0R{c))

(12)

is an autotopism and so is its inverse,

{e-\R{c)-^e-\R{c)-^0-^),

Since L is a loop, there exists d £ L such t h a t dO - c = 1. Since 0 is a pseudo-automorphism with companion c, for any x G //, xO = {xe)l so (9 = R{d)0R{c).

= {x0){d0 • c) = {xd)0 ' c,

It foUows t h a t R{c)-^0-^

= 0-^R{d).

Comparison with

(12) shows t h a t 0~^ is a pseudo-automorphism of L with companion d.

D

60

II. LOOP THEORY AND M O U F A N G LOOPS

2. I n v e r s e p r o p e r t y l o o p s In this section, we identify and discuss properties of a class of loops which contains all the loops of interest to us in this monograph. 2.1 D e f i n i t i o n . A loop L is an inverse property

loop if every x £ L has a

unique two-sided inverse, which we denote a:~^, and if, for all x^y ^ L^ the loop satisfies x~^{xy)

= y,

the left inverse

property,

(yx)x~^

= y,

the right inverse

and property.

Inverse property loops have many useful properties, the most basic of which we record in the following proposition. 2.2 P r o p o s i t i o n . Let L be an inverse property loop and denote by I and J, respectively,

the identity

for X E L.

and inverse maps on L; that is, xl = x, xj = x~^

Then

(i) J^ = I; that is, (x~^)~^

= x for all x G L;

(ii) R{x)-^

= R{x-^)

and L{x)-^

(iii) {xy)~^

= y~^x~^,

for all x,y e L;

(iv) JL{x)J

= R(x-^)

PROOF.

= L{^~^) V

and JR{x)J

= L{x-^)

« ' ' a: G L;

for all x e L.

Since every element of L has a unique two-sided inverse, and

since xx~^ = x~^x = 1, clearly (x~^)~^

= x, so we have statement (i). For

(ii), we write x~^(xy)

= y as yL(x)L{x~^)

so L{x)~^

Similarly, R{x)~^

= L{x~^).

inverse property.

= y.

Thus L{x)L{x~^)

— R{x~^)

follows from the right

For (iii), set z = xy and use the inverse properties to

deduce t h a t x = zy~^ and y~^ = z~^x^ so t h a t z~^ = y~^x~^. (iv), we note t h a t aJL{x)J aJR{x)J

= /,

= {xa~^)~^

= ax~^ = aR(x~^)

= aL{x-^),

Finally, for and, similarly, D

In view of Propositions 2.2 and 1.2, it is clear t h a t , in an inverse property loop, the subloop test is the same as it is for groups.

2. INVERSE PROPERTY LOOPS

61

2.3 Corollary. A nonempty subset H of an inverse property loop L is a subloop of L if and only if H is closed under products and inverses. In Section 1, we saw that a loop has a left, middle and right nucleus, and a nucleus which is the intersection of the three. 2.4 Proposition. LetAf\,M^ andMp denote^ respectively^ the left, middle and right nuclei of a loop L; let M{L) = M\ fl M^ fl Mp denote the nucleus of L. If L is an inverse property loop, all these nuclei are equal:

Let x G M\. Since M\ is a group, we have x~^ 6 M\{L) also. Hence, for any a, 6 € L, [x~^a~^)h~^ = x~^{a~^b~^). Taking the inverse of each side and using part (iii) of Proposition 2.2, we obtain b(ax) = (ba)x. Thus x G A/'p, so Afx C Afp, A similar argument gives the reverse inclusion and hence the equality of A/A and Afp, Now let X G A/"^. Then, for any a^b £ L, we have {ax)b = a{xb) and hence, by the left inverse property, PROOF.

(13)

b = {ax)-\a

• xb) = (x~^a"^)(a • xb).

Now let a^c ^ L. Since L is a loop, there exists b such that c = a - xb. By (13), 6 = ( x - ^ a - i ) c . On the other hand, x'^ia'^c) = x'^xb) = b. Thus {x~^a~^)c = x~^(a~^c), which shows that x~^ G Af\. Since A/A is a group, x G A/A. Thus Af^ C A/A- For the reverse inclusion, let x G A/A- Thus, for any a^b £ L, x(ab) — (xa)b^ so (14)

b = {xa)-\x

• ab) = (a~^a;"^)(x • ab).

Now let a^c ^ L. Since L is a loop, there exists b ^ L such that x - ab = c^ so, by(14), b = {a~^x~^)c. On the other hand, a~^[x~^c) — a~^{ab) = b. Thus (a~^x~^)c = a~^(x~^c) showing that x~^^ and hence also x^ are in A/"^. Thus A/A Q M^ and the result follows. D 2.5 The index of a subloop. Suppose that L is an inverse property loop and that H \s d, subloop of L. Since {h~^ \ h e H} = H^ it follows from part (iii) of Proposition 2.2 that the map Hx H^ X~^H is well defined and a bijection between the families of right and left cosets of H. Thus we can define [L: H] to be the minimal number of right cosets (or, equivalently, the

62

II. LOOP THEORY AND MOUFANG LOOPS

minimal number of left cosets) of H which cover L, if this number is finite, and infinity otherwise. We call [L\ H] the index of H in L. If L is the disjoint union of right cosets, then this definition coincides with its usual definition in group theory. Furthermore, Proposition 2.2(iii) shows that (3) is equivalent to {xh)H = xH

for all x G L and all h e H.

Thus L is the disjoint union of right cosets of H if and only if it is the disjoint union of (the same number) of left cosets of H and, if L is finite, we have [L: H] = \L\/\H\. Let ^ be a pseudo-automorphism of an inverse property loop L. Setting X = y = 1 in (9), we see that 16 = 1, Then, with y = x~^ ^ we obtain {xe){x-^e-c) = ie'C = c, hence x - ^ ^ - c ^ (x(9)-i-cand so x-^(9 = {x0)-^. Thus pseudo-automorphisms of inverse property loops share some of the properties of automorphisms. 2.6 Proposition. Let L be an inverse property loop and let 0 be a pseudoautomorphism of L. Then the restriction of 0 to the nucleus Af of L is an automorphism of Af. Let c be a companion of 6. Then, for any a^b £ L and any n G A/", we have {na)b = n(ab). Applying 9 and multiplying by c, we obtain PROOF.

{(na)e}{b0 . c} = {n0){{ab)9 - c) = (ne){{a9)(be • c)}. Since L is a loop and 9 is bijection of L, for any d £ L^ there exists b ^ L such that b9 ' c = d. Thus, for any a^d ^ L and any n G Af^ we have (15)

{na)9'd

=

{n9){a9'd).

Setting d — 1, we see that (na)9 = {n9){a9). Thus (15) becomes {n9a9)'d=

{n9){a9 - d),

showing that n9 G Af. Given any n G A/', we have n — x9 for some x ^ L. Since X — n9^^ and 9~^ is also a pseudo-automorphism of L, we have x G N] thus 9\j^ is a bijection of Af. Finally, for rij, n2 G A/", {n\n2)9 • c = {ni9){n29 • c) = [ni9n29)c since n\9 G Af. Thus (ni7i2)^ = (^i^)(^2^) so that 9\j^ is an automorphism of A/", as claimed. D

3. MOUFANG LOOPS

63

3. M o u f a n g l o o p s In this work, almost every loop encountered will be a Moufang loop. Such loops are named after Ruth Moufang who introduced them in a geometrical context in 1935 [Mou35]. 3.1 Theorem. In any loop, the following identities are equivalent: (16)

{{xy)x)z = x{y{xz))^

the left Moufang identity,

(17)

((xy)z)y = x{y(zy))^

the right Moufang

(18)

(xy){zx) = {x(yz))x^

the middle Moufang identity.

identity,

If L is a loop satisfying any of these identities, then L is an inverse property loop which also satisfies {yx)x = t/x^,

the right alternative

x(xy) = x^y^

the left alternative

{xy)x = x{yx)^

the flexible identity.

identity, identity,

P R O O F . Suppose L satisfies the left Moufang identity. Setting z = I in this identity gives the flexible identity, {xy)x — x{yx). In particular, {xx^)x = x{x^x) = a:, so xx^ = 1. Thus x^ = x^ so that x has a unique (two-sided) inverse, x~^. Setting x = y~^ in (16) gives y~^z = y~^{y -y'^z) for any y and z. For any y^w £ L, the equation y~^z = w can be solved for z. So we obtain w = y~^{yw) and hence the left inverse property. With z = x~^ in (16) we get {xy • x)x~^ = xy and hence the right inverse property. Thus L is an inverse property loop. Setting y = 1 in (16) gives the left alternative identity which, in an inverse property loop, is equivalent to the right alternative identity. (See part (iii) of Proposition 2.2.) Similarly, taking the inverse of each side of the left Moufang identity yields the right Moufang identity. Replacing z by x~^z in the left Moufang identity gives {xy ' x){x~^z) = X ' yz^ so

(x ' yz)x = {{xy • x){x~^z)}x — {xy){x{x~^z

• x}

= {xy){{x • x~^z)x} = (x2/)(zx);

by the right Moufang identity by the flexible identity

64

II. LOOP THEORY AND M O U F A N G LOOPS

thus we obtain the middle Moufang identity. Suppose that // satisfies the middle Moufang identity. Setting y = 1 in this identity gives x(zx) = (xz)x^ the flexible identity, which, with zx = 1, gives X = (xz)x and hence xz = 1. Thus each element x of L has a unique two-sided inverse, x~^. Setting y = x"^ in (18) gives zx = {x • x~^z)x and hence z = x - x~^z^ giving the left inverse identity, and setting z = x~^ in (18) gives xy = {x'yx~^)x = x{yx^^ -x), by flexibility, so that (yx~^)x = y. Thus L also satisfies the right inverse property. As an inverse property loop, L satisfies {xy)~^ = y~^x~^; thus (18) implies zx = {y~^x~^)(x ' yz ' x) where the last factor on the right here is unambiguous by flexibility. Replacing y by t/~^, we obtain zx = (yx~^)(x • y~^z • x), then, replacing z by yz, we obtain (yz)x =

{yx~^){xzx)

and finally, replacing y by yx, we obtain (yx • z)x = y{xzx) — y{x • zx) which gives the right Moufang identity. Lastly, if L satisfies the right Moufang identity, straightforward arguments, as above, show that L is an inverse property loop and hence, applying part (iii) of Proposition 2.2 to the right Moufang identity gives the left Moufang identity. D 3.2 Definition. A loop L is Moufang if it satisfies any of the three (equivalent) Moufang identities. The right (or left) alternative identity shows that x^ is well defined in any Moufang loop. By induction, it follows that a Moufang loop L is power associative; that is, x^ is well defined for any x £ L and any integer n. Any group is a Moufang loop and, unless we specify to the contrary, all statements in this book about Moufang loops apply equally to groups. In a group, the inner maps R(x^y) = R{x)R{y)R{xy)~^ and L{x^y) = L{x)L{y)L(yx)~^ reduce to the identity while T(x) = R(x)L(x)~^ is just

3. MOUFANG LOOPS

the m a p a \-^ x~^ax, known as conjugation

65

by x. Thus, in a group, all inner

maps are automorphisms. While such need not be the case in general in a Moufang loop, it is useful to know t h a t inner maps are at least pseudoautomorphisms and, hence, t h a t any inner m a p ^ is a

semi-automorphism

in the sense t h a t (xyx)e = for all

{x0){ye)(x0)

x^y.

3.3 T h e o r e m . IfL (i) R{x^y)

is a Moufang

= L(x~^,y~^)

ion the commutator, (ii) T(x)

loop, then

is a pseudo-automorphism

compan-

(x^y);

is a pseudo-automorphism

of L with companion

(iii) every inner map on L is a pseudo-automorphism (iv) every pseudo-automorphism PROOF.

of L with

of L is a

x~^;

of L and

semi-automorphism.

For x,t/ G L^ we have

L{x-\y-')

= L{x-')L{y-')L{y-'x-'r'

=

L{x-')L{y-')L{xy),

by Proposition 2.2. Thus a L ( x ~ \ y ~ ^ ) = (xy){y~^{x~^a)}. aL(x"\y"^)

Hence

'{xy)

— {{^y)y~^}{{^~^^){^y)} = x{{x~^a){xy)}

^y the middle Moufang identity by the right inverse property

= {{x ' x~^a)x}y

by the left Moufang identity

= ax • y

by the left inverse property,

from which it follows t h a t aL{x~^,y~^) t h a t is, L{x~^^y~^)

—

= {ax - y){xy)~^

=

aR{x,y)',

R(x^y).

Note t h a t the middle Moufang identity—xy - zx — {x - yz)x—is lent to the assertion t h a t A{x) = (L{x)^ R{x)^ L(x)R{x))

equiva-

is an autotopism

of L, for any x. Since the autotopisms of a loop form a group under coordinatewise composition,

Aix-')A{y-')Aiy-'x-')-'

= ia,P,'f)

66

II. LOOP THEORY AND MOUFANG LOOPS

is also an autotopism of L, where

a = Lix-')L{y-')L{y-'x-')-'

=

L(x-\y-')

and 0 =

R{x-')Riy-')R{y-'x-')-\

Since ( a , / 3 , 7 ) is an autotopism of L, we have (19)

{xa){yfi)

=

{xy)j

for any x^y e L. With x = 1^ and noting t h a t la = 1, we have yP = yj for all 2/, hence /3 = j . Thus (19) becomes (xa){yf3) y = 1, (xa)(lf3) (x^y).

= {xy)l3 and then, with

= x/3. Thus /? = aR{c)^ where 0=1/3

= x~^y~^

• x?/ =

Since (a,/3,7) = (a,ai2(c),ai?(c))

is an autotopism, a = L ( x ~ \ y ~ ^ ) is a pseudo-automorphism with companion c = (x^y)^ which gives (i). For (ii), we use the right alternative identity and right inverse property to note t h a t {bx)x~^ (20)

= {bx 'X~^)x~^

bT{x) . x-^ = {x-^ '

bx}{x-^x-^}

= x~^{bx ' x~^)x~^

=

= bx~^ and thus

by the middle Moufang identity

x-\bx-^)x-\

(Throughout these calculations, we use the flexible identity to avoid the excessive use of parentheses.) Thus {aT{x)){bT{x)'X-^)

= {x-^

= x~^{(ax){x~^

'ax}{x-\bx-'^)x-^}

' bx~^)]x~^

— a;~^{[(ax • x~^)b\x~^}x~^ = x~^{ab

by the middle Moufang identity by the right Moufang identity

'x~^}x~^

= {ab)T{x)'X-^

by (20).

Thus T{x) is a pseudo-automorphism with companion a:~^, as claimed. For (iii), we recall t h a t the inner mapping group Inn(L) is generated by maps of the form R(x^y)^

L{x^y)

and T{x) and so, in a Moufang loop.

3. MOUFANG LOOPS

67

just by maps of the form R[x^y) and T(x), by part (i). Since the pseudoautomorphisms of a loop form a group (Theorem 1.12), aU inner maps of a Moufang loop are pseudo-automorphisms, by part (ii). Finally, suppose 0 is a pseudo-automorphism of L, with companion c. Then, for x^y ^ L, {x • yx)e • c = {xe}{{yx)e

• c} = {x0}{{ye){xe

' c)} = {{xO . ye)xe}c

by the left Moufang identity. Cancelling c and remembering that Moufang loops satisfy the flexible identity, we see indeed that 0 is a semiautomorphism. D Since any inner map of a Moufang loop is a pseudo-automorphism, the nucleus of Moufang loop is invariant under inner maps, by Proposition 2.6. The next result is therefore clear. 3.4 Corollary. The nucleus of a Moufang loop is a normal subloop. 3.5 Remark. While Lagrange's Theorem holds for finite Moufang loops of odd order [Gla68], it is not known whether or not this theorem is true for Moufang loops in general. Nevertheless, if L is a Moufang loop, then L is an inverse property loop and so we can still talk about the index of a subloop of L. (See §2.5.) If a and x are any elements of L, it is not hard to show that a'^{a'^x) = a'^'^'^x for any integers n and m. Thus, if H = (a) is the cyclic group generated by an element a, we have H{hx) = Hx for all X G // and h ^ H\ hence, if |L| is finite, \L\ is divisible by the order of a, as shown in §1.3. Conversely, we note that if L is a Moufang loop of finite even order and if/? is a prime which divides |L|, then L need not contain an element of order p, (This cannot occur for \L\ odd, however. See [Gla68].) In this section, it has become clear that Moufang loops share many of the familiar properties of groups. Perhaps that result which best typifies the close connection between groups and Moufang loops is "Moufang's Theorem", which is the analogue for Moufang loops of the Generalized Theorem of Artin for alternative rings. As we shall see in §5.3, Moufang's Theorem is a consequence of Theorem 1.1.8 for those Moufang loops which are contained in alternative rings. (Such will always be the case in this work.) For

68

11. LOOP THEORY AND MOUFANG LOOPS

a proof of the general theorem, we refer the reader to the Pflugfelder's book [Pfl90, Section IV.2]. 3.6 T h e o r e m ( M o u f a n g ) . If a^ b and c are (not necessarily ements

of a Moufang

generated

loop which associate

distinct)

el-

in some order, then the subloop

by a, b and c is a group.

Since Moufang loops satisfy the right alternative identity (yx)x

= ^x^,

it follows immediately from Moufang's Theorem t h a t 3 . 7 C o r o l l a r y . A Moufang

loop is

diassociative.

Diassociativity, which we often use implicitly, is an extremely i m p o r t a n t property of a Moufang loop.

4. H a m i l t o n i a n l o o p s A Moufang loop is hamiltonian

if it all its subloops are normal. Readers

familiar with the theory of hamiltonian groups will note t h a t our definition (standard in loop theory) conflicts with the usual definition which requires a hamiltonian group to be nonabelian. The quaternion group Qs of order 8 is hamiltonian; moreover, any hamiltonian group

is the direct product

of Qs-, an abelian group of exponent two and an abelian group all of whose elements have odd order. In the theory of Moufang loops, there is a special loop, known as the Cayley loop^ which shares many of the properties of Qg. In particular, any hamiltonian Moufang loop is either an abelian group, a hamiltonian group, or the direct product of the Cayley loop, an abelian group of exponent two and an abelian group all of whose elements have odd order. This result of D. A. Norton is the primary focus of this section [Nor52]. We begin by defining the Cayley loop. 4.1

The Cayley loop,

MIG{QS)*

Let {l,z,j,/;;,£, z ^ , j ^ , H } be the usual

basis of the Cayley numbers (see §1.2). These elements multiply as shown in Table 1.

T h u s , it is clear t h a t the sixteen elements consisting of these

eight, together with their negatives, form a loop which is a subset of the Cayley numbers. Since the Cayley numbers form an alternative ring and hence satisfy the Moufang identities, this loop is a Moufang loop. It is called

4. HAMILTONIAN LOOPS

69

T A B L E 1. A multiplication table for a basis of t h e Cayley numbers

1

i

j

k

i

a

3^

ki

1

1

I

3

k

\ i

a

ji

ki

i

i

-1

k

-3

a

-i

-ki

ji

3

J

-k

-1

i

ji

ki

-i

-a

k

k

J

-i

-1

ki

-3^

a

-i

i

i

-i£

-ji

-ki

-1

i

3

k

a

a

i

-kl

3^

— i -1

ji

ji

ki

£

-a

-3

ki

kl

-j(

il

i

-k

-k

3 1

k

-1

—i

-3

i

-1

the Cayley loop and denoted Mi6(Q8) following Orin Chein, who classified and introduced notation for all Moufang loops of order less than 64 (which are not groups) [ C h e 7 8 ] . In preparation for Norton's result, we gather in a series of lemmas some facts about hamiltonian Moufang loops. The material which follows uses ideas from Norton's paper as well as Section 12.5 of Marshall Hall's book [Hal59] where the structure theorem for hamiltonian groups is proven in careful detail. 4.2 L e m m a . Suppose

L is a hamiltonian

Moufang

loop and a^b^c G L.

Then 1. ab — bai = bia for some ai G {a) and some 6i G (b); 2. the commutator

(a, 6) G (a) fl (6);

3. {ab)c = ai{bc) = a{bic) = a(bc^) for some ai G {a), bi G (b) and ci e (c); 4. a{bc) = {a2b)c == {ab2)c = {ab)c2 for some a2 G (a), 62 € {b) and 02 G ( c ) ;

70

II. LOOP THEORY AND MOUFANG LOOPS

5. the associator PROOF.

(a, 6, c) E (a) H (6) fl (c).

Our proof makes extensive use of the inner maps T ( x ) ,

R(x^y)

and L(x, y) defined in (8) and of the fact t h a t a subloop of a loop is normal if and only if it is invariant under each of these maps and their inverses. 1. By the definition of loop, ab = bai for some ai ^ L, In a Moufang loop, ai = b~^ab = aT(b) G (a), since L is hamiltonian. Similarly, ab = bia for some 6i G (b). 2. By part 1, diassociativity and the definition of loop, ab = bai = b(aa2) = (6a)a2, where a i , a 2 G (a), so, by definition of the c o m m u t a t o r , (a,6) = a2 E (a). Similarly, (a,6) G (b). 3. We have (a6)c r:. aR{b)R{c)

^ aR(b)R{c)R(bc)-^R{bc)

=

aR{b,c)R{bc).

Setting ai = aR(b^c) G (a), we obtain (ab)c = ai{bc) as required. Also {ab)c = cL{ab) = cL{b,a)-^L{b,a)L[ab) With ci = cL(b^a)~^ (ab)c = -

=

ciL{b)L{a).

G (c), we obtain (ab)c = a(6ci), as required. Finally,

bL{a)R{c) bT-\a)R{a)R{c)

= biR{a,c)R{ac)

with 6i = bT-\a)

= b2T{ac)L{ac)

with 62 = 6ii?(a,c) G (6)

= 63L(c,a)"^L(c,a)L(ac)

with 63 = b2T{ac) G (6)

=• b4L{c)L{a) -

G (6)

with 64 = 631(0, a ) " ^ G (6)

64T-\c)/2(c)L(a)

= bsR{c)L{a)

with 65 = & 4 r " \ c ) G (6)

= a(65c).

4. We show how the first of these equations can be derived from part 3 and leave the others to the reader.

Since a Moufang loop is an inverse

property loop, for any x and y in L, we have {xy)~^ a{bc) = {(c-H-^)a-^}-^

=

= y~^x~^,

{c~\b-\-^)}-^

Thus

4. HAMILTONIAN LOOPS

71

for some a^^ G (a~^) = (a), by part 3. So a{bc) = (a2b)c as desired. 5. Using parts 1, 3 and 4, and the definition of loop, we have (ab)c = ai(bc) = (a2a){bc) = as{a • be) = (a • 6c)a4, with all a^ G (a), thus (a, 6, c) = a^ G (a). For similar reasons, {ab)c = a{bci) = a{b • CC2) = a{bc • C3) = (a • 6c)c4, all c^- G (c), so (a, 6, c) = C4 G (c). Finally, (a6)c = a(bic) = a{b2b c) = a{bs • be) = a(be • 64) = (a • 60)65, ^lU 6^- G (6), so (a, 6, c) = 65 G (6). 4 . 3 L e m m a . If L is any diassociative tator (x, y) commutes Moufang

loop),

loop, ifx^y^L

and if the

with x and with y (in particular,

if L is a

commu-

hamiltonian

then

1. (a;^,t/^) = ( x , y ) ^ ^ and for all 7n,n G Z. PROOF.

We require the fact t h a t (a, 6) = a~^b~^ab and (a,6)~^ = (6, a)

for any elements a and 6 in a diassociative loop. Since (a:, 2/) commutes with X, the element y~^xy

= x{x^y)

x~^; so ( x , y ) = x~^y~^xy

also commutes with x and hence also with

= y~^xyx~^

= (y^x~^).

Since (x^y)

commutes

with both X and y, by iteration, we get (21)

{x,y)

= {y,x-')

= {x-\y-')

=

{y-\x).

To prove statement 1, it is sufficient to prove only

(22)

{x^,y) = {x,yr.

To see why, we note t h a t (22) implies, in particular, t h a t the commutator of x^ and y commutes with both x"^ and y. Hence, for any m G Z, ( x ^ , y ^ ) = ( 2 / ^ , x ^ ) - ^ = iiy.x'')'^)-^

= [x'^.y)'^

= {x.y)'^'^,

using (22) twice. Thus

the first statement does follow from (22), as asserted. We turn our attention to the proof of (22). Observe t h a t the case n = 0 is trivial and, if the case n = A; is true, the case n = /? + 1 is true because {x.yf^^

= {x,y)\x,y)

= {x, y)^ {x~^ y"^ xy) =

= x~^{x^,y)y~^xy

x~^{x,y)^y~^xy

= x~^x~^y~^x^yy~^xy

=

Thus (22) is true for all n > 0. For n > 0, {x,y)-^

= {(x^y)^)-'

= (x^^y)-'

= (t/,x-) =

{x~^,y)

D

(x^'^^.y).

72

II. LOOP THEORY AND MOUFANG LOOPS

(using (21) at the last step), so (22) is true for all n G Z. Turning to statement 2 of the lemma, we note that the case n = 0 is again trivial. Also, if (xy)^ = a:^t/^(y, x)^(^~^)/^, then

{xyf+' = ixy)''xy =

x''y'iy,x)'^''-'y^xy

^x'+'y>'-'\y,x)''(''+'y\ giving the result for all n > 0. The result for all integers now follows from

(xy)-^ =

iixyrr'

= x-"2/-"(y-",a:-")(2/,x)-"("-i)/2 = x - " y - " ( y " , a;")(y, x)-"^"-')/^

by (21)

= a;~"j/~"(y,x)"^~'"^"~^^/^'

using statement 1 twice

Finally, the parenthetical remark in the statement of the lemma is a consequence of the fact that a Moufang loop is diassociative and, if it is hamiltonian and x^y £ L, then (x^y) G (x) D {y) (Lemma 4.2) so (x^y) commutes with x and with y, D In the proof of the result which follows, we use the fact that if a group G = (a, b) is generated by two commuting elements of prime power order and o(a) > 0(6), then there exists Some 61 G G such that G = (a) x (61). This is a consequence of the theory of finite abelian groups. 4.4 Lemma. Let L be a commutative hamiltonian Moufang loop. Then L is associative and hence an abelian group. We show first that x^ is in the nucleus of L for every x ^ L. Since L is commutative, for any x £ L^ the inner map T(x) is the identity. PROOF.

4. HAMILTONIAN LOOPS

73

By Theorem 3.3, T{x) is a pseudo-automorphism with companion x~^.

It

follows t h a t a:~^ is in the nucleus of L; hence, so is x^. Now suppose a^b^c £ L and the associator ( a , 6 , c ) ^ 1. We claim a, b and c have finite order. For this, write {ab)c = (a • bc)s with s = (a, 6, c) = a^ = 6^ = c^ for some nonzero integers i^j^ky by Lemma 4.2. Let m = ijA;. Then (23)

(ab-c)"^

=

{a'bc)'^s'^.

By commutativity and diassociativity {ab • c ) ^ = (ab)'^c'^ = {a'^b'^)c'^

= s^^s'^s'^

= ^ij+jk-\-ik

and, similarly, (a-bc)'^ = ^«i+jA:+i/: From (23), we obtain s'^ = I and hence ^tm _ f^jm _ ^/:m _ j Xhus a, 6 and c have finite order, as claimed. Write o(a) = 3^m, where 3 J^ m, and choose integers k and ^ such t h a t fc3^ + £m = 1. Then a = a^^"^ a^"^ with the first factor in the nucleus of L (and hence in the centre). It follows t h a t {a^b^c) — (a^'^^b^c)

and so,

replacing a by a^^, if necessary, we may assume t h a t the order of a is a power of 3. Similarly, we assume that the order of 6 is a power of 3. In view of the remark preceding this lemma, we can write (a,6) = (a) X (6i) for some bi. In particular, for any integers i and j , we have (a^) 0 (6^) — {1} so, by Lemma 4.2, {a\b'[yc)

— 1. Now b — d^b\ for some integers u and v.

By diassociativity, ab — a^+^ft^, so {ab)c = (a^+^6y)c = a''^\b\c)

= a^a"" • 6^c) = a{a''b\ • c) = a(6c),

which contradicts ( a , 6 , c ) 7^ 1. The result follows. 4.5 L e m m a . Let L be a hamiltonian elements

of L which do not commute.

the subloop which they generate contains PROOF.

Moufang

D loop and let a and b be

Then a and b have finite order and Qg-

By Lemma 4.2, there exist integers u and v such t h a t c =

(a, 6) = a~^b''^ab

= a^ = b^. In particular, c commutes with a and with b.

By Lemma 4.3, 1 = (c,6) = (d'^-fb) — (a^b)'^. Thus c, and hence also a and 6, have finite order. Consider the set of all pairs of elements x^y ^ (o^,&) which do not commute and choose a pair for which o(x) + o{y) is minimal.

74

11. LOOP THEORY AND MOUFANG LOOPS

As above, the commutator (x, y) commutes with x and with y. Let n = o{x) and m = o{y). Suppose p is a prime dividing n and p ^ n. Then x^ and y commute by minimality, so 1 = {x^',y) = (^^vY showing that (x^y) has order p. It follows that p is the only prime divisor of n and, similarly, that p is the only prime divisor of m. Let n = p'^^ and m — p^^ and assume, without loss of generality, that n\ >_ m\. As before, (x,2/) = x^ for some i. Since p = o{x^) = p'^^/gcd(p'^^^i)^ we have £ = jp'^^~^ for some j ^ 0 (mod p). So {x,y) = x^^""^ and, similarly,

for some k ^ 0 (mod p). Let t = —jp"'i-^i and yi = x^y^. X and yi do not commute since gcd(k,p'^) T/[

Note t h a t

= 1 implies (y) = (t/i). Thus

7^ 1, by minimality. By Lemma 4.3,

= x^^"'^~\^^^^~\y^,x^y^^~^^^'^'~^~^^/'^ = x~^^^' ~' y^^"^~^

(t/, x)^^^^'~'

(^"^"^"^)/^

= (x,y)-'(a:,y)(x,y)'=^^"'-(^'"'-"-^)/^ and thus (x,y)^'^^"^ (^"^^ that mi > 1 and, since ni j are both odd, and ni = and t/ have order 4, (a;, y)

-i)/2 ^ i whereas (x,y) has order p. It follows > m j , that ni > 1 too. Therefore p = 2, so A; and 2. Since ni > mi > 1, also mi = 2. Therefore x = x'^ = y'^ has order 2, and (x, t/) = QgD

4.6 Corollary. Let L be a hamiltonian Moufang loop which is not commutative. Then every element of L has finite order. By Lemma 4.5, there exist a,b e L such that a"^ = 1, a'^ = b'^ = (a, 6) and (a, 6) = Qs- Let PROOF.

C{L) = {g £ L \ gx = xg for all x ^ L}. If c ^ C(Z/), then c has finite order by Lemma 4.5. Suppose c G C{L) and write (24)

(ab)c = (a • bc)s

4. HAMILTONIAN LOOPS

75

for some s £ {a)n {b) fl (c), by Lemma 4.2; so s = I or s = a^. In the latter case, c has finite order because 5 G (c), so assume s = 1. Squaring each side of (24) gives (25)

( a 6 ) V = (a • bcf.

If a and 6c do not commute, then be has finite order, hence so does c (because 6 and c commute).

If a and be do commute, then (ab^e'^

= a^{be)'^ =

oP'ilP'e^) and so, since o? — {ah)^ in Q^^ b^i? = (? and b^ — 1, contradicting the fact t h a t b has order 4.

D

4 . 7 L e m m a . Le^ L be a hamiltonian tative.

Moufang

loop which is not

eommu-

Then L = LQ x A, where LQ is a 2-loop and A is an abelian group

all of whose elements PROOF.

The set

have odd order in the centre of L. LQ

of aU elements of 2-power order in L is a subloop

by Lemma 4.3. Let A be the set of elements of odd order in L. If a i , a 2 G A and ( a i , a 2 ) ^ 1, then the group G — ( a i , a 2 ) contains an element of order 2, by Lemma 4.5. This cannot be the case, however, since the fact t h a t ( a i , a2) is a power of a\ makes it is easy to see t h a t the elements of G have the form d^a^-, which has odd order by Lemma 4.3. Thus the elements of A commute, hence A is commutative and an abelian group, by Lemma 4.4. Since every element of L has finite order (Corollary 4.6), we have L = LQA.

Any element of LQ commutes with any element of A, by Lemma 4.2,

and, for the same reason, if x,y^z {x^y^z}

are three elements of L with one of

in LQ and another of { x , y , z } in A, then also (x^y^z)

= 1. Let

ci,C2 G Z/o, cti, a2 G i4 and observe t h a t (ciai)(c2a2) = {ciai • C2)a2 = (ci • aiC2)a2 = (ci • C2ai)a2 = (ciC2 • ai)a2 =

(c\C2)(aia2).

It follows t h a t L = LQ X A and A is central.

D

4.8 T h e o r e m ( N o r t o n [Nor52]). Let L be a Moufang hamiltonian

loop.

Then L is

if and only if

1. L is an abelian group, or 2. L — Qs X E X A, where Qs is the quaternion is a (possibly trivial) elementary

group of order 8, E

abelian 2'group and A is a (possibly

76

II. LOOP THEORY AND MOUFANG LOOPS

trivial) abelian group all of whose elements

consist of finite odd order,

or 3. L = MieiQs)

X E X A, where Mie(Qs)

is the Cayley loop and E and

A are as in 2. PROOF.

First we show t h a t the classes of loops identified in the state-

ment are hamiltonian.

This is obvious for an abelian group.

L = LQ X E X A where LQ = Mie{Qs)^

Suppose

E is an elementary abelian 2-group

and A is an abelian group all of whose elements have odd order. Note t h a t LQ has a unique square (different from the identity). Call this element s. If ^ is a subloop of L and H is central, then H is normal. On the other hand, let h ^ H be an element which is not central. Write h = qea with q e Lo^ e G E^ a G A and note t h a t q must have order 4 (otherwise, it is central). Thus q^ = s^ h"^ = sa'^ and, with n = o ( a ) , /i^^ = s'^a'^'^ = s since n is odd and ^^ = 1. Thus s e H, Now let h e H and x G L. Then xh = hx or xh = (hx)s

since s is the only c o m m u t a t o r in L other than the identity.

Since s G H and s is central, {hx)s = {hs)x G Hx. It follows t h a t xH = Let h £ H and a:,t/ G L. Then {hx)y

= h(xy)

or {hx)y

— [h • xy^s

Hx. since

s is also the only associator in L other than the identity. It follows t h a t {Hx)y

— H{xy)

and, similarly, t h a t x{yH)

= {xy)H^ so H is normal. Thus

L is hamiltonian. A similar argument shows that any group as described in statement 2 is also hamiltonian. The main import of this theorem is necessity, to which we now turn our attention. If L is commutative, then L is an abelian group by Lemma 4.4 so, for the rest of the proof, we assume t h a t L is not commutative. Moreover, in view of Lemma 4.7, we also assume t h a t L is a 2-loop. Suppose first t h a t L is a group. By Lemma 4.5, there exist a, 6 G Z/ such t h a t a"^ = 1, a-^ = 6"^ = (a, 6), so t h a t Q = {a,b) is a group isomorphic to the quaternion group of order 8. Let C(Q) = {c e L \ gc = eg for all g G Q}. U g G L does not commute with a, then g'^ag

— a ~ \ since g~^ag G (a) has

order 4. Since b~^ab — a ~ \ we see t h a t gh~^ commutes with a. Similarly, \i g G L does not commute with 6, then ga~^ commutes with 6. We claim

4. HAMILTONIAN LOOPS

77

that L = C{Q)Q and consider four possibilities, li g ^ L commutes with both a and 6, then g G C{Q) C C{Q)Q. If g commutes with b but not with a, then gb~^ commutes with both a and 6, so ^^6"^ G C{Q), hence g = (gb~^)b G C{Q)Q. Similarly, if p commutes with a but not with 6, ^ G C(Q)Q. Finally, if ^f commutes with neither a nor 6, then ^6~^ commutes with a but not with 6, so gb~^a~^ commutes with both a and 6. Therefore, g{ab)-^ G C(g), so ^ G C{Q){ab) C C ( g ) g . Thus, indeed, L = C(Q)Q. Let c G C((5) have order 4. Then be and a do not commute and (bc)"^ = 1. If be has order 4, then a"^(6c)a is an element of order 4 in (6c), which is cyclic of order 4. Thus a~^(6c)a = (6c)~^, an equation which also holds if be has order 2. In all cases, then, a~^(be)a = {be)~^ = e~^b~^ = b~^e~^. On the other hand, a~^(be)a = 6~^c, so c = e~^ has order 2, a contradiction. Thus C{Q)^ containing no elements of order 4, is an elementary abelian 2group. Clearly C{Q) is central. Since {a'^) is the only subgroup of Q which does not contain an element of order 4, we also haveC(Q)nQ = {l^a^}. Let £ be a maximal subgroup of C{Q) not containing a^. Thus E H Q = {1}. If X G C{Q) has order 2 and x ^ E^ then a^ G {E^x) implies a'^ = 6X, for some e G ^ , so x = a^e G Q ^ . Thus L = Q X E. Assume L is not associative. Let a and b be any two elements of L which do not commute and let G be the group generated by a and 6. As just shown, we must have G = Q8>^ E for some elementary abelian 2-group E. Thus G/G' is an elementary abelian 2-group which can be generated by two elements, so \G/G^\ < 4; therefore, G = Qs- We have learned that any two elements a and 6 of L which do not commute must generate a group isomorphic to Qg] in particular they have order 4 and a? — b'^ — (a^b). Now fix a, 6 G Z/ such that Q = {a,b) is isomorphic to Qg- Let C{L) = {e ^ L \ ex = xe for all x G L}. We claim that (26)

C{L) = {eeL\e^

= l},

Suppose c G L, c^ = 1 and c / 1. Then, for any x G L, we have xe — e\x with ci G (c) = {l,c} by Lemma 4.2, so ci = c. Thus e G C(//). Conversely,

78

II. LOOP THEORY AND MOUFANG LOOPS

suppose c G C{L). Write {ab)c = (a • bc)s^ where 5 G (a) fl (6) D (c). Hence s = 1 OT s = a^. Since a^ = (aby G Q and L is diassociative, a^c^ = {abfc^

= {ab • cf

= (a • 60)^52 = (a • 6 c ) ^

If a and 6c commute, then a^c^ = 0^(60)-^, so c^^ = i^^)^ = ^^^^ implying b^ = 1^ which is not true. Thus a and be do not commute, so they generate a group isomorphic to Qg- So a^ = (bc)'^ = b^c^ = o?c^ and c^ = 1. This verifies (26). Let C{Q) = {c e L \ ex - xc for all x e Q} and note that the preceding argument in fact showed t h a t , if c G C(Q)^ then e^ = I. Thus

C{Q) = CiL). Also, we see t h a t a^ = b^ is the only nonidentity square in L, for if c G C((5), then c^ — \^ while if c does not commute with some q £ Q^ then c^ = q^ = c?, From this fact, and because L is a 2-loop, it follows, in particular, t h a t any x G //, x 7^ 1, must have order 2 or 4. We claim t h a t C(L) = 2 ( L ) , the centre of L, For this, we have only to show C(Z/) C M{L)^ the nucleus of L, because 2 ( L ) = M{L) n C(L). Let c G C(L), c / x , y G //, (x^)c = x(yei)^

ci G (c) = { l , c } since c'^ = 1. Since c /

must have Cj = c, so (x^y^e) L is Moufang, e G M{L).

1. Then, for any 1, we

= 1. So c is in the right nucleus and, because

(See Proposition 2.4 and Theorem 3.1.)

Suppose x^y £ L and xi/ = yx.

We claim (x^y^z)

= 1 for all z £ L.

Since C(//) = Z{L)^ this is clear if either a:-^ = 1 or y^ = 1, by (26). Suppose x^ y^ I and i/-^ /

1. Then x ^ C{L)^ y ^ C{L)^ so x and y have order 4,

x^ = ?/^ and y = yx"* = (yx^)^ with (yx^)^ = y^x^ = 1. Thus y = ex with c = yx'^ G 2{L). easily t h a t (x^y^z)

By diassociativity and the definition of centre, it follows = 1 for all z.

Since a^ is the only nonidentity square in L and since x~^y~^xy

=

x~^(a:y~^)'^?/^, it follows t h a t a^ is the only nonidentity commutator in L. Suppose x , y , z G // do not associate.

As we have just shown, x and y

4. HAMILTONIAN LOOPS

79

cannot commute, so x and y each have order 4 and x^ — y^ — a^. Moreover [x^y^z)

G {x)n(t/) and this intersection must be {l,a:^}. Thus (x^y^z)

= a?.

So a? is also the only nonidentity associator in L. Replacing a and 6, if necessary, there exist a^b^c e L such t h a t (a, 6, c) / 1. Since a and b do not commute,

g = (a,6) is a group isomorphic to QsM — {a,b,c)

Our observations to this point show t h a t

is the Cayley loop. ( T h e elements a, 6, a t , c, ac, 6c, a6 • c

correspond, respectively, to the elements i, j , A:, £, i£, j £ , H which define MieiQs)-)

Let (J(7lf) = {c 6 i/ I cm = mc for all m e L}.

Clearly C{M) C C(Q) = Z{L),

so C{M) = 2 ( / / ) is an elementary abelian

2-group. We claim t h a t L = Z{L)M. 4. If ^f-^ = 1, then g G Z{L),

Let g £ L^ g ^ I, Thus ^ has order 2 or Suppose g has order 4. If g commutes with

an element m G M of order 4, then (^m"^)-^ = g'^m~'^ — 1 since g^ = Tn-^. Thus ^m~^ G >2(L) and ^ = {gm'^)!!!

G Z{L)M.

On the other hand,

suppose ^ commutes with no elements in M of order 4. If ^ associates with two noncommuting elements m i , m 2 G M of order 4, then ^, rrii and m2 generate a group G (Moufang's Theorem) and, as in the proof of the already completed hamiltonian group case, G = QQE where QQ = (^1,7712) — Qs and E is an elementary abelian 2-group. Thus g — me with m e M and e G £" C Z{L)^ so ^ G Z{L)M.

So we now further assume t h a t g associates

with no two of a, 6, c. If ga^ b and c associate, then ga G Z{L)M^ so ^ G Z{L)M.

as above,

On the other hand, if (ga^b^c) ^ 1, remembering t h a t a-^

is the only nonidentity associator in L and t h a t this element is central, we obtain {ga){bc) = {{ga • b)c}a'^ = {{g • ab)c}a'^ = {g{ab • c)]a^ — {g{a • bc)}a^ — g(a • be) which shows t h a t g, a and be associate. Since a and be are noncommuting elements of order 4 in M , this implies ^ G Z{L)M. t h a t L = Z{L)M.

So we have established

Since Z{L) n M = { l , a ^ } , letting £^ be a maximal

80

II. LOOP THEORY AND M O U F A N G LOOPS

subgroup of Z(L)

not containing a^^, it follows t h a t L = M X £ as in the

group case. This completes the proof.

D

5, E x a m p l e s o f M o u f a n g l o o p s In this section, we introduce some important examples of Moufang loops which are not groups. 5.1

M{G^ 2) l o o p s . Let G be a nonabelian group and let u be an inde-

terminate. Let L = G 0 Gu be the disjoint union of G and Gu and extend the binary operation on G to a binary operation on L by the rules

(27)

g{hu)

=

{hg)u

{gu)h

=

{gh-')u

(gu){hu)

=

h-^g

for g^h E G. Then L is a Moufang loop, denoted M ( G ' , 2 ) , which is not a group. Moufang loops of this type were discovered by Orin Chein [ C h e 7 4 ] . The case in which G is the symmetric group 5*3 on three letters gives rise to the Moufang loop ^ ^ ( 5 3 , 2 ) , of order 12, which is significant because it is the smallest Moufang loop which is not a group [ C P 7 1 ] . 5.2

M ( G , * , gfo) l o o p s . We generalize the class of loops defined in the

previous section. Suppose G is a nonabelian group, t h a t go ^ G is central and t h a t ^ »-> ^* is an involution

of G (that is, an antiautomorphism of G

of order 2) such t h a t ^Q = go and gg"^ is in the centre of G for every g £ G. (Note t h a t this implies t h a t g and g* commute.)

Let L — G (j Gu and

extend the binary operation from G to L by the rules

g{hu) {gu)h {gu){hu)

— {hg)u — {gh^)u =

goh^g

for g^h £ G. A straightforward case by case argument shows t h a t L is a Moufang loop. If it were a group, then, for any g^h E G, we would have (gh)u

== g{hu)

= {hg)u^ giving gh = hg.

Thus G would be abelian, a

contradiction. The requirement t h a t G be nonabelian assures us, therefore, t h a t the loop L is not associative. This loop is denoted M ( G , *,^o)-

5. EXAMPLES O F MOUFANG LOOPS

81

T h e notation M{G^ —^^9o) is useful when the involution is the inverse m a p g \-^ g~^.

T h u s , M ( G , —1,1) is the loop M ( G , 2 ) introduced in §5.1.

W i t h G — Qsi the quaternion group of order 8, and go the generator of the centre of Qg? it is not difficult to see t h a t M{Qs^ —l,^o) is the Cayley loop described in §4.1. (The element u is the element i of §4.1 and go is —1.) 5.3

L o o p s of u n i t s . Suppose R is an alternative ring with unity.

element x ^ R is called invertible

An

or a unit if there exist elements y and z

such t h a t xy = zx = 1. Let ZY(i?) denote the set of invertible elements of R, We wish to show t h a t U(R) is a Moufang loop. First, if xy = zx = 1^ then using Proposition L l . l l , the alternating n a t u r e of the associator and equation (1.7), we have [x, y, z] = {xy)[x, y, z] = y{x[x, y, z]) = y(x[z, x, y]) = y[zx, x,y] = 0 and so z = z{xy)

= {zx)y = y. Thus, if an element x of an alternative ring

has a right inverse and a left inverse, these elements are equal: we denote this element

x~^,

If X is invertible and a is arbitrary, appealing again to (1.7) and to Proposition L l . l l , we find (28)

[ x ~ \ x , a ] = ( x x ~ ^ ) [ x ~ \ x , a ] = x~^ {x[x~^ ^ x ^ a]) = x~^[x~^XjX^a]

— 0.

By the Generalized Theorem of Artin, we see t h a t x, x~^ and a generate an associative subalgebra of R, In particular, this shows t h a t if x is an invertible element in an alternative ring, then either of the equations ax = 6x, xa = xb implies t h a t a — b. We also see t h a t for any a, 6 G i?, the equations ax = b and xa = b have unique solutions. Thus U{R) is a loop provided it is closed under multiplication, and a Moufang loop since the Moufang identities hold in R, Continuing our assumption t h a t x is invertible, but with a and 6 arbitrary, and denoting by / the Kleinfeld function (see Definition 1.1.4), we see t h a t / ( a , 6, x^x~^) 0 =

— 0 for any a, 6 G R. Thus, if [x, a, 6] = 0, then

[x~^x^a^b] = x[x~^^ a, 6] + [x^a^b]x~^ + f(x~^^x^a^b)

= x[x~^^ ct,6];

II. LOOP THEORY AND MOUFANG LOOPS

82

hence

[x"^a,6] = 0.

(29)

Finally, suppose that x and y are invertible. Since [x^y^xy] — 0, we have [x~^^y^xy] — 0 and hence [x~^^y~^^xy] = 0. Thus {xy){y~^x~^) — 1. This shows that if x and y are in U{R)j then so is xy^ with inverse y~^x~^. Thus, as claimed, the set U{R) of all invertible elements of /? is a Moufang loop called the (Moufang) loop of units or the unit loop of R. Note that any subloop oiU{R) is also a Moufang loop. We conclude this section by establishing Moufang's Theorem for such subloops. 5.4 Theorem (Moufang's Theorem for subloops of tl(R)). Ifa^b^c are invertible elements of an alternative ring R which associate, then the subloop ofU{R) which they generate is a group. Let A - {a,a~^^, B = {b,b~^} and C = {c,c~^}. It is clear from (28) that [A,A,R] = [B,B,R] = [C,C,R] = 0 and, from (29), that [A^ B^C] = 0, so the result follows from Theorem 1.1.8. D PROOF.

5.5 The general linear loop. Let i2 be a commutative associative ring with unity. The general linear group, denoted GL(2,/2), is the group of 2 x 2 invertible matrices over R. The special linear group, SL(2, R) is the subgroup of GL(2, R) consisting of those matrices of determinant 1. Each of these groups has a Moufang loop analogue. Recall that Zorn's vector matrix algebra over R is the set 3(-K) of 2 x 2 matrices of the form

a,6 G i2, x,y G i2^, with entrywise addition and multiplication defined by

al

Xl

(12

X2

[yi

bi

_y2

&2

(See §1.3.5.)

Ci\CL2 + Xl • 72 (i2yi + &iy2 + Xl X X2

aiX2 + 62X1 - Vl X y2 6162 + yi • X2

5. EXAMPLES OF MOUFANG LOOPS

83

In §1.3.5, we noted that the function det: 3 ( ^ ) -^ R defined by det

a

X

y

b

ab — X' y

is multiplicative. Thus, just as with ordinary matrices, an element A

a

X

y b

of 3 ( ^ ) is invertible if and only if its determinant is an invertible element of i2, in which case A-^ =

(detA)-^

-X

a

The loop of invertible elements in 3 ( ^ ) is a Moufang loop analogous to the general linear group GL(2, R) of degree 2 over R. We call this loop the general linear loop and denote it GLL(2, i2); that is, (30)

GLL(2, R)= {Ae 3{R) \ det ^ is a unit in R}.

The special linear loop is the subloop SLL(2, R) of GLL(2, R) consisting of those matrices of determinant 1: (31)

SLL(2, R)= {Ae

GLL(2, R)

\detA=\}.

Chapter III

Nonassociative Loop Rings

1. L o o p r i n g s In this chapter, we first recount some of the early results about loop rings in general. We then turn our attention to loop rings which are alternative over Moufang loops of the form A/(6',*,^o) ^^^

to a description of the

corresponding groups. This work is based primarily upon [ B r u 4 4 , B r u 4 6 ] and [ G P 8 7 , G o o 9 1 ] Let L be a loop and let i? be a commutative and associative ring. 1.1 D e f i n i t i o n . The loop ring of L with coefficients in iZ, denoted RL^ is the free /2-module with basis L and multiplication given by extending the multiplication in L via the distributive laws. Except in Chapter VI, we always assume t h a t R has a unity and identify the element li of RL with the loop element i. Thus we think of L as a subset of RL and the identity element of L as the unity for this ring. An element of RL is a formal sum ^i^iC^e^-,

where the a£ are elements of /2, all but

finitely many of which are 0. The definitions of addition and multiplication take the form

ieL

eel

£eL

and

leL

leL

£eL 85

hk=i

86

III. NONASSOCIATIVE LOOP RINGS

T h e loop elements which actually appear (that is, with nonzero coefficients) in a formal sum have a name. 1,2 D e f i n i t i o n . If a = ^ a.il is an element of a loop ring, then the support of a , denoted s u p p ( a ) , is the set oi i E L for which a£ / 0: supp(E«^^) = { ^ e I | a ^ 7 ^ 0 } . As a free i2-module with basis L, the coefficients of each element of RL are unique: (1)

J2oi£^=Y^(id

implies

ae = Pi for all i e L.

Group rings—the case where L is a group—have been studied in depth for a long time. T h e subject is full of interest and intriguing problems. We cite, for instance, the two classical textbooks of D. S. Passman [Pas77] and S. K. Sehgal [Seh78] and a more recent book by S. K. Sehgal [Seh93]. By contrast, with only a few exceptions, the general nonassociative loop ring did not receive much attention until the early 1980s. R. H. Bruck did establish a semi-simplicity result about arbitrary loop rings in 1944 [Bru44] and two years later a theorem about the centre of a loop ring [Bru46] which we wish to describe here. Recall from Chapter II t h a t the inner mapping group of a loop L is the group Inn(Z/) generated by all maps of the form T{x) = R{x,y)=

R{x)L{x)-^ R{x)R{y)R{xy)-'

L{x,y)=L{x)L{y)L{yx)-' for x^y G L. We say t h a t elements a and b of L are conjugate

if b = a6 for

some 0 G Inn(L) and note t h a t conjugacy defines an equivalence relation on L. The equivalence classes associated with this relation are called classes.

Since, in a group, R{x^y)

and L{x^y)

conjugacy

each reduce to the identity

m a p while T{x) maps a to x~^ax, the definition of conjugacy in a loop is a generalization of its definition in group theory. In a loop ring, a (finite) class sum is the sum of all the elements in a finite conjugacy class of L.

With

this background, and recalling the definition of the centre of a nonassociative ring in (1.1.1), we can state Bruck's 1946 result [Bru46].

1. LOOP RINGS

87

1.3 T h e o r e m . The (finite) class sums of a loop ring RL form an for the centre of

R-basis

RL.

Since the elements of L form an i2-basis for i?L, an element a

PROOF.

is in the centre of RL if and only if ax ' y = a ' xy, for all x.,y £ L.

xa - y — x - ay.,

xy - a = x - ya

and

ax = xa

It is easy to see t h a t in the presence of the remaining

conditions, the second one here can be omitted. In other words, a is central if and only if (2)

ax ' y = a ' xy,

xy - a = x - ya

and

ax = xa

for all a;, y G i>. Extending linearly the inner maps of L to RL, the conditions in (2) are equivalent to the single condition aO = a for aU 0 G I n n ( L ) . It follows immediately t h a t any finite class sum is central. On the other hand, if ^ = ^£eL ^^^' then, for 6 G I n n ( L ) ,

£eL

Thus a G Z{RL)

£eL

iieL

implies t h a t a^Q-\ — ai for all d G Inn(L); t h a t is, the

coefficients of conjugate elements in the support of a are equal. Since the support of a is finite, the conjugacy class of each element in the support of a must be finite and so a is a linear combination of finite class sums. Thus the finite class sums span Z{RL)

as an 7Z-module. Since these sums have

disjoint supports and the elements of L are linearly independent, the finite class sums are also linearly independent and hence form a basis of as asserted.

Z{RL)., D

1.4 R e m a r k . The term "unique nonidentity c o m m u t a t o r " will occur frequently in this book. It refers to an element 5 /

1 in a group or loop L

with the property t h a t every commutator in L is 1 or 5. More generally, we shall call an element 5 7^ 1 of a loop L a "unique nonidentity commutatorassociator" if every commutator in L is 1 or 5 and every associator is 1 or s. It is convenient to record here two results which will be important later. 1.5 C o r o l l a r y . Let L be a (possibly associative) a unique nonidentity

commutator-associator

s.

Moufang

loop which has

Then s is central of order

88

2.

III. NONASSOCIATIVE LOOP RINGS

The centre of a loop ring RL is spanned

elements

those

of RL which are of the form i + si, £ E L. Since s"^ ^ 1 and since s~^ is also a c o m m u t a t o r , we have

PROOF.

s~^ = s.

by the centre of L and

Thus 5^ = 1. If is :^ si for some i e L^ then is = {si)s^

so

i — si and 5 = 1, a contradiction. Thus s commutes with every element of L, Also, if {ki^s ^ k(is)

for some k^i E L, then {ki)s

= (k • is)s.

Again,

after suitable cancellation, we reach the contradiction 5 = 1 . Thus s is also in the nucleus of L and hence in the centre of L, For i^x^y

G L^ we have iR{x^y)

either i - xy or (i - xy)s

= [ix • y){xy)~^

and, since ix - y \s

with s central, it follows t h a t iR{x,y)

G

{i,si}.

Similarly, iO G {^, si} for any inner m a p 0, so the class sums in RL are of the form i, for i G Z{L),

and i + si, for i ^ Z{L).

A nonassociative ring is power associative

D

if each of its elements gener-

ates an associative subring. In 1955, Lowell Paige showed t h a t most commutative power associative loop algebras are group algebras, thereby suggesting the probable complexity of a loop algebra which is not associative. Recall t h a t the characteristic

of a ring R with unity 1, denoted c h a r i J , is

the smallest positive integer n such t h a t n\ — 0, if such n exists; otherwise, by definition, the characteristic is 0. 1.6 T h e o r e m ( P a i g e [Pai55]). Let L be a loop and let R be a tative and associative relatively prime associative

ring of characteristic

zero or positive

characteristic

to 30. Then the loop ring RL is commutative

if and only if L is an abelian

commu-

and

power

group.

P R O O F . One direction is obvious. For the other, we present the essence of Paige's original proof, though Paige overlooked some problems with characteristic t h a t were later noticed by J. M. Osborn [Osb84]. Assume t h a t RL is commutative and power associative and linearize the identity x^x^ = (x^x)x.

First, replacing x by x + y, we obtain

4x'^{xy) + 2x^y^ + \{xy){xy)

+

\{xy)y^

- {xP'x)y 4- {x^y)x + {x^y)y + "l^xy • x)x + 1{xy • x)y + 2{xy ' y)x + 2{xy • y)y + {xy'^)x -f {xy'^)y +

{y'^y)x.

1. LOOP RINGS

89

Now replace y hy y + z. We obtain 4x^{yz) + 8{xy){xz) + 8{xy){yz) + A{xy)z'^ + A{xz)y'^ +

8{xz){yz)

= {x'^y)z + {x'^z)y + 2(xy • x)z + 2{xz • x)y + 2{xy • z)x +2(xz ' y)x + 2{xy • y)z + 2(a;2/ • z)y + 2(a;t/ • z)z +2{xz • y)y + 2{xz • y)z + 2(a:z • z)y + 2{x • yz)x +{xy'^)z + 2{x • yz)y + 2(x • i/.2r)2: + {xz'^)y +{y'^z)x + 2{yz • y)x + 2{yz • z)x + {z'^y)x. Replacing x by 2x leads to 16(x2/)(x2:) + 8x^(yz) = 2{x^y)z + 2{x^z)y + A{xy • a;)z + A{xy • >2r)x + A{xz • x)y + 4(a:z • y)x + 4(yz • x)x. Now let X, 7/ and z be elements of L. Thus each side of the above equation is a linear combination of loop elements. In characteristic 0 or positive characteristic prime to 2, we may divide by 2, obtaining (3) 8{xy){xz) + Ax'^iyz) = {x'^y)z + {x'^z)y + 2{xy • x)z + 2{xy • z)x + 2{xz ' x)y + 2{xz • y)x + 2{yz • x)x and then, with y = x, we obtain 12x^{xz) = 3(x^x)z + 3(a:'^z)a: + 6(xz • x)x^ which holds for all x^y^z G L. As before, our assumptions on the characteristic permit us to divide by 3, giving (4)

4x'^{xz) = {x'^x)z + {x'^z)x + 2{xz • x)x

for any x and z in L, Since Ax'^{xz) ^ 0, each side of (4) is a nonzero element of RL with just one element of L in its support. By the restrictions on the characteristic, {xz • x)x and {x'^z)x are in the support of the right side, so they must be equal. After cancelling x, we have (5)

xz ' x = x^z

for all a:,z G L, With x,y,z e L and using the identity (5) to conclude that x'^y = xy • x^ x^z — xz - x and x^{yz) = (x • yz)x = (yz • x)x, equation (3)

90

III. NONASSOCIATIVE LOOP RINGS

becomes (6) 8{xy){xz) +

2x\yz) = 3{xy ' x)z + 3(xz • x)y + 2(xy • z)x + 2{xz • y)x.

Now suppose K;, X and z are three elements of L and we want to prove (wx)z = w(xz). Since L is a loop, there exists y £ L such that w = xy. Rewriting (6), we have (7)

8w{xz) + 2x^{yz) = 3{wx)z + 3{xz • x)y + 2{wz)x + 2{xz • y)x.

Because of the restrictions on characteristic, either {wx)z = w(xz)^ or (wx)z — x^(yz)^ or chari? = 7 and {wx)z = (wz)x = (xz • y)x. Suppose {wx)z = x'^(yz). Then 8w{xz) = (wx)z + 3{xz ' x)y + 2{wz)x + 2{xz • y)x. Now w{xz) is the only element in the support of the left side, so it is the only element in the support of the right side. Since [wx)z is in the support of the right side, we must have (wx)z = w(xz). It follows that either {wx)z = w{xz) or char R = 7 and {wx)z = {wz)x = (xz • y)x^ with w = xy. Suppose the latter is the case, {wx)z — {wz)x = {xz ' y)x and apply our observation to the elements xz^ y, x. Either {xz • y)x = {xz){yx) or char/Z = 7 and {xz • y)x = {xz • x)y. The first possibility gives {wx)z = {xz ' y)x = {xz){yx) = w{xz). On the other hand, the second possibility shows that all loop elements on the right hand side of (7) are equal and the coefficient is not zero. Thus all terms in the equation are equal and again {wx)z = w{xz). D 2. A l t e r n a t i v e l o o p rings As a consequence of Paige's Theorem, one is not going to find loop rings of characteristic relatively prime to 30 (which are not associative) within the variety of Jordan rings (since these are power associative and commutative) nor do there exist Lie loop rings, since Lie rings, which satisfy the identity x'^ = 0, cannot contain invertible elements. There is a third important

2. ALTERNATIVE LOOP RINGS

91

variety of nonassociative rings, however, the variety of alternative rings, and here, as first shown in [ G o o 8 3 ] , the situation is more promising. 2.1 T h e o r e m . Let R be a commutative

and associative

let L = M(G^ *,5'o) be a Moufang loop constructed as described in ^11,5,2. Then the following 1. RL satisfies

the right alternative

2. RL satisfies the left alternative 3- 9 -\- 9* is i'^ Z{RG)j

ring with unity

from a nonabelian

statements

are

and group

equivalent.

identity. identity.

the centre of RG, for each g £ G.

4. If chd^r R ^ 2, then, for each g E G, h'^gh

G {g^g""} for all h G G;

z / c h a r i ? = 2, then, for each g £ G, either g = g* or else h~^gh G {9^9"^} for all h PROOF.

eG.

First note t h a t any element in RL can be written in the form

X + yu^ with x , y 6 RG.

With elements of RL represented in this way,

multiplication in RL is reminiscent of multiplication in a Cayley-Dickson algebra (c.f. §1.3) since, for x^y^z^w

G RG^

{x + yu){z + wu) = {xz + gow'^y) + {wx +

yz*)u.

Let a = X + yu and b = z -\- wu and compute {ab)b - {xz^ + gow*yz + goiw'^wx + u^*y^*)} + {wxz + goww'^y + wxz* +

y(z*)^}u

and ab^ = {xz^ + g^xw^w + goiz'^w^y + + {wzx + wz^'x + y{z*)^ +

zw*y)} goyw*w}u.

Remembering t h a t [s^t] denotes the (ring) commutator of 5 and ^, and t h a t [r, 5,^] denotes the (ring) associator of r, s and /, we obtain [a, 6, b] = go {[w'^y, z + z''] + [w^'w, x]} + {w[x, z + 2:*] + go{ww*y - yw^'w)} u = 9o {['^*y, 2: + z*] + [w^'w, x]) + {w[x, 2: + 2r*] + golww"^, y] + gQy[w, it;*]} u because ww*y — yw*w = [ww^^^y] + yww'^ — yw*w = [ww*^y] + y[w^w*]. Suppose RL satisfies the right alternative identity, so t h a t [a, 6,6] = 0 for

92

III. NONASSOCIATIVE LOOP RINGS

all a, 6 € RL. If, for some w G RG^ w'^w were not central in the group ring RG^ then, for some x G RG^ we would have [w'^w, x] ^ 0. Setting y = z = 0^ we would have [a, 6,6] = ^ o [ ^ * ^ ? ^ ] / 0, a contradiction. Such arguments show t h a t RL satisfies the right alternative identity if and only if z + z* e Z{RG), for all 2r, ti; G RG.

ww"" e Z{RG)

and ww"" = w^'w

Since centrality oi w -\- w* implies, in particular, t h a t w

and tt;* commute, RL satisfies the right alternative identity if and only if z + z* and zz^ G for all z G RG.

Z{RG)

Thus it is clear t h a t if RL satisfies the right alternative

identity, then ^^ + ^* is central for all ^ G G. Conversely, if ^ + ^* is central for all ^ G G, then z + z* is central in RG for all z G RG and hence zz* is central for all z G RG as well, since zz* is a linear combination of terms of the form ^^*, g E G (which is central by definition of M{G^ *?5'o)) ^ind of the form gh"" + hg'^ = gh*-\-{gh*Y ^ g^h ^ G. Thus statements (1) and (3) are equivalent. Again setting a = x + yu and b = z + wu with x, y^z^w E RG^ we have a^6 = {x'^z + goy'^yz + goi'w^yx +

w'^yx*)}

+ {wx^ + gowy*y + yxz* +

yx*z*}u

and a(a6) = {x^z + goxw'^y + goix^w'^y + + {wx'^ + yz^'x + yz*x* +

zy*y)} goyy''w}u

so t h a t [a, a, 6] = go {[y*y, z] + [w^'y, x + x*]} + {y[a: + x*, z*] + go^wy^y - yy^w)}

u

^ 50 {[2/*2/, z] + [tt;*t/, X + x*]}

from which it follows as before t h a t statements (2) and (3) are equivalent. For the equivalence of statements (3) and (4), we recaD t h a t the centre of a group ring RG is spanned by the finite class sums, a class sum being

2. ALTERNATIVE LOOP RINGS

93

the sum of all the elements in a conjugacy class of G (see Theorem 1.3). It is then immediate t h a t (4) implies (3). On the other hand, since h~^{9 + 9")h = h-^gh

+

h'^g^'h

for any g^h e G/ii g + g* is central for all g £ G^ then (8)

g + g^'^h-'gh

+ h-'g^'h

for all g and h in G. Suppose the characteristic of R is not 2. Then the support of the left hand side of (8) is {^r,^*}.

So, for any ^ G G, h~^gh

is g or g* for all

h. If the characteristic of i? is 2 and g y^ g*^ then there are two different elements in the support of each side of equation (8).

So again, for any

g ^ G^ h~^gh G {g^g*} for all h. These remarks establish statement (4) and complete the proof.

D

We investigate condition (4) of Theorem 2.1 in the special case t h a t * is the inverse map on a nonabelian group and the characteristic of the coefficient ring is different from 2. Suppose then t h a t G is a nonabelian group and, for any ^ G G, h~^gh G {g^g~^}

for all h £ G. This condition implies t h a t all subgroups of G are

normal; thus G is hamiltonian and hence the direct product Qg X E x A of the quaternion group Qg of order 8, an abelian group E of exponent 2 and an abelian group A all of whose elements have odd order.

(See

Norton's Theorem, Theorem II.4.8.) Let qi and q be elements of Qg such that q^^qqi

= q~^ ^ q and let a be any element of A.

Then, in C =

Qs X E X A, (^, l , a ) is not fixed under conjugation by ( ^ i , l , a ) , so this conjugate is (^, l , a ) ~ ^ .

Thus a is both fixed and mapped to a~^ under

conjugation by a, so a = a~^. Since the nonidentity elements of ^ have odd order, we conclude t h a t a = 1. Thus ^ = {1} and G = Qg X E. Conversely, \i G — QgX E for some abelian group E of exponent 2, then it is readily verified t h a t h~^gh G {g'>g~^} for all g and h. So, in view of Theorem 2 . 1 , we obtain 2.2 T h e o r e m . Let R be a ring of characteristic be a nonabelian

different

from 2, let G

group and let L = M{G^ —l^ffo) for some central go G G.

Then the loop ring RL is alternative

if and only if G = QgX E is the direct

94

III. NONASSOCIATIVE LOOP RINGS

product of the quaternion

group of order 8 and an abelian group of

exponent

2. In particular, this theorem shows t h a t the loop ring of the Cayley loop is alternative since, as noted in §11.5.2, the Cayley loop is M{Qs-, — l , ^ o ) , where go is the generator of Z{Qs)'

Similarly, the loop M((58?2) = M^Qs^, —1,1)

(see §11.5.1) also has an alternative loop ring.

3. T h e L C p r o p e r t y In this section, we characterize those nonabelian groups which have involutions satisfying statement (4) of Theorem 2.1 in the case of characteristic different from 2; t h a t is, those nonabelian groups G which have an involution such t h a t h~^gh G {^,^*} for all ^ , / i G G. In the process, we identify a certain property of a loop which plays a dominant role in this book. In any diassociative loop, elements x and y will commute if any one of x^ y^ xy is central. If these obvious situations are the only ones in which elements commute, it is reasonable to assert t h a t the loop has a certain lack of commutativity. 3.1 D e f i n i t i o n . A (possibly associative) Moufang loop L has the lack of commutativity

property

("LC", for short)

if it is not commutative and if,

for any pair of elements x , y G L^ it is the case t h a t xy = yx if and only if x 6 Z{L)

OT y e Z{L)

or xy G

Z[L).

It is obvious, but useful to note, t h a t squares are central in a Moufang loop with LC. Hence commutators are also central because {x^y)

=

x~'^{xy-^)'^y'^. T h e quaternion group Qs of order 8, the dihedral group D4 of order 8 and the Cayley loop are all Moufang loops with LC. Each of these loops also possesses a unique commutator-associator 5 / 1. T h a t these conditions are independent is illustrated by the following example. 3.2 E x a m p l e . Let G be the group presented as follows: G = (a:i,X2,X3 I Xi"^ = (xi^^Xj) Then G/Z{G)

= {(xi^Xj)^Xk)

= 1 for distinct

i^j^k).

has exponent 2 and two noncentral elements of G commute

if and only if they lie in the same coset of the centre of G. It follows t h a t G

3. THE LC PROPERTY

95

has LC. On the other hand, G has three nonidentity commutators (xi,X2), {xi,x^)

and (x2,X3).

Our reason for drawing attention to the LC property is the following theorem which, in the light of Theorem 2.1, shows how to construct an abundance of loops with alternative loop rings. 3 . 3 T h e o r e m . Let G he a nonahelian g ^-^ g* with the property if G has property

group.

Then G has an

that h~^gh £ j^^,^*} for all g^h £ G if and only

LC and a unique nonidentity

group, the involution

involution

commutator.

For such a

is given by g

(9)

if g is central

9* = { sg

where s is the nonidentity PROOF.

commutator

otherwise of G.

Assume first that G has a unique c o m m u t a t o r s ^

\.

By

Corollary 1.5, s is central and of order 2. Define g* by (9). Then (^*)* = g for any g £ G because 5^ = 1. Now assume that G also has property LC. We show t h a t ^ H^ ^* is an antiautomorphism (and hence an involution) by considering two cases. Case I: gh ^ hg. In this case, hg = sgh. /i*^* = {sh){sg)

Also, neither g nor h nor gh is central.

= hg = sgh =

So

{ghy.

Case 2: gh = hg. Here there are four possibilities corresponding to whether or not each of g^h is central.

If both these elements are central, then so is gh and

/i*^* — hg — gh — {ghy.

If neither g nor h is central, then gh must be

central because of LC and we have /i*^* = {sh){sg)

= hg = gh = {ghy.

If

g is central, but h is not, then gh is not central, so /i*^* = {^h)g = sgh = {ghy.

The last case—h central, g not central—is similar. Thus ^ i-> ^r* is

an involution and certainly h~^gh G {^,^*} for all g^h £ G. Conversely, assume t h a t G is a nonabelian group with an involution g ^-^ g"" which is such t h a t h~^gh G {5^,5'*} for all ^ , / i G G. li g = ^* for some g £ Gy then g is central. In particular, gg* is central for any g. From this, we also learn t h a t g* commutes with g and with g~^, for any g £ G.

96

III. NONASSOCIATIVE LOOP RINGS

Now suppose g^h ^ G and gh ^ hg. Then g~^hg = /i*, so hg = gh^. Also, h'^g'^h

= g (otherwise, h would commute with ^* and with ^^*, hence with

^ ) , so hg = g*h = gh*. Remembering t h a t h~^ commutes with /i*, we have We claim t h a t the element s — g~^g* is independent of noncentral g. For this, Rx g^h e G with gh 7^ hg and let x be any noncentral element of G, If xg 7^ ^ x , then, as we have shown, x~^a:* = g'^g'^-

Similarly, if

xh 7^ /ix, then x~^x* = h~^h* — g'^g"". In the case x^ = gx and x/i = /ix, then h~^(gx)h

= gx

while also, h~^{gx)h

or

h~^{gx)h

= {h~'^gh)(h~^xh)

= (gx)* — x*g^ = y*a:* = g*x.

Thus g = g* or x = x*,

either of which yields a contradiction because neither g nor x is central. This establishes the claim. Noting t h a t if gh / hg^ then g^^h^^gh

= g~^g*^ we see t h a t s = g'^g""

is a unique nonidentity commutator and also t h a t ^* = h~^gh = sg. If ^ is central and h is not, then gh is not central and we have (5/1)^* = h'^g* = (^/i)* =z s(gh) = shg^ so ^* = g. It follows t h a t g* is defined by (9). To see why G has LC, recall t h a t a unique nonidentity commutator in a group is central and of order 2. If gh = hg^ but g ^ 2(G) then ^* = sg and /i* = 5/1, so {ghy

= h^g"" = {sh){sg)

and /i ^ 2 ' ( G ) ,

= hg = gh. Thus gh

is central.

D

If G is not abelian, then the loop L = M{G^ *?5'o) is not associative (see §11.5.2) and thus RL is not associative. In view of Theorems 2.1 and 3.3, the following corollary is immediate. 3 . 4 C o r o l l a r y . Let R be a commutative, of characteristic

ring with unity

different from 2. Let L be the loop M ( G , *,^o) for

nonabelian group G, some involution go in G. Then RL is an alternative if G has LC and a unique involution

associative

g ^-^ g'^ of G, and some central ring which is not associative

nonidentity

commutator

s.

and some

element

if and only

In this case,

the

on G is given by (9).

With the next corollary, we commence gathering information loops whose loop rings are alternative.

about

3. THE LC P R O P E R T Y

3.5 C o r o l l a r y . Let R be a commutative and of characteristic alternative^

97

and associative

ring with

different from 2 and let L — M{G^^^go).

unity

If RL

is

then

(i) V = G' has order 2; (ii) M{L)

= Z{L)

(iii) L has property

=

Z{G); LC;

(iv) the centre of RL is spanned by Z{L)

and those elements

of RL of the

form £ + si, i e L. Recall t h a t L = G U Gu is the disjoint union of G and a coset

PROOF.

Gu and t h a t multiplication in L is given by g(hu)

=

{hg)u

{gu)h

=

{gh'')u

{gu){hu)

=

goh^'g

for g^h e G, By Corollary 3.5, G has a unique nonidentity commutator, 5, and G' = { 1 , 5 } . It is routine to verify t h a t (10)

{g,hu)~

g~^g*

and

{gu.hu)

=

{gh'")~\gh*y

for any g^h E G. By Corollary 3.4, k~^k* G {l,'^} for any fc G C , so the only commutators in L are 1 and s. Direct calculation shows t h a t any associator in // is a c o m m u t a t o r or the product of two commutators. Specifically, for ^, /i. A; G G, we have (g,h,ku) {gu,h,k)

= {g,h),

{g,ku,h)

= {k~^,h~^),

= {h~\g*~^),

{gu,h,ku)

{g,hu,ku)

= {k^g.h''),

=

{gu,ku,h)

{k^h.g), =

[k^g.h)

and {gu,hu,ku)

=

{g*-\h-^){{hg*)-\k*-').

Thus L', which is the subloop of L generated by the commutators and associators, is { l , ^ } = G'. This gives (i). Let g^h^k

G G. The equation (gu^h^k)

= (k~^^h~^)

not in the nucleus of L, while the equation (g^h^ku)

shows t h a t gu is

— (g^h)* shows t h a t

^ G G is in the nucleus of L if and only if g is in the centre of G. Af{L) = Z(G).

Thus

The equation (g^hu) = g'^g"" shows t h a t if ^ is in the centre

of G, then it commutes with all elements of L, so (ii) follows.

98

III. NONASSOCIATIVE LOOP RINGS

Since the central elements of G are precisely those k for which k'^ = k^ the relations in (10) also show t h a t an element of the form hu commutes with g e G if and only if g is central, and t h a t elements gu and hu commute if and only if gh* is central; t h a t is, if and only if the product (hu){gu)

is

central. Thus L has property LC, which is (iii). Part (iv) follows immediately from part (i) and Corollary 1.5.

D

Theorem 3.3 has focused attention on groups with LC and a unique nonidentity commutator. Such groups are easily characterized. 3.6 P r o p o s i t i o n . A group G with centre Z{G) has the lack of ity property and a unique nonidentity

commutator

commutativ-

if and only if G/Z{G)

=

C2 X C2. PROOF.

We have previously observed t h a t squares are central in a group

with LC, so GIZ[G)

is an elementary abelian 2-group. Suppose this group

contains the direct product {aZ{G))

X {hZ{G))

x

{cZ{G))

of three copies of the cyclic group C2 of order 2. Then no two of a^b^c can commute, because of LC. Since, in any group, there is the identity (xy, z) — (x, z){{x^ z), y)(y, z)^ and since commutators are central in a group with LC, we have (xy, z) — (x, z){y^ z) for all x^y^z

£ G. Now suppose t h a t

G also has a unique nonidentity commutator s.

By Corollary 1.5, 5 is a

central element of order 2, so [ab^c) = (a,c)(6,c) = s^ = I and, because of LC, either ab or c or abc is central, none of which is true. It follows t h a t G/Z{G)

contains the direct product of at most two copies of C2 and, since

G is not abelian, exactly two; t h a t is, G/Z(G)

= C2 X C2.

Conversely, if G is a group such t h a t G/Z{G)

= {aZ{G))

X

{bZ{G))

with each factor isomorphic to C2, then a and b cannot commute (because G is not abelian). Therefore two noncentral elements of G commute if and only if they both lie in the same one of the cosets aZ[G)^

bZ{G),

abZ(G).

From this, lack of commutativity in G is evident. Moreover, denoting by s the commutator of a and 6, it is easy to see from the relation (xy^z) (x^z)(y^z)^

=

which holds because commutators in G are central, t h a t s is the

only commutator in G other than 1.

D

3. THE LC PROPERTY

99

3.7 E x a m p l e s . Using the notation of M. Hall Jr. and J. K. Senior [ H S 6 4 ] , we enumerate the groups of order less than 32 which have a unique commut a t o r and property LC. • Qs = {a^b\ a"^ = l^b'^ = a^^ba = a~^b) • D4 = {a.bla"^ = b'^ = 1, ba =

a'^b)

• 16r2C2 = (a, 6 I a^ = 6^ = 1, (a, b) = a?) . 16r2&= {a,b,c\ • UV2d = {a,b\a^ • QsX

a^ = l , ( a , c ) = ( 6 , c ) = \,o? = 6^ ^ c^ = (a,6)) = b^ = 1, (a, b) = a^)

C2, Qs X C3, /^4 X C2 and Z:)4 x C3.

In §V.2, we shall see why this list is complete. If G is any of these groups and s is the unique nonidentity commutator of G, the m a p g ^-^ g^ given by (9) is an involution.

Moreover, if R

is any commutative and associative ring with unity and of characteristic not 2 (this restriction is, in fact, not necessary—see Theorem IV.3.1 and Corollary IV. 1.2), and if go is any element in the centre of G, the loop ring RL of the loop L = M ( G , * , ^ o ) is an alternative but not associative ring, by Corollary 3.4. 3.8 R e m a r k . Both Qg and D4 have the LC property and, in both these groups, the element a'^ is a unique nonidentity central commutator. Thus the loop Mi6((58,2) = M ( g 8 , - l , l ) and the Cayley loop, M{Q8^—li(i^)',

are loops with alternative loop rings.

=

MIG{QS)

They are not iso-

morphic because, for example, a^ is the only element of order 2 in the Cayley loop whereas there are nine elements of order 2 in loop

MIQ{QS',2)

MIG{QS^2).

The

is, however, isomorphic to A/(Z?4, —1, a^). Conditions un-

der which M ( G , * , ^ o ) is independent of ^0 are known [ C G 8 5 , Theorem 3]. In [ C G 8 5 ] , it is also shown how to determine whether or not, given two nonabelian groups Gi and G2 (each with LC, a unique nonidentity commutator and containing central elements ^1,^2 respectively), RA loops of the form M ( G ' i , * , ^ i ) and M(G2?*^^2) are isomorphic. 3.9 E x a m p l e s . Let N = 2'^ with n > 1 and let G be the group generated by two elements a and b with a^^ = b"^ = I, ba = a^'^^b. (a^) is cyclic and G/Z{G)

Then Z(G)

=

= C2 X C2. Thus G has property LC and a

unique nonidentity commutator so, by Corollary 3.4, any loop L of the form M{G^ *?^o) = GuGu

with * given by (9) has an alternative loop ring.

100

III. NONASSOCIATIVE LOOP RINGS

Note t h a t |L| = 21^1 = 8iV = 2^+^. Now it is not difficult to show t h a t the group G described here is generated by any pair of elements which do not commute. This observation allows us to establish a significant property of L: all proper subloops of L are associative. The argument, briefly, is as follows: if x i , X2 and x^ are any elements of L and they all belong to G, then obviously the subloop (a^i, 0:2, 2:3) generated by the Xi is associative. If xi and X2 are in G, if X3 is in Gu and if x\ and X2 do not c o m m u t e , then Xi and X2 generate G, so quite clearly {x\^X2^x^)

= L. The possibility t h a t

just xi is in G while both X2 and x^ lie in Gu reduces immediately to the case of two generators in G since ( x i , 0:2, a^a) = (a^i, 0:23:3, 0:3); so does the final case, all Xi in Gu^ since (a:i,X2,X3) =

{x\X2->X2Xz^x^).

The point is t h a t we now have, for each n > 4, at least one loop of order 2'^ which has an alternative loop ring t h a t is not associative, but which contains properly no other such loop. This gives some indication of the richness of the class of loops around which much of this book is based.

4. T h e n u c l e u s a n d c e n t r e Let /Z be a commutative and associative ring with unity and of characteristic different from 2. Let G be a nonabelian group with the LC property and a unique nonidentity (necessarily central) commutator, s. Let ^ H-^ ^* be the involution on G defined by

(U)

/ J ' '"'^^"^^ \sg

(See Theorem 3.3.) Let g^ G Z{G)

\fgtZ{G). and let L be the loop M ( G , * , ^ o ) de-

scribed in §n.5.2. The involution on G extends linearly to a (ring) involution of the group ring RG which we also denote *: (12)

( E , e G « . 5 r = E,6G«p5*-

By Corollary 1.5, the centre of RG is spanned by the centre of G and elements of the form ^ + 5^, so it follows t h a t both g -\- g* and g + sg are in Z{RG)

for any g eG.

(13) for any x G RG.

Thus

x + x"" e Z{RG)

and (1 + s)x 6

Z{RG)

4. T H E NUCLEUS AND CENTRE

101

Now, if X G RG^ \{ sx = X and if g is any element in the support of x, then the coefficients of g and sg must be the same, so x is central: (14)

x G RG and sx — x

implies

x G

Z{RG).

Suppose x* = X. Writing x = XQ + x\^ where a:o has support in

Z{G)

and supp(a:i) fl 2 ( G ) = 0, we have x"" — XQ + sxi^ so xi = sxi^ Xi is central and hence x also is central. On the other hand, if x is central and written as a linear combination of class sums of the form g^ g ^ Z{G),

and g + sg^

^ G G, it is clear t h a t x* = x. Thus (15)

X G 2:(i2G) if and only if x* = x.

As noted in the proof of Theorem 2.1, any element of RL can be written in the form x + yu^ where x , ^ G RG^ and, representing elements this way, multiplication in RL takes the form (x + yu){z + wu) - {xz + gow*y) + (wx + where x^y^z^w

yz*)u,

G RG, In what follows, it is convenient to refer to x and y

as the RG- and i^Gi^-components of the loop ring element x + yu. 4.1 P r o p o s i t i o n . Let L be a loop of the form M{G^ '^'>9o) o,^^d suppose is alternative.

Let J\f{RL)

denote the nucleus and Z{RL)

RL

the centre of RL.

Then M{RL)

= Z{RL)

^ {x + yu\x,y

^ Z{RG),

= {x + yt/ I X G Z{RG), PROOF.

sy = y] sy =

y).

Observe t h a t

{x + yi/ I X, y G Z{RG),

sy = y] = {x-^ yu\ x e Z{RG),

sy = y]

because of (14). We first show that this set is the nucleus of RL. Let a — X -\- yu^ f] — z -\- wu and j — p -\- qu he elements of RL^ where x.,y^z^w.,p^q

are in RG. Direct calculation shows t h a t the associator

[a,/?,7] can be written in terms of commutators in RG: (16)

[a,/3,7] = ^ o { K 2 / , p ] + [q'^.x]

+ [^*y,^*]}

+ {(l[x, z] + ti;[x, ;?*] + y[2:*, p*] + g^lq, w'']y + ^ o k * g , y]]u.

102

III. NONASSOCIATIVE LOOP RINGS

Suppose t h a t sy = y (and hence t h a t y is central). For any t = Y^g^o^gS RG^ we have t-f"

= {I-

^

s) YlgdziG) ^gd ^^^ ^^^^ 2/(^ - T ) = 0. Using this

and the fact t h a t i/* = y, by (15), we have

{tyy = 2/*r = ye = y{e -t + t) = yt = ty; thus ty is central for all ^ G RG. Moreover, since gh — hg = 0 OT gh — hg = (1 — s)gh for any g^h ^ G^ sy = y implies also t h a t (gh — hg)y = 0 for any g,h and hence [t^v]y = 0 for any t^v £ RG.

If then b o t h x and y are

central in RG and 5^/ = t/, it follows easily from (16) t h a t [a,/3,7] = 0; t h a t is, a e

Af{RL).

Conversely, suppose t h a t a G M{RL)

so t h a t

[Q^,/3,7]

7 in iJL. For any t and i; in i?G, the centrality oit-\-t*

= 0 for aU /? and

and t; + t;* implies

t h a t [ r , t ; ] = [^,^*] = -[^,'^] and [r,t;*] = [t^v]. Referring to (16), it follows that [a, /?, 7] = goiiw^'y, p] + [gr*it;, x] + [y*^, z]}

+ {^[a^, ^] - '^[a^, j^] + y[z,v] - 9o[q, ^]y + ^o[^*g, y]}u = o. Suppose X ^ Z{RG).

Then choosing it; G i2G such t h a t [w^x] ^ 0 and

setting q — I and p = 2^ = 0, the iZG-component of [a,/3,7] is ^ o [ ^ , ^ ] 7^ 0. Therefore x must be central and

[«,/?, 7]

If 2/ ^ Z{RG)^

choose p so t h a t [y^p] 7^ 0 and set it; = 1 and z = 0. Then

the iJG-component of [a,/J, 7] is ^'oiy^p] 7^ 0. Thus both x and y must be central elements of RG and [a,/3,7] = ^ o { k * y , p ] + [y*^,^]} + {2/[^,p] - 9o[q,w]y}u Taking ^ = 0, it follows t h a t y[z^p] = 0 for any z^p £ RG,

= 0. With z and

p noncommuting elements of G, we have zp — pz = (1 — 5)2:^ as before. Then y{l — s)zp — 0 implies y(\ — s) — ^ because zp is invertible. Thus the nucleus of RL is the required set. Since 2(6*) = Z{L)^ from Corollary 3.5 we see t h a t the centre of RL is spanned by

Z{G)u{i

+ si\ie

L},

5. THE NORM AND T R A C E

Thus Z{RL) = Af{RL).

103

D

4.2 Corollary. Let L be a loop of the form M(G,*,^o) ^^^ suppose RL is alternative. Then Z{RL) = {re RL\re

= ir for all i G L],

Suppose r G RL commutes with all ^ G Z/. Write r — x -\- yu^ x^y e RG. Let g £ G, Since rg = xg + {yg*)u and gr = gx + {yg)u^ it is clear that xg = gx and yg = yg* for all ^ G C, so x is in Z(RG). Let g be a noncentral element of G. Then yg = yg* — syg and hence y = sy. By Proposition 4.1, r = x + yu e Z{RL). D PROOF.

For r — X + yu e RL., define (17)

r* = X* + syu.

Then * is an involution on RL which extends the involutions on G and on RG and which induces an involution on L defined mutatis mutandis like that on G: (18)

r = < l^si

^ ^ i{i^Z{L).

Proposition 4.1 and observation (15) give immediately the following result. 4.3 Corollary, r G Z{RL) if and only if r"^ = r. In particular, for any r G RL, r + r"^ and rr* (~ r^'r) are central elements of RL. 5. T h e n o r m and t r a c e Remember that an algebra A over a commutative and associative ring $ with unity is quadratic if every x E A satisfies an equation of the form x^ — t{x)x + n{x)l = 0 with t{x),n{x) G $. (See §1.4.) Let L be a loop of the form M{G^ *55'o) for which RL is alternative and, for r G RL, define respectively the trace t{r) and norm n{r) of r by (19)

t(r) = r + r*

and

n{r) = rr'^.

104

III. NONASSOCIATIVE LOOP RINGS

The trace and norm are functions RL -^ Z(RL)

whose basic properties are

summarized in the next proposition. 5.1 P r o p o s i t i o n .

(a) Every element

of RL satisfies

t{r)r + n ( r ) = 0. Thus RL is a quadratic

the equation r^ —

algebra

over its

centre

Z(RL), (b) The trace is a linear function

and the norm a quadratic form on RL,

(c) The trace is commutative in the sense that t(qr) = t{rq) for all q,r ^ RL, p,q,r (d) IfR

and associative in the sense that t(pq • r ) = t{p • qr) for all e RL, has no additive 2'torsion,

then the norm is

nondegenerate.

P R O O F . T h e proofs of (a) and (b) are direct verifications. For part (c), we use the facts t h a t L has the LC property and a unique nonidentity commutator, 5, which is also a unique nonidentity associator (see Corollary 3.5). Commutativity of the trace is equivalent to qr + r*^* = rq + q*r*;

t h a t is,

[q^ r] = [^*, r*]

for all q^r ^ RL, Now [^*,r*] = q*r'' — r'^q* = (rq — qr)* = [r, ^]*, so we must show t h a t [r, ^]* = —[r, ^] for any q^r £ RL.

Since the c o m m u t a t o r

is bilinear, it is sufficient to prove t h a t [r, g]* = —[^, ^] for r^q £ L. This is obvious if either r or g' is central. If neither r nor q is central, then r* = sr and (7* = sq and so [r,qY = [^*,r*] = q*r%- r^'q* = {sq){sr)

- {sr){sq)

=

-[r,q]

because s is central and s^ = I. This establishes commutativity of the trace. Associativity is equivalent to pq ' r + r'^ ' q'^p'^ = p - qr + r*q'^ • p*;

t h a t is,

[p, ^, r]* = — [j9, q^ r]

for all p^q^r e RL and, as above, it is sufficient to establish this property in the case t h a t p^q^r e L. \i p or q or r is central, or if p , q and r associate, then clearly [p^q^r] = 0. So assume these elements do not associate and that none is central. Then p* = sp,, q* = sq^ r* — sr and

b, q-, rY = {pq • ^)* - {v • q^T = ^* • ^ V - ^*^* • p* = (^r) • {sq){sp) - {sr){sq) • [sp) = s{r'qp-rq'p)

=

-s[r,q,p].

5. T H E NORM AND T R A C E

105

Since the associator is an alternating function in RL^ we get [p, g,r]* = s[p^q^r\.

Now |j9, g, r] = pq-r — p-qr = (p'qr)s

— p'qr

= (s — l)(p'qr)^

since

5 is a unique nonidentity associator. Therefore s[p^ q^ r] = (s'^ — s)(p • qr) = (1 — s){p • qr) = —[p,^,^] as desired. This establishes associativity of the trace. For (d), we must prove nondegeneracy of the bilinear function / associated with q. Note t h a t f(r^p)

= n{r + p) — n(r) — n{p) = rp* + pr* =

Suppose r G RL and f(r^p)

— 0 for all p G RL.

t{rp^).

W i t h p — 1, we obtain

r* = —r. Writing

leZ(L)

^tz(L)

£ez(L)

^iZ{L)

we have

and, since R has no 2-torsion, a^ = 0 for all ^ G >2(//). Now / ( r , p) = 0 for all p implies / ( r , p\p) — 0 for all p, pi G -RL. Since / ( p , r ) = /(pr*) and the trace is associative, / ( r , p i p * ) = ^{Tpp\) — f{rp^pi)

= 0 for all p , p i G RL.

By

what we have already learned, this means t h a t (rp)* = — (^p) for all p G ^ L and hence, for any p G KL, rp has no central element in its support. Note, however, t h a t if i is in the support of r, then ri has P in its support and i'^ is central because L has LC, by Corollary 3.5. It follows t h a t r = 0.

D

Chapter IV

R A Loops

In this chapter, which is based primarily on [ G o o 8 3 , C G 8 6 ] , we investigate and describe those loops whose loop rings, in characteristic different from 2, are alternative, but not associative. It is convenient to have a name for such loops. 0.1 D e f i n i t i o n . An RA {ring alternative)

loop is a loop whose loop ring

RL over some commutative, associative ring R with unity and of characteristic different from 2 is alternative, but not associative. Since an alternative ring satisfies the Moufang identities and since a loop ring RL contains L, it is clear t h a t an RA loop is a Moufang loop.

1. B a s i c p r o p e r t i e s of R A l o o p s We begin with one of several possible characterizations of RA loops. 1.1 T h e o r e m . A loop L is an RA loop if and only if L is not and has the following (i) if three elements

associative

properties: of L associate in some order, then they associate

in

all orders; (ii) if g^h^k £ L do not associate, PROOF.

then g - kh — gh - k — h - gk.

First, suppose t h a t L is an RA loop and R is some ring of

characteristic different from 2. Since L C RL and RL is an alternative ring, statement (i) holds by the Generalized Theorem of Artin, Corollary 1.1.9. To obtain (ii), let ^f, /i and k be three elements of L which do not associate and put X =^ h-\- k^ y — g m the right alternative identity, yx - x — yx^. We 107

108

IV. RA LOOPS

obtain gh ' h + gh ' k + gk ' h + gk ' k = gh? + g - hk + g - kh + gk^ and, since gh - h — gK^ and gk - k = gk'^^ gh • k + gk ' h = g • hk + g ' kh. Because chari? / 2, ^/i • fc is in the support of the left side and thus in the support of the right. So, \i gh - k ^ g • hk^ then gh- k = g - kh. Similarly, by considering the left alternative identity, we obtain gh - k = h - gk. Conversely, suppose t h a t L is a loop which satisfies statements (i) and (ii) of the theorem. Let iZ be a commutative and associative ring with unity. For X = "^ agg and t/ = ^ I3gg in RL., yx-x

— yx^ is a linear combination of

terms of the form gh-k — g - hk. li h = k^ then gh-h = g -hhhy

associativity

or (ii), so yx ' X — yx^ reduces to a sum of terms of the form [gh ' k — g • hk) + (gk - h — g - kh) with h ^ k and this is 0 by (i) and (ii). This shows t h a t the right alternative identity holds in RL. The left alternative identity follows in a similar way.

n Since statements (i) and (ii) of Theorem 1.1 involve only the loop L and not the ring R and because the second half of the proof of Theorem 1.1 required no assumptions about the characteristic of the coefficient ring, we note the following. 1.2 C o r o l l a r y . If L is an RA loop, then RL is an alternative coefficient

ring for any

ring R.

Since an alternative ring is power associative, Paige's Theorem (Theorem III. 1.6) shows t h a t most commutative alternative rings are group rings. This also follows from Theorem 1.1. 1.3 C o r o l l a r y . If L is an RA loop and if h and k are elements commute, alternative

then gh - k — g - hk for all g ^ L. A commutative loop ring in characteristic

of L which loop with an

different from 2 is a group.

P R O O F . By Theorem 1.1, either gh - k = g - hk or gh - k = g - kh = g - hk. Thus gh ' k — g ' hk m every case. The second statement is clear.

D

1. BASIC PROPERTIES OF RA LOOPS

109

1.4 C o r o l l a r y . The direct product L x K of loops is an RA loop if and only if precisely one of L and K is an RA loop while the other is an abelian group. PROOF.

It is straightforward to check t h a t the direct product of an

abelian group and a loop satisfying statements (i) and (ii) of Theorem 1.1 is a loop which also satisfies (i) and (ii). On the other hand, if L x K is an RA loop satisfying (i) and (ii), then both L and K satisfy (i) and (ii) and at least one of these loops is not associative. Without loss of generality, assume that L is not associative (and hence R A ) . We complete the proof by showing t h a t K is an abelian group. It is sufficient to show t h a t K is commutative since associativity will then follow by Corollary 1.3. To show t h a t K is commutative, let a, 6 G K and let g, h and k be three elements of L which do not associate. Since (^, 1), (/i,a) and (A;, 6) cannot associate in L X Ii,, we must have

[ig,l)ih,a)]ik,b)

=

ig,l)[ik,b){h,a)],

by Theorem 1.1. Thus ab — ha. F. Fenyves has called a loop extra

D if it satisfies the identity {xy • z)x —

x{y • zx) and, furthermore, shown t h a t an extra loop is Moufang [ F e n 6 8 ] . 0 . Chein and D. A. Robinson later characterized extra loops as precisely those Moufang loops in which all squares are in the nucleus [ C R 7 2 ] . 1.5 C o r o l l a r y . An RA loop is an extra loop. PROOF.

Let L be an RA loop. If x, y and z are three elements of L

which associate, then, by Moufang's Theorem, these elements generate a group, so (xy • z)x = x(y • zx) is clear. If x, y and z do not associate, then neither do xy^ z^ x (again by Moufang's Theorem), so (xy • z)x — by Theorem 1.1 and, since x, y, xz do not associate, {xy){xz)

{xy)(xz)

= x{xz • y).

Repeated use of Theorem 1.1 gives xz ' y = z ' xy — zy ' x — y ' zx and so, again, [xy • z)x = x(y • zx).

D

We wish next to describe some fundamental characteristics of an RA loop, but first several points should be noted.

no

IV. RA LOOPS

1.6 R e m a r k . A Moufang loop of exponent 2 is an abelian group. To see why, let L be such a loop and note first t h a t because L is diassociative, xyxy

= 1 implies xy = yx] thus L is commutative. Then {x ' yz)x = (yz • x)x = yz

by commutativity, diassociativity and the fact t h a t x^ — \. Also {xy • z)x = {yx - z)x = y(x • zx) = y{zx ' x) = yz Thus (x ' yz)x

by the right Moufang identity using diassociativity and x^ = 1.

= (xy • z)x and, after cancelling x, we get x - yz — xy - z.

Thus L is associative. 1.7 R e m a r k . If ^ is a proper subloop of a Moufang loop L, then L is generated by the complement of H. This follows by fixing a £ L\

H and

noting t h a t if h ^ H^ then h is the product of ha and a~^, both of which are in the complement of H. 1.8 T h e o r e m . Let L be an RA loop.

Then

(i) g'^ e J^{L) for each g G L; (ii) Af{L) = Z{L)

= {a ^ L \ ax — xa for all x G L];

(iii) (^, h) — \ for g^h ^ L if and only if (^, h^k) = 1 for all k G L; (iv) if g^h^k ^ L and (^, h^k) ^ 1, then (^, /i, A:) = (^, /i) = (^, k) = (/i, /c) Z5 a central element

of order 2;

(v) L', the commutator-associator

subloop of L, is a central group of

order 2. PROOF.

Throughout this proof, we assume t h a t L is contained in an

alternative loop ring RL and t h a t char R ^ 2. As previously noted, statement (i) holds because L is an extra loop, but we present here a proof which does not make use of this fact. Linearizing the middle Moufang identity, xy ' zx — {x ' yz)x^ we see that the alternative ring RL satisfies xy ' zw + wy - zx — [x - yz)w + [w • yz)x for all x^y^z.w

G RL.

Let ^, h and k be three elements of L and set

x = y = g^z = h and w = k. We get g'^ ' hk + kg ' hg = {g ' gh)k + {k • gh)g = g'^h'k

+ {k'

gh)g,

1. BASIC PROPERTIES O F RA LOOPS

111

using the left alternative identity to rewrite g - gh d.s g^h. Suppose g^j h and k do not associate. Then g^h - k ^ g^ - hk^ so g^h - k = kg - hg. Also, neither triple g^ h^ k nor triple g^ h^ kg can associate, by Moufang's Theorem. Repeated use of Theorem 1.1 now gives kh-g'^

- (kh • g)g = {h • kg)g = kg - hg = g^h • k = g^ ' kh = kg^ - h — k - hg^;

t h a t is. A;, h and g^ associate. This contradiction shows t h a t g^ is indeed in the nucleus of RL^ for any g ^ L. (ii) Denote by C{L) the set {a £ L \ ax = xa for all x ^ L} and remember t h a t Z{L)

= C{L) nAf{L),

If a G C(L), then (a, x,y)=

by Corollary 1.3, so a G AfiL). remains to show t h a t M{L)

1 for all x,y e L

Thus C(L) C A/'(L), so Z{L)

C Z{L).

= C{L).

It

Since the nucleus of a Moufang loop

is normal (Corollary II.3.4), we can form the quotient loop L/Af{L)^

which

is a Moufang loop of exponent 2, by (i), and hence an abelian group by Remark 1.6. Clearly, M{L)

is contained in the nucleus of RL.

Let n G M{L)

and let ^, h and k be any three elements of L which do not associate. Then gh ' k — n\{g • hk) for some n^ G J^{L) gn — (ng)n2 for some 712 G M{L)

(since L/Af(L)

(since LjM^V)

is a group) and

is commutative) and so

the (ring) associator \g^h^k\ and (ring) commutator [^,n] are \g,h,k\

^ (ni - \){g'hk)

and

[g,n\ = [ng){n2 - 1).

By Proposition 1.1.7, [g-inWg^h^k] — 0, so {{ng){n2-\)}{(n,-l){g-hk))

= d.

Since 712 — 1 is in the nucleus of RL^ we have {ng){{n2-\){{n,-\){g-hk))]

=Q

and, since ng is invertible, (712 — l ) { ( ^ i — ^)[9' hk)} = 0. Since ni — 1 is in the nucleus of iZL, {n2 - l ) { ( n i - 1)(^ . hk)} = {(712 - l ) ( n i - 1)}(^ • hk), so (712 —

1){TII

— 1) = 0 because g - hk \s invertible. So we have 7ii + 712 =

1 + ^2^1 3.nd, since 7^l /

1 and we are in characteristic different from 2,

necessarily 712 = 1, so ^7z = TI^. This argument shows t h a t n commutes with all elements not in the nucleus. Since L is not associative, N{L)

is a

112

IV. RA LOOPS

proper subloop, so its complement generates L, by Remark 1.7. It follows t h a t n commutes with every element of L and is therefore central. (iii) By Corollary 1.3, (g^h)

= 1 implies (g^h^k)

Conversely, let g^h e L and suppose t h a t (g^h^k)

= 1 for all fc G L.

= 1 for aU /: G L.

have to prove t h a t gh = hg. By (ii), we can assume t h a t h ^ J^iL).

Clearly,

[g^h^x] = 0 for all x G RL so, by Proposition 1.1.7, we have [g^gh] G Also gh ' g = n{g • gh) for some n G J^{L) so gh ' g = ng^h.

(since L/Af{L)

Hence [g^gh] = (1 — n){g^h)

We

Af(RL).

is c o m m u t a t i v e ) ,

G M{RL).

Hence, for all

a,6 G i/, {(1 - n){g^h)]a Now g'^ G M{L) n)h}(ab)

= Z{L)

• 6 = {(1 - n ) ( / / . ) } ( a 6 ) .

and g^ is invertible, so {(1 - n)h}a

for all a^b e L. Since h ^ Af(L) and because L/Af{L)

there exist a and b such t h a t ha-b — ni(h'ab) Therefore (1 — n)ni{h

- b = {{I

-

is associative,

for some ni G Af(L)^ rii ^ 1.

• a6) =:= (1 — n){h • a6). So (1 — n)ni

= 1 — n, giving

m + n = 1 + n n i . Since ni / 1, necessarily n — I. Therefore gh - g = g^h and, cancelling g, we get /i^ = gh, as required. (iv) Let n = (g^h^k),

note t h a t n G N{L)

= Z{L)

(because L/Af{L)

is

a group) and suppose n 7^ 1. By Theorem 1.1, kg ' h = g • kh = gh • k = n{g • hk) = n{gk - h) = (n - gk)h, so kg = ngk. Therefore n = (k^g) — {g,k)~^.

Since g^ G M{L)

= Z{L)

and

n is central, we have g^k — kg

— kg - g — ngk - g — ng - kg — ng - ngk = {ng)^k = n^g^k,

so n'^ = 1. Thus {g,k)~^

= (5^,^)5 so (g^h^k)

— n — (g^k).

Interchanging

the roles of h and A:, we have also (g, k, h) = (^, h). Since g, h and k do not associate. Theorem 1.1 gives gk-h

— g- hk and g-kh

— gh-k.

Since gh-k

=

(g ' hk){g, h, k) (by definition of (^, /i. A:)), we have g - kh — [gk • /i)(p, /i, A:), so gk ' h = (g ' kh){g,h,k)~^

— {g • kh)(g,h,k)

order 2. This shows t h a t {g,k,h)

= (g^h^k)

because (g^h^k) = {g^h) - {g,k).

= n has Similarly,

hg ' k = g ' hk and h - gk — gh - k implies (^, /i, fc) = (/i, ^, fc) = (/i. A;). (v) First note t h a t parts (iii) and (iv) together show t h a t every commutator in L is an associator, t h a t every associator in // is a c o m m u t a t o r , and that the commutator-associator subloop V is a central group of exponent

1. BASIC PROPERTIES OF RA LOOPS

113

2. We show IL'I = 2 by establishing that any two nonidentity associators are equal. We already know this to be the case if the associators in question have two elements in common; for example, if (^,/i, a) 7^ 1 7^ (^,/i,6), then (g^h^a) = (g^h) = (g^h^b) by (iv). Next, suppose that two nonidentity associators have just one element in common. Specifically, suppose that (g^h^k) 7^ 1 7^ (g^a^b). If (a^g^h) ^ 1, then, since this associator has two elements in common with both (g^a^b) and (g^h^k)^ we have (g,a,b) = (a,g,h) = {g,h,k). Similarly, if {b,g,h) 7^ 1, we have {g,a,b) = {b,g,h) =: {g,h,k). If {ab,g,h) 7^ 1, then (ab.g) = {ab,g,h) = {g,h,k), while ab-g = a- gb = ga-b = {g ' ab){g, a, b) implies that {ab^g) = (g^a^b). Thus (g^h^k) = (g^a^b). All this shows that (^, /i, k) — (^, a, 6) if any of the associators (a, g^ /i), (6, g^ h), (ab^ g^ h) is not 1. We now consider the case that each of these associators is 1. Denoting the Kleinfeld function by / , as usual, we have / ( a , b,g,h)=

[ab, g,h]-

6[a, g, h] - [6, g, h]a.

Since [a^g.h] = [b^g^h] — \ab^g^h\ = 0, we have f{a^b^g^h) = 0 and hence also (1) 0 = f{a,h,g,b)

= [ah,g,b]-

h[a,g,b]-

[h,g,b]a = [(^h,g,b]-

h[a,g,b],

because / is alternating. Since (g^a^b) 7^ l-^ g-, a and b do not associate, so [g^a^b] 7^ 0 and therefore [a,5,6] 7^ 0. Thus (1) shows that [a/i,p,6] 7^ 0; that is, (ah^g^b) 7^ 1. Similarly, we may assume that f(h,k^g^a) = 0 and conclude that (ah^g^k) 7^ 1. Thus {g,h,k) = {ah,g,k)

= {ah,g,b) = {g,a,b).

We conclude that any two associators which are not the identity and which have at least one element in common are equal. Finally, suppose that (g^h^k) 7^ 1 7^ (a^b^c) and that the sets {g^h^k} and {a^b^c} are disjoint. Considering f{g, h, k, a) = [gh,fc,a] - h[g, k, a] - [h, k, a]g,

114

IV. RA LOOPS

we see that each associator on the right has an element in common with both (g^h^k) and (a^b^c)^ so, if any of these associators is not zero, we obtain (g^h^k) = (a^b^c) as above. Thus we may assume that each of the associators [^/i, fc,a], [g^k^a] and [h^k^a] is zero and hence that f{g^h^k^a) = 0 too. Since / is an alternating function, 0=

f(g,a,h,k) = [ga, h, k] - a[g, h, k] - [a, h, k]g = [ga, h, k] - a[g, /i, k].

Since [g^h^k] ^ 0, we have [ga^h^k] ^ 0, so (ga^h^k) ^ 1. Similarly we can show that (ya, b^c) ^ 1 and so (a, 6, c) = (ga, 6, c) = (ga, h, k) = (g, h, k), giving the desired result.

D

In §11.1.3, we observed that a finite loop L is the disjoint union of right cosets of a subloop H if and only if H{hx) = Hx for all x G /> and all /i G H. Whether or not Lagrange's Theorem is true for Moufang loops is an open question, but the issue is clear for RA loops. 1.9 Corollary. If H is a subloop of an RA loop L, then L is the disjoint union of cosets of H and hence, if L is finite, the order of L is divisible by the order of H; thus, RA loops satisfy Lagrange's Theorem. Let h and hi be elements of H and let x ^ L. Then either hi{hx) = {hih)x^ or else h\{hx) = {hh\)x^ by Theorem 1.1. In either case, hi(hx) C Hx. Thus H(hx) C Hx. Conversely, let h^hi ^ H and X ^ L. If (/ii/i~\/i,x) = 1, then, by diassociativity, hix — {h\h~^ • h)x = {hih~'^){hx) e H{hx). On the other hand, suppose (/ii/i~\/i, x) 7^ 1. Then {hih~^){hx) = {h ' hih~^)x by Theorem 1.1. Also (h^hi) ^ 1; otherwise, (/i,/ii,x) = 1, by Theorem 1.8, so, by Moufang's Theorem, h, hi, x would generate a group, a contradiction. By Theorem 1.1, L' = {l,'^} for some central element s. Since (/i,/ii) 7^ 1, we have (/i,/ii) = s e H. Using the centrality of s, we obtain hix = s(hhih~^)x = s{hih~^){hx) € H{hx). D PROOF.

1. BASIC PROPERTIES OF RA LOOPS

115

A loop identity of the form {xy)(zx^) = (x • yz)x^^ where A; is a positive integer, is called an Mk-law, Any Moufang loop satisfies the Mi-law, which is just the middle Moufang identity. 1.10 Corollary. An RA loop satisfies the M^-law: {xy){zx^) = {x • yz)x^. PROOF.

Let x, y and z be three elements of an RA loop L. Then

(x ' yz)x^ = {(x • yz)x}x^

by diassociativity

= (xy ' zx)x^

by the middle Moufang identity

= (xy)(zx ' x^)

since x^ is in the nucleus of L

= {xy)(zx^)

by diassociativity.

D

0 . Chein has shown that Lagrange's Theorem holds in any finite loop which satisfies the Ma-law [Che73]. Thus we have further confirmation of the validity of Lagrange's Theorem for finite RA loops. Recall that in loop theory, a subloop ^ of a loop L is normal if, for any x^y ^ L^ Hx — xH and any two of the relations {Hx)y = H{xy),

{xH)y = x{Hy),

x{yH) = {xy)H

are valid. (See Definition II. 1.6 and ensuing remark.) The following corollary is therefore of interest. 1.11 Corollary. A subloop H of an RA loop L is normal if and only if xH = Hx for all x G L; equivalently, if and only if H is central or L' C H. P R O O F . We show that for RA loops, the relation (Hx)y = H{xy) for all x^y follows from Hx = xH for all x, and leave it to the reader to modify our argument in order to verify that x{yH) = {xy)H also holds for all x^y. So assume Hx = xH for all x and let x^y ^ L and h £ H, By Theorem 1.1, either {hx)y = h{xy) or {hx)y — x{hy) = {xy)h — h'{xy) for some h' G H, In either case we obtain {Hx)y C H(xy), Similarly, if h{xy) ^ {hx)y^ then h{xy) — {xh)y = {h'x)y for some h' G H^ so H(xy) C (Hx)y. Let ^ be a subloop of L. If H is central, clearly H is normal. Suppose V C H. By Theorem 1.1, V = {l,^} for some central element s. For h £ H and X G //, write hx = {xh)c^ where c = (h^x) G H. Since c is central, by

116

IV. RA LOOPS

Theorem 1.8, {xh)c = x{hc)^ we obtain Hx C xH. To obtain the other inclusion, write xh = (hx)c = c(hx) = {ch)x^ by centrality of c. Conversely, if H is normal in L but not central, let x ^ L and h £ H he such that s = (/i, x) 7^ 1. Then hx = (xh)s = x(hs)^ since s is central. By normality, hx G xH^ so hs and hence also s are elements of ^ . Since L^ = { l , ^ } , the result follows. D Loops (Z/, •) and (M, •) are said to be isotopic and each is a loop isotope of the other, if there exist bijections a^/3^j: L ^^ M such that {xai^y/3) = (x • y)j for each x^y £ L, The notion of isotopy does not arise in group theory because if two groups are isotopic, then they are isomorphic. (See, for example, [Pfl90, Lemma IL3.1, p. 48].) There are, however, loops which are not groups but which are also isomorphic to all their loop isotopes. In the hterature, such loops have often been called "G-loops". Thus, a G-loop is a loop which is isomorphic to all its loop isotopes. Since a Moufang loop which satisfies an M^^-law with k ^ \ (mod 3) is a G-loop [Pfi90, Theorem IV.4.11, p. 105], we obtain the following additional property of RA loops. 1.12 Corollary. An RA loop is a G-loop. 1.13 Theorem. Let L be a loop which is not associative. Then L is an RA loop if and only if it contains a central element s of order 2 such that, for all x^y^z ^ L, (i) if x.,y.,z associate in some order, then they associate in all orders; and (ii) if (x, y,z)^l, then (x, t/, z) = (x, y) = (x, z) = {y, z) = s. Suppose L is an RA loop. Statement (i) is an immediate consequence of Moufang's Theorem. For statement (ii), note that if x, y and z do not associate, then, by Theorem 1.8(iv), (x,y) ^ 1, (x^z) ^ 1 and (y^z) 7^ 1. By Theorem 1.8(v), there exists a central element s of order 2 such that L' = {1,^}, so PROOF.

(x, y,z)i^\

implies (x, y, z) = (x, y) = (x, z) = (y, z) = s.

Conversely, suppose L is a loop which is not associative and which contains a central element s of order 2. Suppose that statements (i) and (ii) hold for all x^y^z e L. In order to show that L is an RA loop, it is enough, by

1. BASIC PROPERTIES OF RA LOOPS

117

Theorem 1.1, to show that \i {x^y^z) ^ 1, then y -xz = xy - z = x - zy. But xy'Z

= (x' yz)(x, y, z) = (x - yz)s = (x • zy)s^ = x- zy,

the third equality holding since (y, z) = s is central. Also, xy ' z = (yx ' z)s = (y • xz)s^ = y • xz, completing the proof.

D

There are two other useful identities which hold in any RA loop. 1.14 Theorem. Let L be an RA loop. Then for any w, x, y, z in L, (i) (xy,z) = (x,z){y,z){x,y,z)

and

(ii) (xy, z, w) = (x, z, w){y, z, w). For (i), we proceed as follows, using liberally the fact that all commutators and associators in L are central: PROOF.

xy'Z={x

'yz){x,y,z)

= (xz ' y){x, z, y){y, z){x, y, z) := (zx • t/)(x, z){x, z, y){y, z)(x, t/, z) = (z • xy){z, X, y){x, z){x, z, y){y, z){x, y, z). Thus (xy,z) = (z,x,y){x,z)(x,z,y)(y,z){x,y,z) giving (i) because the associators {z,x,y), {x,z,y) and {z,x,y) are equal elements with square 1. (Using the notation of Theorem 1.13, each is 1 or s). For (ii), we recall that, for any z,w ^ L, the inner map R(z,w), which is defined by R{z,w) =

R{z)R{w)R{zw)-\

is a pseudo-automorphism of L with companion c = (2:, it;), because L is Moufang. (See Theorem II.3.3.) Thus, for any x,y £ L, [xR{z, w)][yR{z, w)'c] = [{xy)R{z, w)]c. Since L is RA, the commutator c is central; thus, upon cancelling c, we see that R(z,w) is an automorphism of L,

118

IV. RA LOOPS

Let x^y^ZyW G L, Since (x^z^w) is central, xR{z^w)

= {xz • w){zw)~^

= (x • zw)(zw)~^{x^z^w)

=

x{x^z^w)

and, similarly, yR{z,w)

= y(y,z,w)

and {xy)R{z,w)

=

{xy){xy,z,w).

Thus {xy){xy,z,w)

= {xy)R{z,w)

=

= x{x,z,w)y{y,z,w)

[xR{z,w)][yR{z,w)] =

{xy)(x,z,w)(y,z,w)

and, after cancelling xy^ we obtain (ii).

D

Recall that a Moufang loop is an inverse property loop (Theorem II.3.1) so, by Corollary II.2.3, the subloop test for Moufang loops is the same as it is for groups. 1.15 Corollary. If L is an RA loop and z^w £ L, then C{z) = {xe

L \{x,z)=

1}

and Af{z,w) = {x e L \ {x,z,w)

= 1}

are subloops of L. If x commutes with ^, so does x~^, If x and y commute with z^ then {x^y^z) r= 1 by Theorem 1.8(iii), so xy commutes with z by Theorem 1.14. Thus C{z) is a subloop. li x^y € M{z^w)^ then {x^z^w) = (y^z.w) = 1, so, by Theorem 1.14, {xy^z^w) = {x^z^w){y^z^w) = 1. Observe also that x, z and w associate if and only if x~^, z and w associate (by Moufang's Theorem) so, because there are only two associators in L, x € Af{z^w) implies x~^ G Af{z^w). Since M{z^w) is closed under products and inverses, it is a subloop too. D PROOF.

2. RA LOOPS HAVE LC

119

2. R A l o o p s have L C In §111.3, we introduced the "lack of commutativity" (LC) property. A Moufang loop L has property LC if, whenever x and y are elements of L which commute, then at least one of x, y, xy is central. If L is an RA loop, y'^ is central, so y~^ € Z{L)y^ thus the condition xy 6 Z{L) is equivalent to the condition x E Z{L)y. The following theorem is the key to a fundamental description of an RA loop [CG86]. 2.1 Theorem. An RA loop has property LC. Let L be an RA loop. As is often the case, we assume throughout this proof that L is contained in an alternative ring RL with char R ^ 2, First, we record the obvious relationship between the ring and loop associators of three elements in L (and use the fact that loop associators are central): PROOF.

(2)

[a,6,c] = ( l - ( a , 6 , c ) ) ( a 6 - c ) .

Let X and y be noncentral commuting elements of L. We prove that xy is in the nucleus of L and hence central. By Theorem 1.14(ii), this can be accomplished by proving that (a:,z,tx;) = (y^z^w) for all z,t(; G //, since there is only one nonidentity associator in an RA loop and this element has order 2. Suppose that z and w are any elements in L with (z,it;,x) ^ 1. Since x and y commute and hence associate with any third element (Theorem 1.8(iii)), it is apparent that f(z, w,x,y)=

[zw, X, y] - w[z, x, y] - [w, x, y]z = 0,

where / denotes the Kleinfeld function (see Definition 1.1.4). Since / is an alternating function and [y, w^ x] = 0, we have 0 = /{z^ y, ti;, x) = [zt/, w^ x] — y[z^w,x]. Using (2) to rewrite this in terms of loop associators and noting that (zy^w^x) = {z^w^x){y^w^x) = (z^w^x) (see Theorem 1.14(ii)), we obtain {zy • w)x + (z. It;, x)y{zw • x) = y{zw • x) + (z^ t/;, x){zy • w)x. Since {zy'w)x is in the support of the left side here, it is also in the support of the right side. Since (z^w^x) ^ 1, we must have (zy • w)x = y{zw • x)

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IV. RA LOOPS

and, since x and y associate with every third element, y{zw - x) = {y - zw)x. Thus (zy • w)x = {y • zw)x and, after cancelling x, we obtain zy -w = y - zw. If 2r, w and y associate, then zy = yz; that is, {z,w,y)

= 1 implies {y,z) = 1.

Since w and z are interchangeable in this argument, we have also (z^ w^y) = 1 implies (t/, w) = 1. Summarizing the situation, we have shown that if x and y are noncentral commuting elements of L and (^, it;, x) / 1 for some z ^ L^ then (t/, 2r) = 1 if and only if {z^ w^y) = 1 if and only if (?/, it;) = 1. We assert that y and z cannot commute and thus (z^w^y) ^ 1. Suppose, to the contrary, that y and z (hence also y and w) do commute. Since y is not central, there exists u £ L such that yu ^ uy^ by Theorem 1.8(ii). Now if (w^u^z) / 1, applying to y and z what we have just learned about commuting elements, we discover that (y. It;) = 1 if and only if (it;, u,y) = 1 if and only if (y, i^) = 1. But (y^u) / 1, while (w^u,y) = 1 by Theorem 1.8(iv), since y and it; commute. The obvious conflict can be resolved only if (it;,^^,^) = 1. This shows that w and z associate with every element u which does not commute with y. Now the set C{y) of elements which commute with y is a subloop of L, by Corollary 1.15, and proper, because y is not central. Since L is generated by the complement of a proper subloop (recall Remark 1.7) and since (a6,it;,z) = (a,it;,>2:)(6,it;,2r), we see that (w^z^u) = 1 for all u £ L^ contradicting (z^w^x) ^ 1. We have indeed proven that y and z do not commute and, as asserted, that (2:,it;,y) 7^ 1. Thus {^z^w^x) 7^ 1 implies (z^w^y) ^ 1. The other implication follows by symmetry of x and y. Since there are only two distinct associators in an RA loop, it follows that (x^z^w) = (y^z^w) for all z and it;, and this is precisely what we wanted to show. D 2.2 Corollary. A Moufang loop which is not associative is an RA loop if and only if it has property LC and a unique nonidentity commutator.

2. RA LOOPS HAVE LC

121

P R O O F . In Theorems 1.8 and 2.1, we have shown that an RA loop has the desired properties. For the converse, let 2/ be a Moufang loop with LC and a unique nonidentity commutator, s. Since s~^ is also a commutator, s has order 2 and s is central because all commutators in a Moufang loop with LC are central. We now prove that L is RA by verifying statement (ii) of Theorem 1.1. (Condition (i) holds in any Moufang loop.) Suppose then that a:, y and z are three elements of L which do not associate. Then no two of these can commute: for example, if xy = yx, then either x G >2(iy) or y G 2^{L) or a; G Z{L)y and, in any of these cases, x, y and z must associate (in the case x G Z{L)y^ by diassociativity), a contradiction. Thus (a:,t/) = {x^z) = (y-,z) = s and no product of any two of x, y, z can be central. Furthermore, no one of x, i/, z can commute with the product of the other two: for example, if xy commutes with z, then either xy or z is central (both possibilities already eliminated) or 2: G Z{L)xy^ again implying that x, y and z associate, because of diassociativity. In particular, (xy^z) = (xz^y) = 5. The left Moufang identity—z{x'zy) = (zx'z)y—yields

z{x • zy) — (x, z){xz • z)t/ = s{xz^^y — sz^{xy) = sz{z ' xy) = s^z(xy • z) = z(xy • z). Cancelling z, we obtain x-zy = xy-z. Similarly, the right Moufang identity— {xy ' z)y = x{y • zy)—gives {xy ' z)y = sx{y • yz) = sy'^xz = sy{y • xz) = y{xz - y) = {y - xz)y, and hence xy - z = y - xz,

D

2.3 Corollary, An RA loop L is an extension of its centre Z by C2 X C2 X C2, where C2 denotes the cyclic group of order 2; that is, LjZ = C2 X C2 X C2. For any elements a^b in L which do not commute and any u E L such that a, b and u do not associate, the subloop of L generated by Z, a and b is a group, G, and L is the disjoint union G U Gu. Since Z contains all commutators and associators of L and since squares in L are central, L/Z is an elementary abelian 2-group. Suppose that X, y, Wy u are four noncentral elements of L and that L/Z contains the direct product (xZ) X (yZ) X (wZ) X (uZ), Because of LC, no two elements of {x^y^w^u} can commute. Furthermore, no three elements of PROOF.

122

IV. RA LOOPS

this set can associate. For example, if x, y and w associate, then (xy^w) = (x^w){y^w) = s'^ = 1 (Theorem 1.14) so, by LC, either xt/, w ox xy - w is central, each possibility contrary to fact. Now repeated use of Theorem 1.14 gives (xy^wu) =

(x^wu){y^wu){x^y^wu)

= {x,w){x,u){x,w,u){y,w){y,u)(y,w,u){x,y,w){x,y,u)

= ^^ = 1.

Thus xy and wu commute, which is not true. We conclude that L/Z is the direct product of at most three copies of C2. If it were the direct product of fewer than three, then L would be generated by its centre and at most two other elements. Then diassociativity (and the definition of centre) would force L to be a group. Thus L/Z = C2 X C2 x C2 as claimed. By diassociativity and the definition of centre, the subloop G generated by Z and any two elements a and 6 is a group. If t/ is a third element such that (a^b^u) ^ 1, each cyclic subgroup (aZ)^ i^^)-) (^^) ^f ^1^ has order 2 and the product {aZ) x {bZ) x {uZ) is direct because of the LC property. So this direct product must be L/Z and it follows that L = G U Gu. D 2.4 Corollary. Let L be an RA loop and let B be a commutative subloop of L. Then, for any x £ L, the subloop ( 5 , x ) generated by B and x is a group. In particular, B is an abelian group. P R O O F . Let 61,62 G B, Since 6162 = 6261, either 61 is central or 62 is central or 6162 is central. In the first two cases, (61,62,2:) = 1 is clear while, if 6162 = >2^ is central, then (61,62,0:) = (z6^\62,x) = 1 because of diassociativity, the definition of centre and the identity

(3)

{xy, z, w) = (x, z, w){y, z, w)

which holds by Theorem 1.14. Now a straightforward induction argument making further use of (3) gives the result. D 3. A d e s c r i p t i o n of an R A l o o p We now show that loops of the form M(G, *,5fo)? which we introduced in §11.5.2 and discussed at length in Chapter III as examples of RA loops, are not as special as it might there have seemed.

3. A DESCRIPTION OF AN RA LOOP

123

3.1 Theorem, / / L is an RA loop with commutator-associator subloop V = {1,5}, then L = M(G,*,5o) where G is any group containing 2(L) and two noncommuting elements of L, go is a central element of G and the involution g ^-^ g* is defined by (4)

9* = {

g

ifgeZiG)

sg

ifgiZ{G).

Converselyf for any nonabelian group G with LC, a unique nonidentity commutator, s, and involution *, the loop M(G^^^gQ) is an RA loop for any go G Z{G)j where * is given by (4). Suppose a and b are two elements of an RA loop L which do not commute and G is a group containing 2{L)^ a and 6. By Theorem 1.8, there exists u £ L such that a, 6 and u do not associate. In particular u ^ G. By Corollary 2.3, L = Gi U G\u^ where Gi is the group {Z(L)^a^b). Since Gi C G, it follows that Gi = G and L = G U Gu. Note that s e Z{L) C G. Let g^h ^ G. We claim that PROOF.

(5)

(p, h) = I \f and only if (g^h^u) = 1.

That {g<,h) = 1 implies (g^h^u) = 1 is part (iii) of Theorem 1.8. On the other hand, if (g^h^u) = 1, then (gh^u) = (g^u){h^u) by Theorem 1.14(i) and, since each commutator here is 1 or 5, at least one of (gh^u)^ (5^?^)^ (/i, u) is 1. Thus u commutes with g or with h or with gh. Since u is not central and none of gu^ hu^ {gh)u is central (because Z{L) C G but u ^ G), either g or h or gh must be central (by property LC); in any of these cases, gh — hg^ di> contradiction. Next, for g £ G, define ^* = ugu~^. If y 6 G is central, clearly, ^* = g. Suppose g is not central. Then, since gu is not central {gu ^ G), gu ^ ug (by LC), so (^, u) = s and g* = sg. All this shows that y* is the element of G given by (4). Moreover, it follows from the LC property and Theorem III.3.3 that g ^-^ g* defines an involution of G. Now we determine how the elements of L = G U Gu multiply. First, for g^h ^ G^ we have g 'hu = {gh- u){g, h, u) = hg - u,

124

IV. RA LOOPS

since (g^h^u) = {g^h) by (5). Secondly, gu'h

= (g ' uh){g,u,h)

= (g - h*u){g,u,h) = {gh^ • u){g, /i*, u){g, u, h) = gh* - u

because h* = h or sh implies that (g^h*^u) = {g^h^u). Finally, {gu){hu) = g{u • hu){g^ n, hu) = g{uh • u){g^ n, hu) =

by the flexible identity

g{h*u^){g,u,hu)

= g{h*u^)(g^u^h)

by Theorem 1.14(ii)

= {gh*)u^(g^u^h)

since u^ is central

= /i*5 • '^^

since {g, K") = (g, h) = {g, u, h).

Setting go = u^ £ ^{G), we have shown that multiplication in L is given by g(hu) {gu)h {gu){hu)

= {hg)u = {gh*)u = goh^'g.

Thus L is precisely the loop M(G, *,^o)The last statement of the theorem is part of Corollary III.3.4.

D

In §V.4, we shall enumerate all the RA loops of order not exceeding 64.

Chapter V

The Classification of Finite R A Loops

In this chapter we classify all finite RA loops which are not direct products of proper subloops and give an explicit presentation for each such loop. As we shall see, this will permit a classification of all finite RA loops, up to isomorphism. We finish the chapter by listing the RA loops of order less than or equal to 64. The results in this chapter are based primarily on work of E. Jespers and G. Leal [JLM95, LM93].

1. Reduction to indecomposables An element of finite order in a (possibly associative) Moufang loop is said to be a torsion element. A Moufang loop is a torsion loop if all its elements have finite order and a 2-loop if all its elements have order a power of 2. Recall that a Moufang loop is diassociative and hence, if it is finite, has order divisible by the order of each of its elements. (See Corollary II.3.7 the discussion in §11.1.3.) It follows that a finite Moufang loop whose order is a power of 2 is a 2-loop. Conversely, if L is a 2-loop, then L is centrally nilpotent by a result of G. Glauberman and C. R. B. Wright [GW68]. Since the central factors of a centrally nilpotent loop are groups, the central factors of L must have 2-power order and hence L also has 2-power order. Thus, as with groups, finite Moufang 2-loops are precisely those Moufang loops whose order is a power of 2. 1.1 Proposition. Let L be a (possibly associative) Moufang loop with central squares and an order 2 commutator-associator subloop. For any prime p, the set Lp of elements of L whose order is a power of p is a normal subloop of L. The set L2' of elements of odd order in L is an abelian group 125

126

V. T H E CLASSIFICATION O F FINITE RA LOOPS

which is central as a subloop of L, If T is any torsion subloop of L, then T ^ T2 X Tr, where T^ = T n L2'. Suppose that x is an element of odd order 2n + 1 in L. Then x"^ = (x^)^ G >2(i/), so the sets i/2' and Lp, for p odd, are central subloops of L and hence abelian groups. Denote by s the unique commutator-associator of L and recall, by Corollary III.1.5, that s is central of order 2. If x and y are any two elements in i/, then, for any n > 0, PROOF.

{xyT

^UyTl

if n = 0 or 1 (mod 4)

sx'^y'^

if n = 2 or 3 (mod 4)

Thus, for t >2, (xy^^ = x'^^y'^\ It follows from Corollary II.2.3 that L2 is a subloop of L, To show that L2 is normal, it is sufficient to show that XL2 = L2X,

(L2x)y - L2{xy)

and

2/(^^2) = (2/^)^2

for any x^y ^ L. (See Definition II.1.6 and ensuing remark.) Each of these equations is an easy consequence of the definition of L2 and properties of s. To prove that {L2x)y — L2{xy)^ for example, let a £ L2 and note that {ax)y = a(xy) or s(a'Xy) = {sa){xy) according as the associator {a^x,y) is 1 or s. Let T be a torsion subloop of L. As above, T2 and T2' are normal subloops of L. Clearly T2 D T2/ = {1} and T = T2T2', so T ^ T2 X T2', U 1.2 Corollary. Let L be a torsion RA loop. Then the set L2 of elements of L whose order is a power of 2 is an RA 2-loop and a normal subloop of L. The set L2' of elements of L of odd order is a central subloop of L. Furthermore L — L2 X L2/. Because of Theorem IV. 1.8, an RA loop satisfies the hypotheses of Proposition 1.1, so virtually everything foUows from this proposition. We note only that since L2/ is central, it is a group, whereas L = L2 X L2' is not. Thus L2 is not associative, so it is an RA loop. D PROOF.

A loop L is indecomposable if it is not the direct product of two proper subloops. 1.3 Corollary. An indecomposable torsion RA loop is an RA 24oop.

1. REDUCTION TO INDECOMPOSABLES

127

1.4 Corollary. An indecomposable torsion group with central squares and a unique nonidentity commutator is a 2-group, 1.5 Remark. It is known that if L is a Moufang loop with central squares and a unique nonidentity commutator 5, and if L is not associative, then s is also a nonidentity associator in L, so V has order 2 [CG90b]. Thus Proposition 1.1, which as stated is sufficient for present purposes, actually holds under somewhat more general conditions. The principle result of Chapter IV, embodied in Theorem IV.3.1, showed that RA loops are loops of the form M(G, *,^o), where G is a nonabelian group, * is a particular involution on G and ^o is an element in the centre of G. 1.6 Proposition. Let L = M(G, *,^o) ^^ ^^ ^ ^ loop and let A be an abelian group. Then {g^aY = (^*,tt) defines an involution on G X A and M{G X A^ *, (^0,1)) i^ o,n RA loop which is isomorphic to M{G^ *?5^o) X A. Conversely, if A is an abelian group such that M(G X A^ *, {go^ 1)) is an RA loop for some nonabelian group G and some go G 2 ( G ) , then the restriction of ^ to G is an involution on G and M{G X A, *, (po, 1)) — ^ ( G , *,yo) X A. P R O O F . For the first part, let s denote the unique nonidentity commutator of G. Clearly G x A is a nonabelian group with LC and a unique nonidentity commutator, (5,1), so M{G X ^4, *,yo) is an RA loop, by Corollary IV.2.2. If V is an indeterminate such that M{G X yl,*,(^o?l)) = (G X yl) U (G X A)v^ then it is a simple matter to verify that the map M{G X >l,*,(^o, 1)) -^ M(G, *,po) X A defined by

(1) {g,a)v ^ (gu.a) is an isomorphism. Conversely, given an abelian group A such that M{G X ^5*5(^051)) is an RA loop, it is clear that G has LC and that the unique nonidentity commutator of G X A is of the form (5,1), where 5 is a unique nonidentity commutator in G. By Theorem IV.3.1, the involution on G X A is defined by I (^f, a) (^?^)

=

if (y, a) is central

\

I (5, l)(y,a)

otherwise.

128

V. THE CLASSIFICATION OF FINITE RA LOOPS

SO the restriction of * to G is an involution on G. Again, it is straightforward to check that the map (1) defines an isomorphism M{G x i4,*,(po, 1)) -^ M{G,^,go)x A. n We now show that, if L = M(G,*,^o) is an indecomposable RA loop, then the group G is "almost" indecomposable. By o(x), we mean the order of an element x in a Moufang loop. 1.7 Theorem. Let L = M(G', *,^o) ^^ o, finite indecomposable RA loop. Then G = D X C where D is an indecomposable 2'group and C is a cyclic group which J if nontrivial, is a 2-group. If C is nontrivial, then go = dc with d e Z{D) and 1 7^ c G C. First write L = G UGu where u ^ G and u^ = go e Z{G). By Corollary 1.3, Z is a 2-loop, so G is a 2-group. Since G is finite nonabelian, we can certainly write G = D X A for groups D and A with D indecomposable and nonabelian. Since G has a unique nonidentity commutator, A must be abelian. If A is trivial, there is nothing to prove, so assume A ^ {1}. Writing ^0 = da^ we note that a 7^ 1; otherwise, by Proposition 1.6, L = M{D X ^,*,(^o? 1)) = ^ ( - D , *,5ro) X A is not indecomposable. Therefore, writing PROOF.

A = {^1) X •••X {tk), for some ti of order 2^' > l , i = l , . . . , / : , we can write go in the form go =

dt{''''tl\

for some 5^ > 0. We claim that ^0 can be chosen in such a way that each exponent Si is 0 or 1. To see this, we consider two possibilities. If s\ is even, let U\ = ut^ ^ and g\ = u\. Noting that the t^ are central in G and hence central in L (see Corollary III.3.5), we have 51 = u ^ i f ' -^> = dt\' • • • tinp If 61 = 2g + 1 is odd, set ui = ut^ gy = uHT"-^'

-^' = dt^.tf,' •••ti".

^ and gi = u{. Then

= dt1'+'f,^ • ••tlHr-''

= dt\tl^ •. -tl'.

1. REDUCTION TO INDECOMPOSABLES

129

In each case, we have found an element Ui ^ G such that gi = uf = dt{H'^^'-fj^ with 6i e {0,1}. Note that L = M(G,*,^i), by Corollary IV.2.3 and Theorem IV.3.1. Repeating for general i the above argument for the case i = 1, we realize our claim. Remember that when ^o is written ^o = da with a £ A^ then necessarily a 7^ 1. Thus there is no loss of generality in assuming that ^o = dti- - -t^^ where I < r < k and o(/i) > ••• > 0(^7.). Let C = {h'-'tr) and B = {/2> X • • • X {tk). Then A = CxB

a^nd G = DxCxB.

Since go e D x C

and B is abelian. Proposition 1.6 shows that L = M(D X C, *,^o) X B. Indecomposability of L implies that B = {l}^ so G = D x C. D The next result plays a key role in the classification theorems of the next section. It also allows us to show that certain groups which arise in the proofs of these theorems are indecomposable. 1.8 Theorem. A finite group G has property LC and a unique nonidentity commutator, s, if and only if G = D x A where A is an abelian group and D — {x^y^Z{D)) is an indecomposable nonabelian 2-group such that . Z{D) =

{ti)x{t2)x{ts),

• o(/,) = 2^» fori=

1,2,3, mi > 1, m2,m3 > 0,

• s = {x,y) = tp~'

e {ti),

• x'^ e (ti) X {^2} 0,'rid •

y^e{t^)x{t2)x{t3).

Suppose that a group G is of the form G = D x A with A abelian and D as described. Then Z{G) = 2{D) X A and G/Z{G) = D/Z{D) = {xZ{D)^yZ(D)). Since x'^ and y'^ are central and D is nonabelian, G/Z{G) = C2 X C2, so G has LC and a unique nonidentity commutator, by Proposition III.3.6. The import of this theorem is the converse. Thus, suppose that G is a finite group with LC and a unique nonidentity commutator; hence, by Proposition III.3.6, G/Z{G) = C2 X €2- Writing G as the direct product of indecomposable groups, it is clear that some factor D is nonabelian. By Corollary 1.4, i) is a 2-group. Also, G — D X A for some group A which is necessarily abelian because G has just one nonidentity commutator. Since Z{G) = Z(D) x A, we have D/Z{D) = G/Z{G) = C2 X C2, so there exist x,y e D such that D = {x,y,Z{D)). PROOF.

130

V. THE CLASSIFICATION OF FINITE RA LOOPS

Clearly the unique nonidentity commutator of G is also a unique nonidentity commutator for D^ and this element is (x^y). We claim that Z(D) is the direct product of cyclic groups with s = (x^y) in the first factor. For this, suppose that (2)

2iD) =

{h)x{t2)X''-x{tk)

is a decomposition of Z{D) in which s is the product of (nonidentity) elements from a minimal number of factors (ti). Without loss of generality, these factors are (^i),... , {U) for some £ > I. Let o(/^) = 2'^\ Since 5^ = 1, we have S — l-^

' ' 'll

,

where, without loss of generality, mi > m2. We show that 1=1. to the contrary that i > 1. Then

Assume

and

so that

contradicting minimality. Thus 5 G (^1), as claimed. Next we assert that we can, if necessary, replace x by an element Xi G xZ{D) and the generators U of Z{D) by new generators f- such that x^ G {t[) X (12)' For this, write x^ = ai • • -a/^ with a^ G (^i). If some ai is not a generator of (^^), then a~^ is also not a generator of (^^), hence a~^ = t^^ for some a. Let xi = /^x. Then xj = ai • • -o^ • • -a^ has no component in (ti). (As is conventional, the ^ denotes an omitted factor.) Repeating this process, if necessary, we see that we may assume that x^ = ai^ - - - a^^, where each {oiij) = {Uj)'> ^ind o(a^j) > • • • > o(a^^). If none of the a^^, I < j < i^ belongs to (ti), then {ti^) X • • • X (ti,) = {ai^ ' '-ai^) X {ai^) X • • • X (a,-,), with x'^ G (o!iiO;z2 * * '^ii)- Thus x^ G (^1) X (^2), with t'^ = a^^a^^ ' ' 'Cti^- In the case that one of the a^ 's belongs to (^i), a similar argument applied to the other factors of x^ gives the same conclusion.

2. FINITE INDECOMPOSABLE GROUPS

131

In a similar way, we can also find y ^ G and a factorization Z(D)

=

{h) X (^'2) X (^3) X • • • X (tk) such t h a t y'^ E {h) X {t'^) X (^'3). Thus Z{D)

=

(^1) X (^2) X (^3) X C for some abelian group C , so i^ = { ^ , y , ^ i , ^ 2 , ^ 3 ) x C . Since D was indecomposable, C — {1} and the proof is complete.

D

2. F i n i t e i n d e c o m p o s a b l e groups In this section, we show t h a t any finite indecomposable group with a unique nonidentity commutator and property LC lies in one of five distinct classes of groups. By Theorem 1.8, if D is such a group, then D =

{x^y^Z{D))^

where • 2(i^)=(/i)x(/2)x(^3), • o{ti) = 2^» for i = 1,2,3, m i > 1, m2, ma > 0,

• x^ = i^H^\

Oii > 0, and

• y' = t^r'4'4\fi,>0. We employ this notation throughout this section and begin by showing t h a t X and y can be chosen in a very convenient way. 2.1 L e m m a . Without

loss of generality,

x2 = f^(«>)t^("^)

and

we may assume

that

j/2 = t^^'^'hl^^'hl^'"'^

where, for an integer n, we define

{ PROOF.

0

if n is even

1

if n is odd.

The proof is similar to t h a t of Theorem 1.7. Defining

x'=

<

xt-^ ^ xt^

if a i is even "^

if a i = 2g + 1 is odd,

we have x^^ = t^"^ or x'^ — tit^'^ respectively.

132

V. THE CLASSIFICATION OF FINITE RA LOOPS

Similar computations apply to a2 and then to y so t h a t eventually, we obtain new elements x\ y' such t h a t D = {x^,y\Z{D))

with x'

and y^ as

stated.

n

2.2 D e f i n i t i o n . The rank of a finite abelian group A, sometimes written />(A), is the fewest number of generators oi A, Theorem 1.8 implies t h a t the rank of the centre of an indecomposable finite group with LC and a unique nonidentity commutator is at most 3. To describe such groups, our strategy will be to consider the three possibilities for central rank. To prove t h a t certain groups are not isomorphic, we will use the fact t h a t the exponent of a Moufang loop is an isomorphism invariant. 2.3 D e f i n i t i o n . The exponent of a Moufang loop L is the smallest natural number n such t h a t x'^ = I for all a: G //, if such n exists; otherwise, the exponent is said to be infinite. 2.4 T h e o r e m . A finite group D is indecomposable identity two

commutator

with LC, a unique

non-

and centre of rank 1 if and only if D belongs to one of

families, V\:

Groups with

presentation {x,y,ti

\x^ = y^ = tp

= 1)

and V2: Groups with

presentation {x,y,ti

\x'^ = y'^ = / i , tl^'

= 1)

where, in each case, ti is central and s = (x^y) = tf"^^ distinct:

no group in V\ is isomorphic

PROOF.

. The families

are

to any group in 7)2-

Let D be a finite indecomposable group with LC, a unique non-

identity commutator and Z{D)

= (^1), o{ti) = 2^^ > 1. Since D/Z{D)

C2 X C'2, if r^i = 1, then |Z)| = 8, so D is isomorphic to either D4 = {x,y I x^ = y^ = l , ( x , t / ) = ^1) or Qs = {x,y\

x^ = 2/^ = ti,{x,y)

= ti).

Clearly D4 is in the family Vi and Qs is in the family V2.

^

2. FINITE INDECOMPOSABLE GROUPS

133

Suppose mi > 1. According to Lemma 2 . 1 , we can choose x and y in such a way t h a t x^ and y^ assume the values 1 or ^i. Thus we have four possible choices: (i) x^ = y^ = 1; (ii) a;2 = i i , t/2 = 1; (iii) x^ = l , y 2 = fj. (iv) x^ = y'^ = ti. Choice (i) gives a group in V^ and choice (iv) a group in 'D2. In case (11), setting y = xy we have y

= sti = q

ti = /^

"^ and, since the

exponent here is odd, Lemma 2.1 shows t h a t we can choose a generator y^^ such t h a t y"

= ^i- So we are dealing with a group in the family V2.

By

symmetry, case (iii) also defines a group in P2Let Z)i be a group in X>i and D2 a group in X>2 with \Di\ = | ^ 2 | - If TTii = 1, it is clear t h a t Di ^ D2 while, if m i > 1, it is easily seen t h a t the exponent of Di is 2^^ while the exponent of D2 is 2^^"'"^ so the groups are not isomorphic. Finally, since Di and D2 both have indecomposable centres, it follows t h a t Di and D2 are indecomposable.

D

2.5 T h e o r e m . A finite group D is indecomposable identity two

commutator

with LC, a unique

non-

and centre of rank 2 if and only if D belongs to one of

families, P 3 : Groups with

presentation

{x,y,tx,t2\x

=ti

=t2

=l,y

=t2)

and V4: Groups with {x,y,ti,t2

presentation \x'^ = ti,y'^ = t2,t'f

= tT^

= 1)

where, in each case, t\ and ^2 are central and s — {x^y) = t\ ^ PROOF.

Let D be an indecomposable group with LC, a unique noniden-

tity commutator and centre Z{D)

= l^ti) X (^2)? where o{ti) — 2^» > 1 for

i = 1,2. Then D — /?/(^2) is a group with LC and a unique nonidentity commutator and it is indecomposable because its centre is indecomposable. Thus D is in one of the two classes V\ or V2 described in Theorem 2.4.

134

V. THE CLASSIFICATION OF FINITE RA LOOPS

Denoting by x and y the cosets xZ{D)

and yZ{D)

respectively, we have

two possibilities, up to isomorphism: (a) x2 = y2 ^ T; (b) X 2 = : y 2 ^ ? i . Because of Lemma 2.1, case (a) can be lifted to D in four different ways: (a.i) x2 = 2/2 ^ i.^ (a.ii) x^ = 1, y2 ^ ^2; (a.iii) x^ = ^2, y^ = 1; (a.iv) x^ = t/2 _ ^2 Case (a.i) gives us the group (x, y , ^ i ) x ( t 2 ) , which is not indecomposable. Cases (a.ii) and (a.iii) give a group in V^, In case (a.iv), setting x' = xy, we have x'

— si\ — t^^

i\, \i m\ > 1,

Lemma 2.1 shows t h a t we can choose a generator x'' such t h a t x "

= 1 , in

which case Z) is in P 3 . If m i = 1, Lemma 2.1 shows t h a t we can choose x^' such t h a t x'' = ^i and D is a group in P 4 . In a similar way, again by Lemma 2.1, case (b) also lifts to D in four different ways: (b.i) x^ = y^ = U; (b.ii) x'^ = ti, y'^ = Ut2\ (b.iii) x^ zz t^t2, y'^ = ti; (b.iv) x^ = y^ = ht2. In case (b.i), the group is the direct product {x,y,^i) X (^2), which is not indecomposable. In case (b.ii), we can write Z{D) and t = t2 otherwise.

t h a t D is a group in V4. x^ — st\t2

= tl ^

— {t) x (ti/2) where / = ^1 if m i < m2

If m i < m2 and t = t i , setting ^2 = hh

shows

If m i > m2 and t = ^2? we take x' = xt/, so

^^t2 and, since m i > 1 in this case. Lemma 2.1 shows

t h a t we can choose a generator x" such t h a t x " = ^2- Setting t[ = ^1/2, we see again t h a t D is in V4. Case (b.iii) is similar to (b.ii). In case (b.iv), we take x' = xy so t h a t x'

= stit2

Lemma 2.1 shows t h a t we can find x" such t h a t x " and x " 2(D)

= ^1 if m i = 1.

= {ti) X {t\t2)

Suppose

TTII

> 1.

— ^1 ^ "'^^^l*

= 1 if m i

> 1

If also m i < m2, then

and D is a group in P 3 while, if mi > m2, then

2. FINITE INDECOMPOSABLE GROUPS

135

Z{D) = {tit2) X (^2) so that D = (x,2/,^i^2) X (^2)5 which is not indecomposable. On the other hand, if mi = 1, then 1 = mi < m2, x'' = ti and Z{D) = {ti) X (^1^2), so D is in V4. To settle the distinctness of P3 and P4, let D3 be a group in V3 and D4 a group in V4, In Z?3, the element x is noncentral of order 2 while there is no such element in D4. Thus the groups D3 and D4 are not isomorphic. Finally, we prove that the groups in V3 and V4 are indecomposable. Let Z? be a group in either of these classes. By Theorem 1.9, if D is not indecomposable, then D = H X K^ where H is nonabelian, Z{H) 7^ 1, K is nontrivial abelian and D' C H, Since Z{D) = 2{H) x A' = (^1) x (^2)? 2(H) must be cyclic. Let /i = ti^t2^ be a generator of Z{H)^ say of order 2 ^ Since 5 = tf' e Z{H) is the unique element of order 2 in Z{H)^ we must have

This implies ^2^ = 1 , so either a2 is even or r — 1 > m2. Since D = Z{D)[JxZ{D)[JyZ{D)[JxyZ{D) and Zf C D is not abelian, it follows that H contains elements from at least two noncentral cosets. In any of the possible cases, it is easy to see that there exists an element yi ^ H which can be written in the form y^ — yz^ with z G Z{D). Hence 1 ^ y2 _ ^^^2 ^ fj f) Z{D) = Z{H). So this element can be written in the form y^ = {t^HTT ^ r some n > 1. Since y'^ = t2 = y\z~^ = (^i'^^2^)^-2:"'^, we see that a2 cannot be even; hence r > m2. Therefore o(h) > 2^^ ^n^^ since (h) is a direct factor of Z{D) = {h) X (^2), it must be that o{h) = 2'^K Therefore ai is odd. Since the element t2Z^ is of the form t2t^^t^'^^ while an element of Z{H) is of the form ^f''^^2'''' we see readily that no element of the form t2Z^ can be in Z{H)^ a contradiction. D 2.6 T h e o r e m . A finite group D is indecomposable with LC, a unique nonidentity commutator and centre of rank 3 if and only if D belongs to the family P5; Groups with presentation {x,y,ti,t2,t3

I x^ = ^2,2/^ = ts.tT'

= tT^ = tT^ = 1),

where ti^ ^2 ^^^ ^3 ^^^ central and s — (x^y) = t^ ^

136

V. THE CLASSIFICATION OF FINITE RA LOOPS PROOF.

Let D be an indecomposable group with LC, a unique noniden-

tity commutator and centre Z{D)

= (ti) X (^2) X (^3) where o(/^) = 2^» > 1

for i = 1,2,3. Consider D = Dl{t^)^

which is a group described by Theo-

rem 2.5. So, denoting by x and y the cosets xZ{D)

and yZ{D)^

respectively,

we have two possible choices, up to isomorphism: ( a ) X'^ rz T, t/2 = t2\

(b) x^ = t,,t

= h-

Because of Lemma 2.1, case (a) lifts to one of the following subcases: (a.i) x^ = 1, 2/2 ^ ^2; (a.ii) x^ = 1, 2/^ = ^2^3; (a.iii) ^2 = ts, y^ = ^2; (a.iv) x'^ = ts, 2/2 ^ t2t3. Case (a.i) leads to the decomposable group {a:,y,/i,/2) X {^3).

Case

(a.ii) leads to the decomposable group (x,i/,/i,^2^3) X i^)? where ^ = ^3 if ^ 2 ^ ^^3? ^i^d t = t2 otherwise. Case (a.iii) gives D in the family P 5 . In case (a.iv), setting y^ = xy^ we have y^ = ^^2^3 = ^Y^^

^2^3- Apply-

ing Lemma 2.1, we see t h a t we can choose a generator y" such t h a t y'^ — ^2 if mi > 1, and y" = ^1^2 if ^ 1 = 1- In the first case, D is in the family P 5 and, in the second case, Z{D)

— {t\) x (^1/2) X (^3), so again we see t h a t D

is in the family P 5 . Case (b) also lifts to four subcases: ( b . i ) x2 = t i , 2/^ = ^2; (b.ii) x'^ = U, y2 3, ^2/3; (b.iii) x2 = ^1/3, 2/2 rzr ^2; ( b . i v ) x^ = ^1^3, 2/2 ^ ^2^3.

Cases (b.i) and (b.ii) define groups which are not indecomposable, these being, respectively, (x, 2/, ^1,^2) X (^3) and (a:, 2/, ^1, ^2^3) X (^), where t = t^ if 7722 > ^ 3 ^ and ^ = ^2 otherwise. In case (b.iii), if mi > m3, we can write Z{D) D is not indecomposable. If m i < m3, then Z{D)

= (hts)

x (^2) x (^3) and

= {ti) x (^2) X (^1^3) and

D is in the family P 5 . T

/I

• \

•

/

1

/

2

In case (b.iv), settmg y = xy, we have y

9

— st\t2t^

2"^1—^-1-1

— ^i

The proof of Lemma 2.1 shows t h a t we can choose y" G xyZ{D) t h a t 2/"

= ^2 if ^ 1 = I9 and 2/"

9

^2^3such

= ^1^2 if ^ 1 > 1- In the first case.

2. FINITE INDECOMPOSABLE GROUPS

137

we are back to (b.iii). To discuss the second case, assume first that mi > m2. Then we can write Z{D) = (^1^2) X (^2^3) X (^3) if 1712 > ms, and 2{D) = {tit2) X (^2) X (^2^3) otherwise. Set Di = (j/,/,^1^2,^2^3). Then either D = Di X {t^) or D =^ Di x {^2)7 respectively, and in both cases D is not indecomposable. Now assume mi < r7^2• If 'n^i > m.3, we can write Z{D) = (^1/3) X (^2^3) X (^3); thus D = {x,y,tit2,,t2h) X (ts) is not indecomposable. In the remaining case, we have mi < min{m2, 7723}, so Z{D) = {h) X {tit2) X {tits) and D is in the family P5. Finally, we shall show that a group D in P5 is indecomposable. Suppose it is not. Then, by Theorem 1.8, D = H X K with H indecomposable, s E H and K nontrivial abelian. A straightforward calculation shows that if nil = 1, then a central element t^t^t^ is the square of an element in D if and only if a = /37 (mod 2) (in which case, t^t^t^ — {x^y^Y)^ while, if TTii > 1, then f^t^t^ is the square of an element in D if and only if a is even (in which case t^t^^tl = {t^^~^^'^^'~'^''^xf^y^f.) Let N be the set of elements in Z{D) which are not squares of elements in D. Then

N

Ui4^3

\f^ + f^l odd}

{^?^2^3 I ^ odd}

if mi == 1 otherwise

and, in both cases, |A^| = 2'^i"*"'^2+m3-i \Ye consider each case separately. Suppose first that mi = 1. Since D — H X K with A^ nontrivial abelian, Z{H) has rank either 1 or 2. If the rank is 1, Z{H) is cyclic and, since 5 = ^1 is an element of order 2 in Z{H) but not a square, it must be that \Z{H)\ = 2. Since \H/Z{H)\ = 4, by Proposition III.3.6, the order of H is eight, so every central element in // is a square. This contradicts the fact that ti £ N, Therefore Z{H) has rank 2 and A^ = (k) is cyclic, say of order 2^. Write Z{H) = (si) X (52) with ^1 G (^i). Since ^1 has order 2 but is not a square, it follows that Si = ti^ so si = s. Thus the unique nonidentity commutator in H is not a square and so, by Theorem 2.5, H £ V3. If 5 is a group in V3 (with ti — s oi order 2), presented as in Theorem 2.5, it is readily seen that the elements of Z{B) which are squares of elements in B are of the form ^2 or ^1^2 ^^^h fi odd. Therefore, the only elements in Z{H) that are not squares of elements in H are those in the set Ni = {tis^ I /? even}. Let N2 = {k^ \ /3 odd}, the set of nonsquares in K.

138

V. THE CLASSIFICATION OF FINITE RA LOOPS

Then N = NiK U {Z{H)N2 \

NiNi),

a set of order \N\ = lo(s2)\K\

+ \Z(H)\2'-'

- io(.2)2^-^

= i 0(^2)2^+ 2 0(52)2^-1 - i 0(52)2^-1 =

lo(s2)2'-\

Since \N\ = 2^i"''^2+m3-i jg ^ power of 2, we have a contradiction. Now suppose 7721 > 1 2ind notice that if a, 6 G N^ then neither ab nor ab~^ is in N. Again K has rank either 1 or 2. If K = (ki) x {k^) is of rank 2, then /ji, ^2 ^ind /;:i/;;^^ are all in TV, a contradiction. On the other hand, if K = (k) has rank 1, then the elements in kZ{H) are not squares in D. Hence, by our remark, for any h G Z{H)^ h = (hk)k~^ is a square. Since k is not a square, k = ^i^'^^^2^3 for some a, /3 and 7. In particular o{k) > 2^^. Write ^1 — hk'^ for some h G Z{H) and n > 0. Since ^ z z ^ p - ^ ^ / , 2 - i - i ^ n 2 - i - ^ G ^ , we obtain A;-2"^^"' G i^; thus A;-^"^^-' = 1. Since o(A;) > 2^^, this implies that 2 | n and hence that t\ is a square, a contradiction. D As an illustration of the results of this section, we conclude by enumerating the groups of order less than or equal to 32 which have LC and a unique nonidentity commutator. Our notation is that of M. Hall Jr. and J. K. Senior [HS64]. (See also [TW8O].) The indecomposable groups with LC and a unique nonidentity commutator are described in Table 1. The others are listed below. Order 16: 16r2ai = D4 X C2, 16r2a2 = Qs X C2 Order 24: D4 X C3, Qs X C3 Order 32: 32r2ai = D4 X C2 X C2, 32r2a2 = Qs X C2 X C2, 32r26 = 16r26 X C2, 32r2Ci = 16r2Ci X C2, 32r2C2 = 16r2C2 X C2, 32r2d = 16r2d X C2, 32r2ei = D^x C4, 32r2e2 = Qg x C4 3. F i n i t e i n d e c o m p o s a b l e R A l o o p s The results of the previous section and a few additional remarks will show how to construct all indecomposable RA loops.

3. FINITE INDECOMPOSABLE RA LOOPS

139

1. The indecomposable groups of order less than or equal to 32 with LC and a unique nonidentity commutator TABLE

[ G D4

1 Qs IGTib l6T2d 16r2Ci 16r2C2 32r25 32T2h 32r2/ 32r2i 32r2ii 32r2i2 32r2A:

_\G\j

Z{G)

8 8 16 16 16 16 32 32 32 32 32 32 32

~^

y'

C2

1

1

C2

h

^1

CA

1

1

CA

h

^1

C2XC2

1

<2

C2XC2

h

<2

Cs

1

Cg

h

C4XC2

1

h h

C4XC2

^1

t2

C2XC4

1

h

C2XC4 C2 X C2 X C2

h h

<2

ir\

tsj

First notice that for a given group G with LC and a unique nonidentity commutator, the RA loop L = M(G,*,^o) = G U Gu is completely determined by the choice of ^0 = '^^' Assume that such a loop is indecomposable. Then, by Theorem 1.7, G = D x C where D is an indecomposable 2-group and C is cyclic. By Theorem 1.8, Z(D) has rank at most 3. Thus Z{G) = ih) X (f2) X (^3) X {w) (where some of the factors may be trivial). as in Lemma 2.1, it can be shown that if v with ai,a2,a3,/3 > 0, then there exists '^i ^p(c.i)^p(«2)^p(c.3)^p(/9) jj^^^^ ^Yie function p is

p{n)

Using the same techniques e L and v'^ = t^'^t'^'^t w^ G v2{L) such that vi"^ defined as before by

0

if n is even

1

if n is odd.

140

V. THE CLASSIFICATION OF FINITE RA LOOPS

In particular, u can be chosen in such a way t h a t v? = g^ =

t^'^t^'^t^^w^

with a i , a 2 , a 3 , / 3 G { 0 , 1 } . Theorem 1.8 shows t h a t if {w) ^ {1}, then the component of ^o in (tt;) must be w\ t h a t is, /3 = 1. Finally, notice t h a t if x^ — ti, then we can choose u such t h a t ai = 0, except in the case where i = 1 and m i = 1. Indeed, if a^ = 1, set v = xu. Then v^ = sx'^v? = stig^. Since s — i^^

, the exponent of ti in v^ is even

(unless I — \ and m\ — 1), so we can make a new choice of u such t h a t ^0 = V? will not contain %{, A similar statement is true if y^ — %{. We are now ready to enumerate all possible indecomposable RA loops. Because of Theorems 2.4, 2.5 and 2.6, a first a t t e m p t gives the list shown in Table 2, but this table contains duplications, as we shall see. Those families denoted with a prime on the subscript—L5/, Lg/, \.\y and L12'— are those which arise from the exceptional case described in the previous paragraph; that is, those where m\ — \ and either x\ — i\ or y\ — i^. We write Di to denote a group in the family Vi. 3.1 T h e o r e m ( C l a s s i f i c a t i o n ) . Any finite indecomposable one of the seven classes of loops listed in Table 3. Furthermore, are distinct:

PROOF.

loops in different

classes are not

RA loop is in these

classes

isomorphic.

We first prove, case by case, t h a t each of the families in Table 2

appears in Table 3. Afterwards, we settle the question of isomorphism. Remember throughout that s — tl^^^

; thus 5 = ^1 if mi = 1.

In our proof, we often make use of the following fact, which is a consequence of the LC property (see Corollary IV.2.3 and Theorem IV.3.1). Suppose t h a t a, b and v are any three elements of L which do not associate. Then no two (distinct) elements of {a,6,t;} commute and G — ( a , 6 , Z ( L ) ) is a nonabelian group with G' — V — {\,s)

and L — M(G,*,t;-^).

Clearly Li C £ 1 . If Z/2 G L2, then {ynf'

— st\.

If m i = 1, then 5 = ^1, so {yuf'

— 1

and, setting u' — yu., we have L2 = M((:r:, 2/,Z(Z/2)), *, i^' ) € >Ci. On the other hand, if mi > 1, let y' — yu. Then y'

= sti is an odd power of ti

so, by Lemma 2.1, there exists ^" such t h a t ?/" = ^i. Hence, in this case, L2 = M{{y",u,Z{L2)),^,x'^)

e L5. We shall soon see t h a t L5 C £1 u £ 2 .

If L3 G L3, then L3 = M{{x,u,Z(Ls)),^,y^)

e £3.

3. FINITE INDECOMPOSABLE RA LOOPS

141

2. Every finite indecomposable RA loop is in one of the classes described below TABLE

1

ZiD)

1

ih)

\

ih) ih) ih) ih) ih) ih) ih)

L2

La L4 L5 L5-

Le

p~ 5l

G

L«^ = 90

^1

i

1

Di

h

1

Di X ( w )

w

1

I>1 X {w)

h h h h

D2

1

D2

^1

D2 X {w)

w

D2 X ( w )

i 1

ih) X {t2) ih) X (i2)

t2

h

D3

u

t2

Dz X (w)

w

Lio

ih) X ih) ih) X ih)

t2

Dz X (w)

Lii

ih)

X (f2)

<2

D4

1

L„'

iU) X (i2)

<2

/)4

L,2

(fl) X (i2)

t2

1)4 X ( w )

h w

<2 1

Z)4 X {w)

L7

Is L9

Li2' I ( i l ) X (^2) Li3 1 ( f l ) X (<2) X (<3)

1 1

ih) X {t2) X ih)

i3

ih) X (i2) X (fa) Lie 1 ih) X (^2) X (ia)

^3

Ll4 L15

^2

h D5 X (w)

^3] D5

w

X (u))

Suppose i/4 G L4. If o(/i) > o[w)^ then s — t^^ — {l\w)^'^'' 6 (^i^^), x^^lp'^v? G {t\w) and i/4 = {x^y^u^t\w) X (t/;) is not indecomposable. If 0(^1) < o('i^), then Z{L^) - (ti) x (tiw) and Z/4 := M((a:, ?/, ^(14)), *, y^) G /:3.

Let Z/5 G L5. If TTii = 1, let x^ = xu and y' = yu. Then x' = t/' = 1 and L5 = M((a:', y', ^(Ls)), *,'U'^) G £1. If mi > 1, we set u' = xu. Then u' = sti and Lemma 2.1 shows that we can find u^^ such that n'' = ^1. So L5 = M ( ( x , y , 2 ( L 5 ) ) , * , w " ' ) e £ 2 .

142

V. THE CLASSIFICATION OF FINITE RA LOOPS

T A B L E 3 . T h e seven classes of finite indecomposable RA loops

1

Z{D)

1 1

(^i) ih)

L^4

{h)x

^2~

1

G

y'

u^ = go

1

1

1

^2

D3

h 1

^1

t2

Z?4

^1

D2

{t2)

]

i

<2

ih) X (i2) X (<3) ^2 C7\ [(«l) X (<2) X (^3) t2

h Ds X {w)

w

Certainly L5/ C £2 and, if Le is a loop in the class Lg, then Le = M{{x,u,Z(Le)),^,y^)eC4. If //g/ G Lg/, then m i = 1 and 5 = / i , so

Z{LQI)

= (ti)

X

{tiw) and it

follows t h a t Z/5/ = M((x,ii,Z(Z/6/)), *,?/'^) G £ 4 . Clearly L7 C £ 3 . Suppose Ls G Lg. Then (xt/)^ = sti. and, with u^ — xu^ Ls — M{{x^y^Z{Ls))^^^u^

If m i = 1, then (xu)'^

= 1

) G £ 3 . If m i > 1, set

x' = x?/. By Lemma 2 . 1 , there exists x " such that x" = ^1. Hence Lg =

M{{x\y,Z{Ls)),^y)eU If L9 G L9, then L9 = M{{y,u,Z{L^)),^,x'^)

G £5.

_ ^2"^iNext, suppose L\Q G LIQ. If 0(^1) > o(it;), then 5 = ^

2 ( L i o ) = (^1^) X (^2) X ( ^ ) and Lio = {x^y^u,t\w,t2) composable. If 0(^1) < o(tt;), writing t h a t Lio = M{{y,u,Z{Lio)),^,x^)

Z{L\Q)

xu.

x (ti;) is not inde-

— {t\) X (^2) X {t\w)^

we see

G £5.

If Lii G Lii, we have {xu)'^ — st\. x'^ = 1 and Lii = M{{x\y,

G (^iti;).

Z{Lu)),^,u^)

If mi = 1, let x' = xu. G £3. If ^ i

Then u^ = sti^ so there exists 1/" such that u"

Then

> 1, let u' =

= ^i. Hence Ln =

Mi{x,y,2iLu)),*,u"^)eC4. It is clear that L n ' C £4 a n d , if L12 is a loop in the class L12, then Lu = M{{y,u,Z{Lr2)),*,x^)

e £e.

We turn our attention to Li2> € \-\2'- Here m i = 1, so o ( / i ) < o{w), Z{Li2') = (ii) X («2) X (iiu;) and Lu' = M{{y,u,Z{Lu')),*,x^)

G A-

4. FINITE RA LOOPS OF SMALL ORDER

143

It is clear t h a t L13 C £ 5 , L14 C CQ and L15 C £ 7 . Finally, suppose t h a t (h) X (^2) X (^3) X (tiw)

L\Q

G

Lie- If 0(^1) < o(tt;), we write

Z{LIQ)

and see t h a t Lie = M{{x,y,Z{Lie)),^,v?)

If o ( t i ) > o(it;), then L\e = {x^y^u^tiw^t2',t^)

=

G £7-

x {w) is not indecomposable.

AU the above shows t h a t , up to isomorphism, there are at most seven classes of indecomposable RA loops, those classes denoted £ 1 , . . . , £ 7 in Table 3. Arguments similar to those of Section 2 show t h a t the loops in this table are aU indecomposable. Finally, we show t h a t loops in different classes are not isomorphic. Elementary consideration of the ranks of centres shows t h a t we need only prove t h a t £1 n £2 = 0, £ 3 n £4 = 0 and £5 D £5 = 0. If m i > 1, these statements are true because s and hence every element in the centre of a loop Li G £ t , i = 2 , 4 , 6 , is a square, while ^1 is not a square in a loop of £ 1 , £3 or £ 5 . If m i = 1, Li G £1 and L2 G £ 2 , then Li ^ Mi6(Q8,2) and L2 = ^leiQs)')

so Li ^ //2. (The loop Mie(Qs)

while there are nine elements of order 2 in

has a unique element of order 2 Mie{Qs^2).)

Suppose mi = 1, L3 G £ 3 and L4 G £4. Then 2(Z/4) = (ti) X (^1^2)- In Z/4 = L4/{tit2),

we have x'^ = u'^ = y'^ = ti and hence L4 = Mie{Qs)-

On

the other hand, all nonabelian quotients of order 16 of L3 are isomorphic to Mie{Qs',2)

because, in any such quotient, we have ^'^ = u'^ = 1 while, in

^leCQs)? the only element of order 2 is central. Similar arguments apply in the last case, m i = 1. If Le G £ 5 , then Le = Le/{{tit2)

x (^1^3)) — ^wiQs)

while, in any quotient of L5 G £5 by a

central subloop, u has order 2.

D

4. F i n i t e R A l o o p s of s m a l l order Table 3 provides a means of determining all indecomposable RA loops of a given order, up to isomorphism. Those RA loops which are not indecomposable are, as we have seen, direct products of an indecomposable RA loop with an abelian group. As an illustration, we enumerate all the RA loops of order 64 or less. In what follows, we assume previous notation. T h u s , when we write an RA loop L as M(G^ *?yo)9 we understand t h a t G is a nonabelian group with

144

V. THE CLASSIFICATION OF FINITE RA LOOPS

a unique nonidentity commutator, 5, go is a central element of G and * is the involution on G defined by

\

if sg

geZ{G)

otherwise.

If L is indecomposable, then G = D x A^ where D is an indecomposable group with D/Z{D)

^ C2 X C2 and A is abelian. If \L\ < 64, then \Z(G)\

<

8, so it is clear from Table 3 t h a t A must be trivial; hence G = D is indecomposable. Moreover,

with o(/^) = 2^^*, mi > 0, m 2 , m 3 > 0. The unique commutator of G is s = tf^^

, which is in the first factor. Therefore, reordering of the second

and third factors leads to isomorphic groups. Once again, we employ the notation of M. Hall Jr. and J. K. Senior [HS64] to denote the groups involved in our construction and label the Moufang loops which arise with the notation of 0 . Chein [ C h e 7 8 ] . T h e smallest possible order for an RA loop is 16. In this case, \G\ = 8, so G = D4 OT G = Qs' There are two loops, both indecomposable: 1. M ( D 4 , * , l ) = M i 6 ( g 8 , 2 ) a n d 2. M ( g 8 , * , ^ i ) = MieiQs).

the Cayley loop.

If an RA loop has order 32, its centre, being of order 4, must be either C4 or C2 X C2. If Z{G) \i x^ = y^ = tuG

= C4 and x^ = y^ = 1, we have G = 16r26, while

= I6r2d.

If Z{G) = C2 x C2, the choice x^ = l,y^

gives G = 16r2Ci, while the choice x^ — t\^y^

= t2

— ^2 gives G — 16r2C2. The

corresponding indecomposable RA loops are 1. M ( 1 6 r 2 6 , * , l ) = M32(£^t,16), 2. M ( 1 6 r 2 ( i , * , / i ) - M 3 2 ( 5 , 5 , 5 , 2 , 2 , 4 ) , 3. M(16r2Ci, *, 1) - M32(16r2C2,16r2C2,16r2Ci, 16r2Ci) and

4. M(i6r2C2,*,/i) = M32(i6r2C2, i6r2C2, i6r24. i6r24). T h e only other RA loops of order 32 are direct products. 5. M32(C,9) = Mie{Qs) X C2 = M ( g 8 , * , ^ i ) X C2 = M{QsX C2,*,(/i,l))and 6. M32(g8 X C2,2) = M32{Q8 X C 2 , * , ( l , l ) ) = M{Qs,^,l) Mi6(g8,2)xC2.

XC2

=

4. FINITE RA LOOPS OF SMALL ORDER

145

4. The indecomposable RA Loops of order less than or equal to 64 TABLE

\w1

p"

y'

G

1

1

D4

z

1 161

i(G) C2

16 j

C2

h

h

Qs

32

C4

1

1

32

C4

h

h

16126 l6T2d

M{lQT2d,*,ti)

1

t2

16r2Ci

M(16r2Ci,*,l)

h

<2

16r2C2

M(16r2C2^*,<0|

32

C2XC2

32 [_

C2XC2

M(Z)4,*,1) _ MiQs,^Ji)

\

M(16r26,*,l)

[64 1

Cs

1

1

32r2ff

M(32r2^,*,i) 1

64

h

u

32r2/i

M{32T2h,*,ti)

64

Cs C4 XC2

1

t2

32r2/

64

C4XC2

h

t2

32r2J

M(32T2f,*,l) M{32T2i,*,h)

64

C2XC4

1

<2

32r2ii

M(32r2i,,*,l)

ti t2

h h tz

32r2i2 32r2A;

M{32T2J2,*,ti) M{32T2k,*,l)

32r2fc

M{S2r2k,*,ti) 1

64 64

C2 XCA C2XC2X C2

64 1C2XC2X

C2

h

The enumeration of RA loops of order less than 64 is complete after the addition of two loops of order 48, these being necessarily direct products: 1. M48(7,7,7,2,2,6) = M^eiQs) x C3 - M{Qs x C3,*,/i) and 2. M48(7,7,7,2,4,6) - M^eiQs.2) x C3 = M{D^ x C3,*,^i). If \L\ = 64, then |>Z(i/)| = 8 and 2(G) must be isomorphic to either Cg, C4XC2, C2XC4 or C2XC2XC2. We obtain seven groups. When 2^(G) = Cg, the choice x^ = y'^ = I gives G = 32r2^ and the choice x'^ = y'^ = ti gives G = 32T2k. With Z{G) = C4XC2, we obtain the group 3 2 r 2 / for the choice a;^ = 1, 2/^ == ^2 2ind the group G = 32r2i for the choice x'^ = t j , y^ = ^2Similarly, Z{G) = C2 X C4 leads to G = 32r2ii and 32r2i25 respectively. Finally, when Z{G) = C2 X C2 X C2, setting x^ = ^2 2ind y'^ = ts yields G = 32T2k.

146

V. THE CLASSIFICATION OF FINITE RA LOOPS

There are eight indecomposable RA loops of order 64, as specified in Table 4. The remaining (eight) RA loops of order 64 are the direct products of Mie{Qs',2) and Mie{Qs) with C2 X C2 and with C4, and the direct products of the four indecomposable RA loops of order 32 with C2.

Chapter VI

The Jacobson and Prime Radicals

In this chapter, we describe the Jacobson and prime radicals of loop rings of RA loops. For alternative loop algebras over fields, we derive a version of the well known theorem of H. Maschke for group algebras and determine conditions under which an alternative loop algebra is the direct sum of simple algebras. Since these simple algebras possess a nondegenerate quadratic form, each is a composition algebra whose structure is therefore known. In our proofs, it is often convenient to work with rings t h a t do not necessarily contain a unity. Therefore, throughout this chapter, except where specifically mentioned, we work with arbitrary rings which may not contain a unity. For any ring A and natural number n, we denote by A'^ the linear span of all possible products, with any arrangement of parentheses, of n elements from A. An element x in a power associative ring A is said to be if a;^ = 0 for some n > 1. An ideal I of A \s nil nilpotent and nilpotent is semiprime

nilpotent

if each of its elements is

if / ^ = 0 for some natural number n > 1. A ring

if it contains no nonzero ideals / with P = 0 and prime

if

IJ zr {0} for ideals / and J implies / = {0} or J = {0}. An ideal / of a ring A is prime if the ring A/1 is prime. If A is an alternative ring and / is an ideal of A^ then I^ is also an ideal oi A. To see this, let x G P , y G / ^ , where j and k are positive integers with j + k = n. Assume inductively t h a t P and I^ are ideals and let a be any element of A. Recalling t h a t [x^y^a] denotes the associator of x, y and a and t h a t the associator is an alternating function of its arguments, the

147

148

VI. THE JACOBSON AND PRIME RADICALS

calculation xy ' a = X - ya + [x^y^a] = X ' ya — [x^a^y] = X ' ya — xa • y + X ' ay^ shows that I^ is a right ideal. Similarly, one can show that /^ is a left ideal. Since A^ = 0 implies A^'^^ = 0 for any ring A, it follows that an alternative ring is semiprime if and only if it contains no nonzero nilpotent ideals. Let ^ be a class of algebras (over a given commutative ring) which is closed with respect to ideals and homomorphic images. A subclass TZ of A is called a radical (class) if 1. 72. is homomorphically closed; 2. each A £ A contains an ideal TZ(A) £ TZ which contains all the ideals of A which, as algebras, are in 72.; 3. n{Ain{A)) = {0} for all A 6 >t. An algebra ^ G -4 is called 72-semisimple if TZ{A) = {0}. In the class of alternative rings, we consider three radicals: P , the prime (or Baer) radical, A/i7, the upper nil (or Kothe) radical and J^ the Jacobson radical, which is also known in the literature as the Smiley, Zhevlakov or Kleinfeld radical. We recall the definitions and some basic properties of these radicals, referring the reader to [Div65] and [ZSSS82] for proofs. The prime radical V{A) of an alternative algebra A is the smallest ideal of A such that A/V{A) is semiprime. It turns out that V{A) is the intersection of all prime ideals of A. The upper nil radical Mil{A) of A is the largest nil ideal of A. It is a basic fact that V{A) = J^il{A) is the set of nilpotent elements if A is commutative and associative. An element a £ A\s called quasi-regular if there exists an element b £ A, called the quasi-inverse of a, such that ab = ba = a + b. An ideal is said to be quasi-regular if aU its elements are quasi-regular. A left ideal / of A is said to be modular if there exists an element e £ A such that ae — a £ I for all a G A. The Jacobson radical J{A) of A is the largest quasi-regular ideal of A. This ideal is also the intersection of all the maximal modular left (or right) ideals of A. If A has a unity, then any left ideal is modular, so J{A) is the intersection of all maximal left ideals. The three above-mentioned radicals are related as follows: V{A) C Mil[A) C

J{A),

1. AUGMENTATION IDEALS

149

Furthermore, these radicals are hereditary; that is, TZ{I) = Ir\TZ{A) for any (two-sided) / of A, where TZ is any of the three radicals P , A/zV, J, (See, for example, [ZSSS82, Theorem 9, p. 167 and Corollary, p. 207].) Let L be an RA loop and R a commutative and associative ring. To determine a radical of the alternative loop ring RL^ we may often assume that R has a unity. We proceed to show why. By R^ we denote the natural extension of i2 to a ring with unity. Thus R^ = Rif R already has a unity; otherwise, R^ = RxZ = {{T^Z) \ r e R^ Z E Z } , with componentwise addition and multiplication defined by (^i, Zi)(r2,>2^2) = (^1^*2 +-^^1^2 + >2:2ri,>2ri2r2), for ^1? ^2 € R and zi^Z2 G Z. In particular, R^/R = Z. Since Z is semisimple for any of the three above-mentioned radicals 72., it follows that TZ{R) = TZ{R^),

1. Augmentation ideals Let i2 be a commutative and associative ring and let L be a (possibly associative) Moufang loop. For a normal subloop TV of L, the canonical map L -^ L/N lifts to an i2-linear ring homomorphism €f\j : RL -^ R[L/N] whose kernel we denote by An{L^N). If R is clear from the context, then we denote this ideal simply A(Z/,7V). In the special case that N = L^ the homomorphism €L:

is called the augmentation map

RL^

R

and denoted simply €. Thus, for a =

The scalar €(a) is called the augmentation of a. The kernel of €, which is called the augmentation ideal^ is denoted A(Z/) rather than A(L,Z/). For rings R without unity, we will sometimes abuse notation by writing rx — ry as r{x — y), and r{xy^ as {rx)y^ where x^y £ L and r ^ R. This should not create any confusion as both notations agree in R^L, We refer the reader to §11.5.2 where the notation M(G, *,^o) was introduced and recall that RA loops are precisely loops of this form (see Theorem IV.3.1).

150

VI. THE JACOBSON AND PRIME RADICALS

1.1 Lemma. Let L be a (possibly associative) Moufang loop whose loop ring RL is alternative and let N be a normal subloop of L. Then

neN

neN

IfL = M(G^ *55^o) i^ «^ R^ loop and N is a normal subloop of L contained in G, then A(L, N) = A(G, TV) + A(G, N)u. Let a = Y^i^iOtii G AR(L,N)^ where each a£ G R. Denote by a and I the images of a and ^, respectively, in the loop ring R[L/N]. Writing L = U ^ e T ^ ^ ' ^^^ some transversal T of TV in L, we have PROOF.

and so, for each x E: T,

£eL/=x

Since (ix~^)x — i for any x^i ^ L (recall that a Moufang loop is an inverse property loop—Theorem IL3.1), it follows that V^

OLii —

i^Lj=x

2_.

{o.iix~^)x

—

£eL,iz=x

2_.

OLiX —

£eL/=x

2_\

OLi{^ix~^ — X)X

£eL,I=x

and, for similar reasons, that y^

a^t, =

£eL,i=x

2_.

a£x{x'~^f,)—

£eL/=x

2_\

OLix — 2_.

£eL,£=x

OLix[x~^t — V),

£eL,£=x

So

^^^

££L/=x

^^^

£eL/=x

Since N is normal, ix~^ and x~^i e N fori =x^ thus aG^(l-n)iiL

and

a G ] ^ i?L(l - n).

neN

So A(L,iV) C Y.neNRH'^ - ") and AiL,N) reverse inclusions are obvious.

neN

C EneAr(l " " ) ^ ^ - The

1. AUGMENTATION IDEALS

151

For the last statement, suppose that L = M(G^ *5 5'o) is an RA loop and that a subset TV of G is normal in L, Since L = G U Gu^ for some u £ L, it is clear that RL = RG + RGu^ and so A(L,7V)= ^{l-n)RL=

^ ( l ~ n ) i ? G + ^{l

neN

neN

= A(G,N)

+

-

n)RGu

neN

A{G,N)u,

The result follows.

D

Now assume that R has a unity. For any finite subloop TV of a Moufang loop L with I TV I invertible in i2, we define the elements TV and TV of RL by TV = y

n

and

TV = r^TV.

neN

'

'

1.2 Lemma. Let R be a commutative and associative ring with unity and let L be a group or an RA loop. If N is a finite subloop of L, then (i) T V ^ ^ |TV|TV;

(ii) TV is central if and only if TV is normal in L; thus, if L is an RA loop with L' = {l,'^}, then N is central if and only if s £ N or N is central. Furthermore, 2/|TV| is invertible in R, (iii) N'^ = N is an idempotent in RL; (iv) if N is normal in L, then {RL)N = R[L/N]; (v) if TV is normal in L, then RL = {RL)N @ RL{\ - N) and RL{1-N) PROOF.

=

AR(L,N),

Clearly nTV = TV for any neN.

Therefore TV^ = |TV|TV, which

is (i). Recall that if L is an RA loop, the map I i-^ i'^ given by i

if ^ is central

si

otherwise

= <

152

VI. THE JACOBSON AND PRIME RADICALS

is an involution.

Moreover, a € RL is central if and only if a* = a or,

equivalently, if and only if a£ = ia for all ^ G Z/. (See §111.4.) We also remind the reader t h a t the test for normality of a subloop of an RA loop is the same as it is for groups, by Corollary IV. 1.11. For the first part of (ii), suppose t h a t N is central. Then, for aU x G i/, N = x~^Nx

= SnGA/^^"^^^' ^^ x~^Nx

= N and N is normal. Conversely,

if TV is normal, then xN = Nx for all x G L, so iV is central. Suppose L is an RA loop. It is clear t h a t if either N is central or 5 G A^, then A* = A , so A* = A and A is central. On the other hand, if A is central but A is not, then there exists n G A such t h a t n* = sn.

The equation A* = A

implies t h a t sn E N^ hence 5 G A . Now suppose t h a t | A | is invertible in R. Clearly (iii) is an immediate consequence of (i). For (iv), suppose A is normal. Then, for any ^ G i>,

where 6 = —^u^

SHGNC^

— n) ^ A ( / / , A ) . It follows t h a t the maps defined

by

are isomorphisms {RL)N

->

RL/AR{L,

A ) -^

R[L/N].

For (v), we note t h a t 1 = A + (1 — A ) i s a decomposition of 1 as the sum of orthogonal central idempotents in RL and so RL = {RL)N

Q RL{1 — A ) .

Clearly RL(l

- N) = RL{\N\ - N)cY^

RL{1 - n) =

AR{L,N).

Since, for any n G A , (1 - n ) ( l - A ) = (1 - n) - (1 - n)N = {I - n) - N + nN = I - n, we obtain RL{1 - N) = AR{L, A ) .

D

As a special case of Lemma 1.2, let L = M{G^ *?5'o) be an RA loop and let N = V = G' = { 1 , 5 } . Then A = i ^2 ,, 1 1 - ^^ A — = i 2^ and we have the following.

2. RADICALS OF ABELIAN GROUP RINGS

153

1.3 Corollary. Let R he a commutative and associative ring with unity and let L = M(G,*,^o) be an RA loop with L' = {l^-^}. If 2 is invertible in R and H = L or G, then the centre of RH is Z{RH) Furthermore, Z{RH^)

= RH^^

=

e

Z{RH^),

R[Z{H)]^,

Because of Lemma 1.2, RH^^ = R[HIH'] is commutative; therefore the first statement is clear. For the second, it is sufficient to show that Z(RH^^) C R[Z(H)]^Y^ since the other inclusion is obvious. So assume z G Z{RH^). Then z € Z{RH) and so, by Corollary III.1.5, z z=i ZQ-\-{\-\- s)zi where ZQ has support in Z{H). Thus z = z^^ = ^o^^ € PROOF.

R[ZiH)]'-^.

n 2. R a d i c a l s of a b e l i a n g r o u p rings

Throughout this section, /2 is a commutative and associative ring. The polynomial ring over R in n commuting variables X i , . . . ,Xn is denoted R[Xi^... ^Xn] and the Laurent polynomial ring in these variables is denoted R[Xi^ ^ r \ • • • ? ^n-) ^n^]' If ^ is a free abelian group of finite rank n / 1, then it is easily seen that RA = R[Xi^X^^^... ^Xn^X'^]. 2.1 Lemma. / / R has a unity and if f — ro + riX -\ h r^X'^ 6 R[X] is a unit, then ro is a unit and ri is nilpotent, i = 1 , . . . , n. Since the prime radical V{R) is the intersection of the prime ideals of R and because R[X]IP[X] ^ {RIP)[X] for any ideal P of R, we may assume that 72 is a domain. Let g = a^ -\- a\X -\f- amX'^ 6 R[X] be the inverse of / . Then PROOF.

1=

Y.

r,a,X'+^.

0
Since i2 is a domain, it follows that r^ = ^n-i = - - - = vi = 0 and that ro is invertible. D 2.2 Proposition. JiR[Xi,...

, Xr,]) = TiR)[Xu

...,Xn]

= V{R[X^,...

, X„]).

154

VI. THE JACOBSON AND PRIME RADICALS

Since R[Xi^... ,Xn] is an ideal in R^[Xi^... the Jacobson and prime radicals are hereditary, PROOF.

JiR[Xr,...

, Xn]) = JiR\Xi,...

,Xn] and because

, Xn]) n R[Xr,

...,Xn]

,Xn]) n R[Xi,...

,X„].

and ViR[Xi,

...,Xn]}

= ViR'lXt,,..

Thus we may assume that R contains a unity. We first deal with the case n = 1. Let J = J{R[X]), li J ^ {0}, let / be a nonzero element of J. Since 1 — fX is invertible in i2[X], Lemma 2.1 implies that each coefficient of / belongs to V{R). Hence J{R[X]) C V{R)[X]. Since V{R) and hence V{R)[X] are sums of nilpotent ideals, we obtain that J{R[X]) = V(R)[X] =

V{R[X]). Since R[Xi^... follows.

^Xn] = {R[Xi^...

^Xn-i])[Xn]^

the general case now D

2.3 Corollary.

j{R[XuX-\...,Xr,,x-'])

=

v{R)[x,,x:^\,,.,Xn^x-'] =

PROOF.

V{R[XuX-\..,,Xn.X-']).

Again we may assume that R has a unity. Let

Ri = R[Xi^...

^Xn]

and

R2 = R[Xi^X^

, . . . , X n , X ^ ].

For every d G i?2^ there exists a monomial a = X^^ • • • ^ ^ " , each mi > 0, such that ad £ R\, Since each such a is invertible in 7?2? it follows that I = R2{I r\ Ri) foT every ideal I of R2. Also note that, if M is a maximal ideal of /?i, then R2M is a maximal ideal of R2 if M does not contain any of the variables Xi] otherwise, R2M = i?2Let d G J{R2)' Then, for any maximal ideal M of /?i, there exists a monomial a = X'!^^ • • • X^"^, each rrii > 0, so that ad G RiXi • • - Xn. Clearly ad G M if M contains one of the variables. On the other hand, if M does not contain any variable, then ad G R2M 0 R\. Since M is also prime, R2M H Ri = M, so ad G M. Since M is an arbitrary maximal ideal of /2i, we have shown that ad ^ J{R\), Thus ad e J{Ri)

= V{R)[X^,,.,

,Xnl

2. RADICALS OF ABELIAN GROUP RINGS

155

by Proposition 2.2, so

de-PiR)[X^,X{\...,Xn,X-']. Since V(R)[Xi^X^^^...

J(R2) =

,Xn,X^^] is a sum of nilpotent ideals, we obtain

ViR)[Xi,Xr\...,Xn,X-'] = PiR[Xr,X{\...,Xn,X-']).

D

To determine the radical of an arbitrary group ring RA of an abelian group A, we first consider finite groups. We need several results which we state as propositions. Recall that a commutative and associative ring T is said to be an integral extension of a subring i2, if each element x G T is integral over R in the sense that x satisfies a monic equation X^ + n x ^ - ^

+ • • • + Vn-lX + rn = 0

with each coefficient n G R. 2.4 Proposition. Let A be a finite abelian group. Then J{R) R andV{R) = V{RA)nR.

—

J{RA)^

P R O O F . We may assume that R has a unity. Since J{RA) fl /2 is an ideal of R and because an element of R is invertible in RA if and only if it is invertible in 72, it is clear that J{RA) 0 R C J'(R). To prove the reverse inclusion, it is sufficient to show that, if M is a maximal ideal of RA^ then J{R) C M D R, For this, note that each element a £ A satisfies al^l = 1. It follows that the field F = RA/M is an integral extension of the domain D = R/{R 0 M). Consequently, if 0 7^ cf 6 /?, then, for some d i , . . . ,dn e D, n> I,

{d-^Y + d^{d-^Y-^ + . . . + dn-l{d-^) + rfn = 0. Hence {di +--' + dn-id''-^+ dnd^-^)d ^ -l.sod-^ e Di Thus i) is a field, so i? n M is a maximal ideal of R. Consequently J{R) C M r\ R, Since V{R) is a nil ideal, so is V{R)A. Hence V{R) C V{R)A H R C V{RA) n R. The inclusion V[RA) DRC V{R) is also clear since V{RA) n R is a nil ideal of R, D

156

VI. THE JACOBSON AND PRIME RADICALS

2.5 Proposition. Let A be a finite abelian group. If r = ^aeA^a^ ^ TZ{RA)j then \A\ra G TZ{RA) for each a ^ A, where Tt is either J or V. In particular \A\ra G IZ{R), Again we may assume that R has a unity and first consider the Jacobson radical. Because of Proposition 2.4, it is sufficient to prove the statement for {R/M)A where M is a maximal ideal of /?; thus we may assume that 72 is a field. Let p be the characteristic of R, li p divides |^|, then \A\r = 0 for any r ^ R and the result is obvious. So we assume p ]^ |^|; hence, either p = 0^ or p > 0 and A does not contain elements of order p. We consider the latter case first. Assume p > 0 and that A does not contain elements of order p. We have to show that J(RA) = {0}. Since RA is a finite dimensional algebra over the field R, J{RA) = V{RA), hence it is sufficient to show that V{RA) = {0}. Let a = YlaeA '^^a e V{RA), Then a^ = 0 for some ^ > 1 and thus a^^ = 0 for some q with p^ > L Since pR — {0}, PROOF.

0 = a^' =

Y,{raay\ aeA

Because a^"^ ^ 1 for a / 1, it follows that r^ is nilpotent and thus r\ — 0. Since aa~^ is also nilpotent, what we have just shown yields that r^ — ^ also. Thus a = 0, so V{RA) = {0}. Now assume that p = 0 and let R denote the algebraic closure of R. Clearly

V{RA) =

V(RA)r\RA,

so, to prove V{RA) = {0}, we may assume that R is algebraically closed. Since char i? = 0, it is known (see, for example, [Coh77, Exercise 8, p. 271]) that R is obtained from a real closed field F by adjoining an element i with i'^ = —1. For any element r = / i + i/2 6 R^ /15/2 € F,, write r = fi — if2. It is then easily verified that the map RA —^ RA defined by

a^A

a^A

is a ring involution. Now the coefficient of 1 in aa is ^^eA ^^^a > 0. Since F is formally real, it follows that a = 0 if and only if a a = 0. Suppose fi e RA and /J^ =:. 0. Let a = /3^. Then a^ = a a = 0 too, so a = 0.

2. RADICALS OF ABELIAN GROUP RINGS

Therefore /3 = 0. Thus RA that is, V{RA) = {0}. Finally, we consider the prove the result for {R/P)A^ assume that i2 is a domain. V{RA) C V(KA) = J{KA),

157

does not contain nonzero nilpotent elements; prime radical and note that it is sufficient to where P is any prime ideal of R, Thus we may Let K denote the field of fractions of R. Since the result follows from the first part. D

For an ideal I oi R and a positive integer n, let In be the set of all elements r E R with nr G / : In = {r e R\nr

e /}.

Clearly In is an ideal of R and, if / is semiprime, then In^ = In for any m > 1. We denote by P the set of (positive) prime numbers. For any Moufang loop L and p G P, by Lp, we mean the set of elements in L whose order is a power of p and, by Lp/, the set of elements of order relatively prime to p. Clearly, if L = yl is an abelian group, both Ap and Apt are subgroups of A, In what follows, we shall make use of the following obvious, but useful observation. If a Moufang loop L is the direct product H X K^ then the loop RL of L over a ring R can be viewed as {RH)K^ the loop ring of K over the coefficient ring RH. 2.6 P r o p o s i t i o n . Let A he a finite abelian group. Then

n{RA) = n{R)A + Y^ n{R)pA{A, Ap), where TZ — J orV.

Furthermore^ IZ{RA)/TZ{R)A

is nilpotent.

Let TZ = J or V. We prove the result by induction on the number of primes dividing the order of A, If A is the trivial group, then the result is obvious. Assume that yl is a (nontrivial) p-group. Because of Proposition 2.4, 7^(i^) C n{RA). Since RA/TZ{R)A ^ {R/n{R))A and because TZ(R/TZ{R)) = {0}, we may assume that TZ{R) = {0}. Write Z = {0}. By Proposition 2.5 and because R is semiprime, TZ{RA) C ZpA. Clearly ZpA is an ideal of RA. Hence, since 72. is hereditary, TZ{RA) — n{ZpA). Now notice that for any a e A, (Zp(l - a))P" C pZpA = {0} for some n > 1. It follows that Azp{A) is nilpotent and thus Azp{A) C PROOF.

158

VI. THE JACOBSON AND PRIME RADICALS

7Z(ZpAp), As ZpA/Azp(A) = Zp and because TZ = {0} and thus also TZ{Zp) = {0}, we obtain that TZ{RA) = Azp{A), as required. For the general case, assume j9 is a prime dividing n = |A|. Since A is finite, we have A = Ap x Api, so RA = {RAp/)Ap is the group ring of Ap over the ring RApf, The induction hypothesis yields 7^(i^A) = n{RAp,)Ap

+ {n{RAp,))^

ARA^MP).

By Lemma 1.1, we know that ARA ,{Ap) = AR{A^AP). Furthermore, if a e {TZ{RAp>)) , then a = /3i + /32, where /Ji = v\Apf\a and /?2 = wpa for some integers v and w. Then /?2 € 7Z(RApf) and pPi G TZ{R)A by the induction hypothesis. Hence a G TZ(RApt) + {TZ{R))pA. The induction hypothesis, applied to RApf^ yields the result. D 2.7 T h e o r e m . Let A be an abelian group. Define V{R) = J{R) torsion and V(R) = V{R) otherwise. Then

if A is

1. (May [May76]; J{RA) = V{R)A + ZpeP V(/2)pA(^, Ap). 2. (Karpilovsky [Kar82]^ V{RA) = V{R)A + Ep6P nR)v^{A,

Ap).

Let TZ be either J or V and first consider the case that A is finitely generated. Write A = At x Aj., where At is the (finite) torsion subgroup of A and Aj is a free abelian subgroup of A., say of rank n. If n = 0, then TZ{RA) has the required form by Proposition 2.6. Now assume n > 1. Since RA = {RAt)Af^ we have PROOF.

RA^(RAt)[XuX^\...,Xn,X-']. Corollary 2.3 implies that TZ{RA) = V{RAt)Af which, by Proposition 2.6, has the required form. For an arbitrary abelian group A, the foregoing implies that TZ{RAi) C TZ{RA2) for finitely generated subgroups Ai C A2 of A. Since A = [jAi^ where the union runs through the finitely generated subgroups Ai of ^ , we have 7Z{RA) — [jTZ(RAi). This fact is easily verified for the prime radical and, for the Jacobson radical, it follows from J{RA)r\RAi C J(RAi). (See the proof of Lemma 3.2.) The result follows. D

3. RADICALS OF LOOP RINGS

159

3, R a d i c a l s of l o o p rings Throughout this section, L = M(G, *9^o) is an RA loop with centre 2 and i2 is a commutative and associative ring, not necessarily with unity. Recall that L can be written L — G \J Gu and that G — {Z^x^y) is generated by Z and two elements x and y, (See §11.5.2 and Theorem IV.3.1.) 3.1 Lemma. Let TZ be either the Jacobson or prime radical and let H be L or G. Then n{R[Z(H)]) C 11{Z{RH)) C n{RH). The result is obvious for the prime radical. The Jacobson radical is hereditary. Furthermore, R[Z[H)\ is an ideal in R^[Z{H)], Z{RH) is an ideal in Z(R^H) and RH is an ideal in R^H. It follows that J{R[Z(H)]) ^ J(R^[Z{H)]) 0 R[Z{H)], J{Z{RH)) = J{Z{R^H)) n Z{RH) and J{RH) = J{R^H) n RH, Hence, to prove the result, we may assume that R has a unity. Let a e J{Z{RH)) and a G RH. Let t{a) rr a + a* and n{a) = aa^ denote the trace and norm of a (see §111.5) and recall that both t{a) and n(a) are central in RH. Then t{a)a — n{a)a^ is in J{Z{RH))^ so this element is quasi-regular. Thus 1 — t{a)a + n(a)a"^ is an invertible element in 2 ( / 2 / r ) , with inverse 6, say. Let d = ba'^n^a) (so that d + b — bt(a)a = 1) and / = —baa + d. Remembering that a^ — t[a)a — n{a)\ and that a, b and d are in the centre of RH ^ we have PROOF.

(aa)f — —b(j?o? + aad — aad — ba^t(a)a + d = aa{d — t{a)ba) + d = aa{l — b) + d = aa + f. Thus aa is quasiregular, with quasi-inverse / . Hence aRH is a nonzero quasi-regular ideal of RH, so aRH C J{RH) and a G J{RH). Thus J{Z{RH))C J{RH). If, in the above, a G J{R[Z{H)]) and a G Z{RH), then aa has quasiinverse / G Z{RH). So J{R[Z{H)]) C J{Z{RH)). U 3.2 Lemma. Le^ H = L or G, let Z = Z{H) and let S be a subloop of H. Then 1. J{RH)nRS 2. J{RH)nRZ 3. ViRH)nRZ

CJ{RS). = J{RZ), = V{RZ).

160

VI. THE JACOBSON AND PRIME RADICALS

Suppose a 6 RS is quasi-regular in RH^ say with quasi-inverse /?. Write /? = /?! + /J2, where supp(/?i) C S and supp(/32) 0 5 = 0. Then PROOF.

ap = a + p and, extracting only the terms with support in 5 , we obtain a/3i z= a + f3i.

Thus a is quasi-regular in RS. Hence a G i?*? is quasi-regular in RS if and only if a is quasi-regular in RH. Since J{RH) H JR5 is an ideal in RS, it follows that J{RH) n RS C J{RS). This proves the first part. By Lemma 3.1, the second part is now also clear. We now turn to the prime radical. Because of Lemma 3.1, V{RZ) C V{RH) n RZ, To establish the reverse inclusion, we show that if a G V{RH) n RZ, then a ^ P, for any prime ideal P of RZ, For this, it is sufficient to show that if P is a prime ideal of RZ, then there exists a prime ideal Q of RH with Q n RZ = P, So let P be a prime ideal of RZ, Then {RL)P = Yl/teT^^-) where T is a transversal for Z m L and 1 G T. A support argument yields that

C^tP)

n RZ = P,

teT

and thus {RL)P D RZ = P. Hence there exists an ideal Q of RL which is maximal with respect to the condition Q D RZ = P. We claim that Q is a prime ideal, therefore completing the proof of 3. Indeed, suppose / and J are ideals of RL properly containing Q. By construction, there exist a, 6 G RZ \ P with a E: I and b £ J, Since P is prime in RZ, ab ^ P, thus abiQ, So IJ % Q. D 3.3 Lemma. Let L = M(G,*,po) f>^ «^ R^ loop with centre Z. L = GuGu andG = {Z, x, y). Let TZ be J or V. If

Write

7 = (ao + OLix -\-ay + a^xy) -f (/?o + P\x -\- /?22/ + fi3xy)u G TZ{RL), with ai,f3i G RZ, then 8a,-,8/3,- G n{RZ)

for i = I,,..

,4. If

7 = ao + a i x + a2y + OL^xy G TZ{RG), thenAai G n{RZ)

for i = 1 , . . . ,4.

3. RADICALS OF LOOP RINGS PROOF.

161

Consider the map / i : RL -» RL defined by a -^ I3u \-^ a — fin

where a^/S ^ RG. It is readily verified that f\ is a ring isomorphism. By Theorem IV.3.1, we also have L ~ M(G2,*,y^) = G2 ^ G2y and L = M(G'3,*,a;^) = G3UG3X, where G2 = {Z{L),x,u) and G3 = {Z{L),y,u). So we also obtain the following isomorphisms RL —> RL: f2'- 7 + Sy^

J - 6y

h'- V + C,x \-^ T] - C,x where 7, ^ G RG2 and ?/, C ^ RG3. Let 7 = (ao + Oi\x + a2y + a^xy) + (/3o + Aa: + ^2y + P3xy)u G Tl{RL), where a^-, A G i22, i = 1 , . . . ,4. Since both the Jacobson and prime radicals are invariant under isomorphisms, it follows that 7 + / i ( 7 ) = 2(ao + c^ix + a2y + a^xy) G Tl{RL) and also that 7 ^ " ' + h{iu-^)

= 2(Po + Pix + f32y + p3xy) G TZ(RL).

Using a similar argument with the map /2, we obtain 4(ao + a i x ) G 7^(i^L)

and

4(^2 + aax) G 7^(i^L)

4(/Jo + Pix) G 7e(i2L)

and

4(/32 + /?3a:) G 7^(i^L).

and also

Repeating the above argument with /a, it follows that 8ai,8/Jt G TZ{RL) for i = 1 , . . . ,4. By Lemma 3.2, 8a,-,8/3, G n{RZ). The final statement, concerning TZ{RG)^ is proved similarly. D Let Z/ be an RA loop or a nonabelian group contained in an RA loop. By Proposition V.1.1, the set Lp of elements of L of p-power order, is a normal subloop of L; in fact, with p odd, Lp is even central. 3.4 Theorem. Let V{R) — J{R) V{R) otherwise. Then

if G (and thus L) is torsion and V{R) —

1. J{RL) = V{R)L + Y:^^^^{R)V^{L,L^). 2. J ( f l G ) = V(il)G+EpGpV(i?)pA(G,Gp). 3. V{RL) = mi{RL) = V{R)L + EpeP^(^)pA(Z,, L^).

162

VI. THE JACOBSON AND PRIME RADICALS

4. r{RG)

= m{RG)

In particular^ if R/V{R) n{RG) + n{RG)u.

= TiR)G

+^^^pV{R)j>A{G,Gp).

has no 2-torsion or if L2 = G2, then 1Z{RL) =

Let H = L or G. By Theorem 2.7, J(R) C J{R[Z{H)]), (Note that H is torsion if and only if Z{H) is torsion since squares in H are central.) Therefore, by Lemma 3.2, we have V{R) C J[RH), Replacing RH by RH/V{R)H ^ (R/V{R))H, if necessary, it foUows that, to prove (1) or (2), we may assume V{R) = {0}. For (3) and (4), we may assume that R is semiprime, because V(R) C V(RH) and (R/V(R))H ^ RH/V(RH). Let 7^ = J or V. Assume that V{R) = {0} if 7^ = J and V(R) = {0} if TZ = V. To prove both parts, it is now sufficient to show that, under these assumptions, n{RH) C EpGP ^ ( ^ ) P ^ ( ^ ' ^ p ) ^ V{RH). By Lemma 3.3, 8n{RH) C n{RZ)H and therefore, by Theorem 2.7, STl^RH) C T{R)H, where T{R) is the ideal of R consisting of the elements of finite additive order. Since R is semiprime, for any p € P, PROOF.

{r e R\ p^r - 0 for some n > Q] = {r e R \ pr - G] = Z^y, where Z = {0}. Since T{R)H is an ideal of RH^ it follows that TZiRH) = 'R(T{R)H)

=

^^^^TliZ^H).

We may therefore also assume that R = Zp. Consider the case p = 2 first. (Thus 1 + 3 = 1 — 5.) Since (1 + sy = 2(1 + 5), {RH{1 + s)y = {0}. Hence RH{1 + s) = RH{\ - s) C V{RH), Clearly RH{\+s)C AR{H,H2). Note that RH/RH{l + s)^ /2[///7/'] and A H ( / / , H2)/RH{l + s) ^ AR{H/H', {H/H')2). Because H/H' is an abelian group. Theorem 2.7 implies that n(RH/RH{l

+ s)) ^ AR{H, H2)/RH{1 + s) = V{RH/RH{1

+ s)).

Hence n{RH)

= RH{\ + 5) + AR{H, H2) = AR{H, H2) =

V{RH),

as required. Consider now an odd prime p. Let 7 = (ao + otix + ay + a^xy) + {(5Q + (5ix + /?2y + P3xy)u

3. RADICALS OF LOOP RINGS

163

be an element of 7Z{RH)^ each ai,/?^ G R2. Of course we assume that each /3i = 0 in case H = G. By Lemma 3.3, 8a^ and 8f3i are each in J{RZ). Since 8 and p are relatively prime, we obtain that each ai and each /J^ is in J{RZ). Because of Theorem 2.7, ai and j3i are in lS.R{Z^Zp), Hence, by Lemma 3.2 and Theorem 2.7, n{RH) C R^HAR(Z,ZP) C V{RH). Since Hp = Zp and because AR(H^HP) = R^HAR(Z^ZP)^ it follows that n(RH) = AH(if, Hp) = V{RH), again as required. Finally, suppose that either R/V{R) has no 2-torsion or that L2 = 02It is then easily seen that, for any prime p, V(R)pA{L,Lp)

C V{R)pA{G,Gp)

+

V{R)pA{G,Gp)u.

By what we have shown, it then follows that n{RL)

= n{RG) + n{RG)u,

3.5 Corollary. Let H = L or G. Then J{RH) if one of the following conditions is satisfied, 1. J(R) = p(^R),

D is a nil ideal if and only

2. H is not torsion. Furthermore, if either of these conditions is satisfied, J{RH)

=

V{RH).

Suppose J{RH) is nil and H is torsion. By Theorem 3.4, V{R) = J{R) ^ J{RH). Hence J{R) is nil and, because R is commutative, J{R) — V{R), This proves one implication. For the converse, assume that J{R) — V{R) or that H is not torsion. In either case. Theorem 3.4 implies that PROOF.

J{RH)

= V{R)H + Y, VWpA(7y, Hp) = V{RL).

D

peP

Parts of the corollaries which follow were proved in [GP86]. Their proofs are almost immediate applications of the description of the prime and Jacobson radicals of alternative loop rings obtained in Theorem 3.4. According to this theorem, for instance, the prime radical of RL is {0} if and only i{V{R) = {0} and, whenever V{R)p ^ 0 (that is, whenever, pr = 0 for some prime p and nonzero r E R)^ then Lp = {1} (that is, L contains no elements of order p). Thus Corollary 3.6 can most easily be verified by recognizing the equivalence of each statement with statement 4.

164

VI. THE JACOBSON AND PRIME RADICALS

3.6 Corollary. Let L = M(G', *,^o) be an RA loop. Then the following conditions are equivalent. 1. 2. 3. 4.

5. 6. 7. 8.

9.

Z{RG) is semiprime, R[Z{L)] = R[Z(G)] is semiprime. RG is semiprime. R is semiprime and, if R contains an element of additive order n > \, then G does not contain central elements of order n (and thus no elements of order n and no normal subgroups of order n). RG has no nonzero nil ideals. Z{RL) is semiprime. RL is semiprime. R is semiprime and, if R contains an element of additive order n > \, then L does not contain central elements of order n (and thus no elements of order n and no normal subgroups of order n). RL contains no nonzero nil ideals.

3.7 Corollary. Let L = M{G,^,go) be an RA loop. Let V{R) = J{R) if G (and thus also L) is torsion and V{R) = V{R) otherwise. Then the following conditions are equivalent. 1. 2. 3. 4. 5. 6.

J{Z{RL))^Q. J{Z{RG)) = Q. J{RG) = 0. J ( i ? L ) = 0. J{RZ) = 0. V{R) = 0 and, if R G does not contain 7. V{R) = 0 and, if R L does not contain

contains an element of additive order n > I, then central elements of order n. contains an element of additive ordern > I, then central elements of order n.

Since, by Corollary IV. 1.9, the order of a subloop of a finite RA loop L divides the order of L, the next corollary, which extends Maschke's Theorem for group algebras, is clear. 3.8 Corollary. Let L = M(G,*,^o) be a finite RA loop, let F be a field and let H — L or G. Then FH is Jacobson semisimple if and only if char F = 0 or char F = p > 0 and H does not contain elements of order p.

3. RADICALS OF LOOP RINGS

165

We have shown t h a t there is strong relationship between the radical of the loop ring and the radical of its centre. In this context the following proposition is of interest [ G P 8 7 ] . 3.9 P r o p o s i t i o n . Let L = M ( G , *,po) be an RA loop and assume that R has no elements

of additive order 2. Let H = L or G. Then every

ideal of RH contains hence central also in PROOF.

a nonzero central element

nonzero

which is central in RH

and

RL.

The last statement follows from Proposition III.4.1 which im-

plies t h a t Z{RG)

C

Z{RL).

Let J be a nonzero ideal of RH and let r be a nonzero element of J. Multiplying by the inverse of a loop element, if necessary, we may assume t h a t 1 G s u p p ( r ) . Write r in the form (1)

r = z + x,

O ^ ^ G Z{RH),

x = ^ a d i

Z{RH)

where T is some finite subset of s u p p ( a ) . We prove by induction on \T\ t h a t J contains a nonzero central element. If \T\ = 0, then r = 2: is the desired element. Now assume t h a t \T\ > 0 and t h a t J contains a nonzero central element whenever it contains a nonzero element which, when written in the form (1), has fewer t h a t | r | elements in the support of its noncentral term. Let r be a nonzero element of J (with 1 in its support) and write r in the form (1). If r G

Z[RH)^

there is nothing to prove. Otherwise, there exist elements to £ T and IQ £ H such t h a t toio ^ 4^0• Write T = To U Tj where To = {t e T \ tio 9^ 4 0 and Ti = T \ TQ. Note t h a t |Ti| < \T\.

Letting s denote the nonidentity

commutator of H^ we have, for any t G To^ t + l^^tio Corollary III. 1.5, is an element oi Z{RH). ri = r + q^rio

where zi = "^^-^^teT

— t -\- st which, by

Therefore J contains the element

= 2^ + ^ at{t + 5t) + 2 ^ a^^ = ^1 + 2 ^ att, ten teTi teTi (^t{t + st) G Z{RH).

Since 72 has no 2-torsion, ri has

1 in its support. Thus r\ is not 0, and central by the induction hypothesis, so the proof is complete.

D

166

VI. THE JACOBSON AND PRIME RADICALS

4. T h e s t r u c t u r e of a s e m i s i m p l e a l t e r n a t i v e a l g e b r a Let F be a field and L a finite RA loop whose order is relatively prime to the characteristic of F, In this section, we apply a theorem of J. Dieudonne about general nonassociative algebras to show that the loop algebra FL is the direct sum of fields and Cayley-Dickson algebras. 4.1 Theorem (Dieudonne). Let A be a nonassociative algebra with unity of finite dimension over a field F. Suppose A contains no nonzero ideals with square 0 and that f: Ax A -^ F is a nondegenerate symmetric bilinear form on A which is associative in the sense that f{xy^z) == f{x^yz) for all x^y^z £ A. Then A is the direct sum of unique ideals A i , . . . , A^ which are simple rings. Any ideal of A is the direct sum of a subset of { ^ i , . . . , An}^ First we show that A can be written as stated. If A is simple, there is nothing to prove; otherwise, let B be an ideal of A of minimal (positive) dimension and note that B-^ = {a G ^ | f{0",b) — 0 for aU 6 G 5 } is an ideal of A., by symmetry and associativity of / . Since B fl B^ C 5 , either B (^ B^ = {{)] or B f\ B^ = B, We show Bn B^ ^ B. By hypothesis, B^ ^ {0}, so the ideal of A generated by B^ must equal B. Thus any b £ B can be written in the form 6 = YL{^t^j)^ij'> where bi^bj G B and each Uij is a product of R(x)s and/or L{x)s for x £ A^ where, as always, R{x): a \-^ ax and L{x): a \-^ xa denote the right and left translation maps on A. Now PROOF.

f{yR{x),z)

= f{yx,z)

= f{y,xz)

=

f{y,zL{x))

and f{yL{x),z)

= f{xy,z)

= f{z,xy)

= f{zx,y)

=

f{y,zR{x))

together imply that, for any product 11 of maps of the form R{x) and Z/(a:), we can write /(yn,2r) = / ( y , ^ n ' ) , where 11' is another product of R{x)s and L{x)s. Suppose that BOB-^ = B. Let b = Y^{bibj)nij be a nonzero element of B and let a be any element of A, Then, for some IIJ-, / ( 6 , a ) = 5 ] / ( ( M , ) n i „ a ) = 5 ] M 6 „ a n ^ ) = Y^{bi,b,{an'-^^)) = 0

4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA

167

since bi ^ B C B^ and hj{a\l\-) G 5 . By nondegeneracy, 6 = 0, a contradiction. We conclude that B fl B^ = {0} and hence, by Proposition 1.4.4, that A = B Q B^ is the direct sum of the ideals B and B^, Also, by Corollary 1.4.6, the restriction of / to B^ is nondegenerate. Since BB^ C 5 n 5-^ = {0} and, similarly, B^B = {0}, any ideal of B is an ideal of A (so 5 is a simple ring) and any ideal of B-^ is an ideal of A. The existence of the required decomposition of A now follows by induction. We have shown that A = Ai Q - • - Q An is the direct sum of simple ideals Ai, Let / be an ideal of A. Since A has a unity, we can write / = (/ n yli) © • • • © ( / n An)' Since / D Ai is an ideal of Ai, each / fl Ai is {0} or Ai, Thus / is the direct sum of certain of the Ai and, if / is a simple ring, exactly one. If A can also be written ^ = 5 i © • • • ® Bm, where the Bi are ideals and simple rings then, by what we have just seen, Bi is one of the Ai, Without loss of generality, Bi = Ai, Similarly, B2 = Ai for some z > 1 (since B^ D B2 = {0}). It follows that n = m and that the sets { A i , . . . , An} and { 5 i , . . . , Bn} are identical. D When an algebra A is written as the direct sum ^ i ffi • • •ffiyl^ of simple ideals, as in the Theorem of Dieudonne, the ideals Ai are called the simple components of A, The following theorem is a consequence of Corollary 3.6; however, there is an easier argument that includes finite groups which we wish to present. Let F be a field and L a finite group or RA loop. For a G FL, the right translation map R{a): FL -^ FL is a linear transformation of FL and the map R: FL -^ Ej\d(FL) is known as the right regular representation of FL. Representing R{a) as a matrix relative to the basis L (with entries in F) and denoting the trace of R(a) by tr R(a), note that 1, Ul j^ ie L, then tr R{i) = 0 and 2. tTR(l)= \L\, 4.2 Theorem (Maschke). Let L be a finite group or RA loop and let F be a field. If the characteristic of F does not divide the order of L, then FL has zero Jacobson radical; in particular^ FL is semiprime. Since FL is a finite dimensional algebra, its Jacobson radical is nilpotent [ZSSS82, Theorem 12.2.2]. Let 0 7^ a := Er=i c^^iA G

PROOF.

J{FL)

168

VI. THE JACOBSON AND PRIME RADICALS

J{FL),

Since J{FL)

is an ideal of F L , multiplying by an appropriate i~^

if necessary, we may assume t h a t a i 7^ 0. By Artin's Theorem, R{aY

= R{a'^) so, since a is nilpotent, R(a)

is

a nilpotent linear transformation. Hence 0 = tr R{a) = S ^ = i Oii tr R((,i) = ai\L\.

Since a i 7^ 0, it follows t h a t the characteristic of F divides |/>|, a

contradiction.

D

4 . 3 C o r o l l a r y . Let H be a finite group or a finite RA loop and let F be a field of characteristic

relatively prime to \L\, Then FH is the direct sum of

ideals which are simple PROOF.

rings.

By Theorem 4.2, FH contains no nonzero ideals with square

0. For X = ^ OLii and y — ^^d

in FE,, define

/(^.2/) ^ X ! «A:/?^ Then / is a bilinear form on FL which is symmetric because ki — \ m L if and only \{ ik — 1, and associative because hk -1 — \ m L \{ and only if h'ki

— 1 too. Moreover, / is nondegenerate because, if x = ^ a^t / 0 and

i G s u p p ( x ) , we have f{x,i-^)

= a^ 7^ 0. The result now follows by the

Theorem of Dieudonne.

D

4 . 4 R e m a r k . In the case of a group ring of a finite group G over a field of characteristic p not dividing |G|, there is a more "classical" proof of Corollary 4.3. Here is a sketch. Suppose / is an ideal of KG.

As a A^-linear m a p , the natural homomor-

phism TT: KG -^ KG 11 splits, so there is a A'-linear m a p ^ : KGjI such t h a t TT o 1/; is the identity m a p on KGjl. A'G-modules and define Q\ KG/1

for X G KG/1.

Consider KGjl

—^ KG

and KG as

—> KG by

It is routine to verify t h a t 0 is a KG-mod\i\e

that TT o 0 is the identity m a p on KG/1,

Hence

KG = ker TT © (^ 0 7r){KG) ^ / ©

KA/I

morphism such

4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA

169

as rings. Since KG has a unity, it follows t h a t / is generated by a central idempotent. Since / is an arbitrary ideal, and because KG is finite dimensional, it follows t h a t KG is a direct sum of simple rings. In particular, J {KG) is trivial. In characteristic different from 2, any alternative loop algebra which is not associative comes equipped with an involution * and a nondegenerate quadratic form n defined by n ( r ) = rr"". (See Sections III.4 and III.5.) 4.5 T h e o r e m . Let L — M ( G , *,po) be a finite RA loop and let F be a field of characteristic

which does not divide \L\. Let H = L or G.

that B is a simple subalgebra of FH, Z{B)

C Z(FH),

multiplicative

Then the map n: B -^ Z(B)

nondegenerate

tion algebra. In particular,

quadratic form

PROOF.

simple component For any a G Z{B)

Thus B is a of FL is a algebra.

of FG is a quaternion

algebra.

composition Each

non-

a'^n{b),

and a* = a for any a € Z{FH).

Since f{x^y)

composi-

and any b ^ B., we have

bilinear form associated with / ; thus, f{x^y) for x , y G FL.

on B.

is a Cayley-Dickson

n{ab) = (a6)(6*a*) = since 66* G Z{FH)

such that B* C B and

defined by n(b) = bb"" is a

each simple component

algebra which, if not associative, commutative

with unity,

Suppose

Let / denote the

= n{x -\- y) — n{x) — n{y)

= xy* + yx*^ f is bilinear, so n is quadratic.

T h a t n permits composition follows from the fact t h a t 66* is central for any 6 G FH.

We show t h a t n is nondegenerate. Thus, let W = BnB^

= {beB\

f{b, x) = 0 for aU x G B}.

We must prove t h a t W = 0. To achieve this, we first use the definition of / and n(pr + qr) = n{p + q)n{r) to obtain n{pr) + n{qr) + f{pr, qr) = [n{p) + n{q) + / ( p , q)]n{r). Thus /(/^r, qr) — f{p^ ^)^(^) for any p^q,r ^ B. Replacing r by r + / gives / ( p r , qr) + f{pr, qt) + f{pt, qr) + f{pt,

qt) = f{p.q)[n{r)

+ n{t) +

f{r,t)]

170

VI. THE JACOBSON AND PRIME RADICALS

and hence f{pr, qt) + f{pt, qr) = f{p, q)f{r, t), for all p,q,r,t G B, With p = w e W and ^ = e, the unity of 5 , we get f{wr^ q) — 0, showing that W^ is a right ideal of B, With r = w ^W and g = e, we see that f(pw,t) = 0, which shows that W is also a left ideal of B. IfW^ 0, then W = B, so e e W, Then 0 = / ( e , e ) = 2e^ = 2e, a contradiction. It follows that W = 0 and that B is indeed a composition algebra. For the final statements of the theorem, observe first that if FH = Ai®'''®Anis the direct sum of simple ideals Ai, then certainly the centre of each Ai is contained in the centre of FH, Writing I = ^ei as the sum of orthogonal central idempotents e^, clearly €{ is a unity for Ai. Moreover e* = €{ implies that Ai 0 A* is a nonzero ideal of Ai. Hence Ai 0 A* = Ai^ so Ai = A*. By the first part, each Ai is a composition algebra. Hence the result follows from Hurwitz's Theorem. D 4.6 Proposition. Let L = M(G, *,po) be an RA loop with V = G' = {1,5} and let F be a field of characteristic different from 2. 1. If FL = A 1 ® • • • © An is the direct sum of simple ideals Ai with unities Ci, then the following statements are equivalent. (i) Ai is commutative. (ii) Ai is associative. (iii) Si = Ci, where Si = sci is the component of s in Ai. 2. / / FG = Ai Q ''' Q An is the direct sum of simple ideals Ai with unities Ci then the following statements are equivalent. (i) Ai is commutative. (ii) Si = Ci, where Si is the component of s in Ai. P R O O F . We prove only the first statement, the proof of 2 being almost identical. The set {ei, 6 2 , . . . , e^} consists of central pairwise orthogonal idempotents whose sum is 1. Let B denote one of the Ai and let TT : FL -^ B denote the natural projection. Thus 7r(s) = sci = Si. Since s is central in FL, Si is in the field which is the centre of B. Since s'^ = 1 and Ci is the unity of B, we have Si = ie^. Now B is generated by {7r(^) | g 6 L}. For g^h^k £ L, the

4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA

171

(ring) associator [^{g)',7r{h)^7r{k)] = 7r([^,/i, A;]) and the (ring) c o m m u t a t o r [7r(^),7r(/i)] = 7r([^,/i]) are of the form (e^- — c)7r{gh • k) and (e^- — d)7r(gh) respectively, where c and d are €{ or Si^ according as (g^h^k)

and (g^h) are

1 or 5, respectively. Now the result follows easily.

D

If Ai is a noncommutative component of FL (and hence ^^ = —e^), if ^ and h are elements of L which do not commute, and if gi and hi are, respectively, the components in Ai of these elements, then gh = shg implies gihi = Sihigi = —higi. Thus gi and hi do not commute in Ai. With a similar argument with respect to elements g^h^k of L which do not associate, we obtain the following rather curious result. 4 . 7 C o r o l l a r y . With F, L and FL as in Proposition g^h £ L commute noncommutative

4-6, two

elements

if and only if their images gi and hi commute

in some

simple component

Ai. Three elements g^h^k ^ L

if and only if their images associate in some simple component

associate

which is not

associative. 4.8 C o r o l l a r y . Let L — M{G^^^go) field of characteristic

^ F[H/H']

PROOF.

with F / / ^ and ^^s

Then

FH^^QFH^

is a direct sum of fields and FH^

is a direct sum of Cay ley-Dickson sum of quaternion

RA loop and let F be a

which does not divide \L\. Let H — L or G. FH =

where FH^^

be a finite

= A ( L , L')

algebras if H = L and otherwise

a direct

algebras.

Because of Lemma 1.2,

^

F[HIH']

FH = FH^^

+

and FH^

=

FH^ AF{L,L').

Since ^^s

=

^

— — ^ ^ , the result now follows from Corollary 4.3, Theorem 4 . 1 ,

Theorem 4.5 and Proposition 4.6.

D

Chapter VII

Loop Algebras of Finite Indecomposable R A Loops

In this chapter, we construct the primitive central idempotents of the rational loop algebra QL of a finite RA loop L and provide an exphcit description of the decomposition of QL into simple rings. This work is mainly based on [JLM945 J L M ] and will be critical to our determination of the loop of units in the integral loop ring ZL in Chapter XII. We need the following two lemmas. 0.1 L e m m a . Let G and L be (possibly associative) Moufang loops and let F be a field. Then F[G x L]^ FG ® F FL, PROOF.

The m a p s / G : FG ^ /^[GxL] a n d / L : FL -> FfGx L] defined

by

are algebra homomorphisms, so there exists an algebra homomorphism FG®FL-* F[G X L] satisfying

(see §1.5.8). Since this map sends g ® t to {g^t) (and hence a basis to a basis), it is an isomorphism. D If F is a field of characteristic diflPerent from 2 and C2 = {0) is cyclic of order 2, then FC2 = F^^ ® F ^ ^ F ® F. Hence the group algebra F[C2 X C2 X • • • X C2] of the direct product of n copies of C2 is isomorphic 173

174

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

to the direct sum of 2^ copies of F , which we denote 2^F. This obvious remark will be useful since, for any RA loop L, L/Z{L) = C2 X C2 X C2. (See Corollary IV.2.3.) 0.2 Lemma. Let L be a loop and let F C E be a field extension. Then E®FFL = EL; in particular, if F has characteristic Q, then F®QQL = FL. We note that the maps fp'. FL -^ EL and fE- E ^ EL defined by fri^) = 2;, fE{oL) = ctfl 3're F-algebra homomorphisms and so there exists an F-algebra homomorphism / : E ®/r FL —> EL mapping a ® x to ax. Since f{l®£)=zi and since {I ® i \ i e L} is dm F-basis for E ® F FL (by Corollary 1.5.6), / is bijective. D PROOF.

Because of Lemma 0.2, a description of QL yields a description of FL over any field F of characteristic 0. When an RA loop L = M(G,*,^o) is indecomposable, we shall import notation from §V.3. Thus, we write G = {Z{L)^x^y) and Z(L) — {t\) x (^2) X (^3) X {w), where • 0(^1) > 2 and 0(^2),0(^3),o(it;) > 1; • 6 = {x,y) e {ti); • x^ e {ti) X (^2) and y^ e {ti) X (^2) x (^3). 1. P r i m i t i v e i d e m p o t e n t s of c o m m u t a t i v e rational group algebras A nonzero (central) idempotent in a ring A with unity is primitive if it is not the sum of two nonzero orthogonal (central) idempotents. If A is the direct sum of simple subrings, then a central idempotent e is primitive if and only if Ae is a simple ring, in which case Ae is one of the simple components. In order to determine the primitive central idempotents of the rational loop algebra of a finite RA loop, we first need to solve this problem for rational group algebras of finite abelian groups. For this, we require some preliminary results. 1.1 Lemma. Suppose H and K are normal subloops of a finite Moufang loop (or group) L with K C H C L, Let f = H and e = K. Then dim(QL)(e - f) = dim(QL)e - dim(QL)/.

1. COMMUTATIVE GROUP ALGEBRAS

175

P R O O F . Note that e and / are central idempotents, by Lemma VI.1.2. Since K C H, we have ef = / . Clearly (QL)e = ( Q L ) / + {QL){e - f) and the sum here is direct because / ( e — / ) = 0. Hence the result is clear. D

If F and E are fields, F C E, we denote by [E: F] the degree of the field extension; that is, the dimension of £^ as a vector space over F, An element ^ G C is said to be a root of unity if ^^ = 1 for some n > 1. A root of unity ^ is a primitive mth root of unity if m is the smallest positive integer with the property that ^^ = 1. The mth cyclotomic polynomial is

the product taken over all primitive mth roots of unity. It is well known that ^rn{x) has coefficients in Q and that this polynomial is irreducible over Q (see, for example, [Coh77, Theorem 2, p. 192]). The degree of ^m{x) is (t>{m)^ where <> / denotes the Euler function; that is, (m) is the number of integers A:, 1 < A: < m, which are relatively prime to m. Let ^rn be a primitive mth root of unity. Then the mapping

($„(x)) '^^^'^> defined by x + ($7^(x)) \-^ (^ is an isomorphism and so

is a field extension of Q of degree (j){m,) known as a cyclotomic field. Since, for n > 1, the polynomial x'^ — I factors

x" - 1 = J] $^(x), the rational group algebra of the cyclic group Cn of order n is

Q c . - - ^ ^ - - e I ^ M ^ - 0 I Q(e^). (x^-i) ^r ($m(^)) ^r 1.2 Lemma. Let p be a rational prime and let A = (a) be a cyclic group of order p'^, m>l. Let A = AoDAiD^''DAm

= {l}

176

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

be the descending chain of all subgroups of A; thus Ai = {a^ ). Then the primitive idempotents of the rational group algebra QA are Co = A

and ei = Ai — A^-i,

I < i < m.

Furthermore^ (QA)e^ = Q(^pi), where ^^i denotes a primitive pHh root of unity, and, for any primitive idempotent e of QA, we have e(l)p^ = [{QA)e: Q], where e{\) denotes the coefficient of 1 in e. PROOF.

Since Q ^ = e™oQ(^p-),

the algebra QA contains precisely m-\-\ primitive idempotents. Since e^^Ai — Ai^ for z > 1, we have coCi = 0 for i > 1. Also, if I < i < i , then AiAj = Ai^ so e^eJ = (A^ - A^.i){Aj

- A^_i) ^^ /I2

/ii/ij

—l

A.'l — l -j- /\i — \ ^^ Jii

Jli/ij — l

which is 0 if i < j and e^ if i = j . It follows that eo, e i , . . . , e^^ are m + 1 pairwise orthogonal (central) idempotents; hence they are primitive. Note that they sum to 1. To prove the last statement of the lemma, we use Lemma VI. 1.2 to observe that {QA)eo = {QA)A = Q and, for 1 < z < m, that (1)

{QA)J,^Q[A/A^]^QB,

where B is cyclic of order p\ By Lemma 1.1, the degree of the cyclotomic field (QA)ei over Q is [{QA){Ji - A^,): Q] = [{QA)'Ar. Q] - [ ( Q ^ ) ^ : Q]

Hence {QA)e^ = Q(^px) for z = 1 , . . . ,m. Also, for i = 1 , . . . , m.

while eo(l)p™ = 1 = [{QA)eo: Q], since (QA)eo = Q- Thus, for any primitive idempotent e of QA, e(l)p'" is as indicated. D

1. COMMUTATIVE GROUP ALGEBRAS

177

1.3 Remark. The proof of Lemma 1.2 strongly depends on the fact that cyclotomic polynomials are irreducible in Q[X]; thus, it can be extended only to group algebras over fields with the same property. The lemma is not true in general; in fact, if F = Fj is the field of seven elements, then X^ - 1 = (X - 1)(X - 2)(X - 4), so the group algebra over F of the cyclic group of order 3 is z.^ ^ nX] ^ F[X] ^ F[X] ^ F[X] ^ - (X3 - 1) " (X - 1) "^ (X - 2) "^ (X - 4)' On the other hand, QC3 ^ Q ® Q ( 6 ) where ^3 denotes a primitive cube root of unity. Thus the rational group algebra of C3 over Q contains only two nontrivial primitive central idempotents while the group algebra of the same group over F contains three, so they cannot be obtained as in Lemma 1.2.

In what follows, we will often use without comment the fact any finite subgroup of a field is cyclic [Coh77, Theorem 1, p. 190]. Also, for r > 1 and unless otherwise stated, the symbol ^r always denotes a primitive rth root of unity. 1.4 Theorem.

1. Let p be a prime and G a finite abelian p-group.

Then the primitive idempotents of QG are G and H — {H^CP"^'^), where H runs through those subgroups of G such that G = {H^c) for some c £ G and G/ H is cyclic of order p ^ > 1. Furthermore, (a) iQG)G^Q. (b) iQG)iH-{H,cP-'))^Qi^,rr.). (c) The number of primitive idempotents e with {QG)e = Qi^d) equals the number of subgroups H with G/H cyclic of order d. 2. [P W50] Let G be a finite abelian group and write G = GiX- - -xGe as the direct product of finite pi-groups Gi, for distinct primes pi^... ^p£. Then the primitive idempotents of QG are of the form ei62 • • *e^, where each Ci is a primitive idempotent of QGi. Furthermore, (a) (QG)eie2 • • -e^ ^ Q{^d) for some d with d\\G\.

178

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

(b) The number

of primitive

equals the number

idempotents

of subgroups

e with {QG)e

=

Q(^d)

H of G with G/ H cyclic of

order d. (c) QG = ®Ai^i(^dQ{^d)f

where ad is the number of subgroups

H

a | \(jr\

of G with G/H PROOF.

cyclic of order d.

1. Suppose G is a finite abelian p-group for some prime p. By

Lemma VI.1.2, {QG)G = Q[G/G] = Q, so the idempotent G is primitive. Let e be a primitive idempotent of QG and assume e ^ G, By Maschke's Theorem and commutativity, the semisimple artinian ring QG is the direct sum of

fields.

In particular, (QG)e is a field and it contains the

finite

subgroup Ge = {ge \ g G G} which is necessarily cyclic, say of order p ^ , m > 0. Let K = {g e G \ ge = e } . Then Ge ^ G/K c e G^ where c denotes the coset Kc.

= (c) for some

Since, again by Lemma VL1.2,

QG = (QG)A' e (QG)(1 - A^) and because Ke = e, it follows t h a t (QG)e = (QG)Ke

is actually a simple component of ( Q G ) A \ Thus e is also a primitive

idempotent of ( Q G ) / ? ^ Q[G/A']. Let (fi: QG -^ Q[G/A^] be the homomorphism induced by the natural m a p G —^ G/K.

Recall t h a t e — Ke »-> ^{e) under the isomorphism

(QG)A' -^ Q[G/A']. Thus ^{e) is a primitive idempotent in Q[G/A']. Writing e — YlfgeG^{d)9'> ^id) ^ Q' ^ ^ woie^ t h a t some e{g) ^ 1/|G| because e 7^ G. Since ke = e for any k G A', it follows t h a t e{g) = ^{kg) for any g E G and k ^ K^ so t h a t the coefficients of e are constant on cosets of A'. Thus

(2)

^= J2 ^^•^^'^' i=0

with each ai G Q and, for some i, ai/\K\ ai ^ l/\G/K\.

^

1/|G|; t h a t is, for some i,

Since

(3)

V(e) = Yl "«^' t=0

it must be t h a t (p{e) ^ G/K.

Hence Lemma 1.2 implies t h a t
{cP"') for some i, 1 < i < m , and t h a t (Q[G/K])(p(e)

^ Q((fp»). So (QG)e =

1. COMMUTATIVE GROUP ALGEBRAS

179

(QG)^e ^ {Q[G/K])(p{e) ^ Q(^p,) too. Then, from (2) and (3), we see that

e = (K^') - {K^-')

=H-

{H\^-')

where H = (/f, c^'). Note that G/H is cyclic of order p' > 1 and that

G={K,c)

= {H,c).

Conversely, suppose c £ G and ^ is a subgroup of G such that G = {H,c) and G/H is cyclic of order p' > 1, Let e = H - (7J,CP*"'). Then

^'

'

heH 0<j
^o<j
So, denoting by (p the natural map QG -^ Q[G/H]^ we see that (p{e) = 1 — (cP*~^) is a primitive idempotent of Q[G/H]^ by Lemma 1.2, and so ( Q [ G / ^ ] M e ) ^ Q ( ^ p . ) - Also

(QG)e = {QG)H(I - ^ J ] c^^'-') - {Q[G/H]Me), 0<j
so e is primitive and (QG)e = Q(^pt). Clearly e = G is the only idempotent with (QG)e = Q = Q(^i) and there is only one subgroup H oi G with G/H of order 1. For e = H — ( ^ , C P * " ^ ) , notice that there are just two coefficients of elements in the support of e and that the larger is l/\H\, Writing e = ^g^G^id)^ ^^^ letting q be the largest rational in the set {e(^) | g G supp(€)}, it is obvious that H = {g ^ supp(e) | e(^) = q}. Thus H is uniquely determined by the idempotent e. On the other hand, under the one-to-one correspondence between subgroups of G containing H and subgroups of G/H^ the subgroup {H^c^' ) corresponds to the unique cyclic subgroup of order p in G/H. Thus the idempotent e is also uniquely determined by H. Hence the number of primitive idempotents e with (QG)e = Q{^d) equals the number of subgroups H with G/H cyclic of order d. 2. We require a certain property about the tensor product of cyclotomic fields: if m and n are relatively prime positive integers, then Q(^m) ®Q Q(^^) =. Q(^mn)- To see this, note that the two algebras in question have the same dimension (since the Euler function is multiplicative on relatively prime integers). Hence, the mapping Q(^rn) ®Q Q(^n) —^ Q(^mn) defined

180

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

by ^m ® 1 »-^ Cmn ^ ^ ^ 1 ® ^n »-^ Cmn ^^ ^ ^^^S epimorphism and, since the dimensions are equal, an isomorphism. By Lemma 0.1, there is a n a t u r a l isomorphism QG = QGi ® • • • ® QG^. For each i, 1 < i < ^, let { e ^ i , . . . ,e^A;,} be the primitive idempotents of QGi,

By part (1), for each i, 1 < i < ^, and for each j ^ , 1 < ji < ki^

{QGi)eij^ =. Qi^iji) for some ^^j., a pith power root of unity. In particular, for i :/: k^ the orders of ^{j. and ^kj^ ^^e relatively prime.

Using basic

properties of the tensor product (see Proposition 1.5.5), it follows t h a t

QG = (8)Li (e^:=i(QG)6,,,) - en,...,^ {^UxiQG)e,„) = e,„...,,, (0-=, Q(eo'.)) = ®4GI "''Q(^'') for some a^ > 1, where ^d is a primitive ofth root of unity. To determine the number a^/, note t h a t , by part (1), each field

QiU) = Q(6,J ® • • • ® Q(6,,) = QGiUL ^in) is uniquely determined by a subgroup H = H\X'''XH£

of G^ where each

Hi is a subgroup of Gi such t h a t Gi/ Hi is cyclic of order di and ^d is a primitive root of unity of order d = d\d2' - -d^. So Gj H is cyclic of order d. Since every subgroup H of G can be written H = {H O G i ) X • • • X [H flG^), and because G / / f is cyclic if and only if each Gi/{H OGi) is cyclic, it follows t h a t ad is precisely the number of subgroups H of G with G/H

cyclic of

order d. It is clear t h a t the set {ei • • -e^ | Ci a primitive idempotent of QGi} accounts for all the primitive idempotents, so the result follows.

D

1.5 R e m a r k . If C is a finite abelian group, then the number ad of subgroups H of G with G/ H cyclic of order d equals the number of cyclic subgroups of G of order d.

In particular, ad(f>(d) = ad[Q{^d)'' Q] is the

number of elements of order d in G, We sketch a brief outline of the proof. Since G is finite abelian, G is isomorphic to G* = Homz(G', Q / Z ) . Let d be a divisor of |G|. Denote by Af the set of all cyclic subgroups of order d in G, and by M the set of all subgroups H of G* with G'^/H cyclic of order d. Consider the map L:Af

^

M

1. COMMUTATIVE GROUP ALGEBRAS

181

defined by H ^

H-^ = {f e G* \ f{h)

= 0 for all h G H},

It is standard to verify t h a t ± is a bijection, giving the result. For details, we refer the reader to the section on character groups in Chapter 10 of [ R o t 95]. 1.6 C o r o l l a r y . Ife is a primitive

idempotent

in the rational group algebra

of a finite abelian group Gj then supp(e) is a subgroup of G. PROOF.

Write G = Gi x • • • x G^ as the direct product of finite pi-

subgroups Gi of G, for distinct primes p i , . . . ,p^. Let e be a primitive idempotent of QG. By Theorem 1.4, e = ci • • • 6^ where each e^ is a primitive idempotent of QG^. It follows t h a t supp(e) = s u p p ( € i ) x - • •xsupp(e^); thus, to prove the result, we may assume t h a t G is a p-group for some prime p. In this case, either e = G, or e = H\ — H2 where H\ and H2 are subgroups of G, H\ ^ H2 and \H2lHi\

— p. In the first case, supp(e) = G is a subgroup. In

the second case, for a given element p G ^ 1 , the coefficient oi g in Hi diflfers from the coefficient of ^ in H2» Hence, supp(e) is the subgroup H2'

•

1.7 E x a m p l e . The support of an idempotent which is not primitive need not be a subgroup. Indeed, let G = (a | a'^ = 1) x (6 | 6^ = 1) and let e be the idempotent e = | ( 3 — a + 2ab + 2ab^). Then e is not primitive since it is the sum of the two orthogonal idempotents ei = ^(1 + a ) ( l + b + b^) and 62 = ^(1 - a)(2 — b — 6^). Also, supp(e) = {l,a,a6,ai^^} is not a subgroup. (Notice, however, t h a t ei and 62 are primitive and the support of each of these elements is G.) As another application of Theorem 1.4, we obtain the following slightly refined version of [ S e h 7 8 , Proposition VI. 1.16]. 1.8 C o r o l l a r y . Let G = Gi X G2 x • • • X G^ be the direct product nite abelian pi-subgroups primitive

idempotent

Gi of G, for distinct primes p i , . . . ,p^. If e is a

in QG, then there exist 2 ^ subgroups Hi^,,.

supp(€) = Hi, w < i, such that e is a linear combination ZTi,... , ^ 2 ^ '^iih coefficients Z'linear

ofG.

combination

of fi-

of the

equal ^o ± 1 . Every idempotent

of idempotents

, H2^ in elements

of QG is a

of the form H, where H is a subgroup

182

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

P R O O F . Since every idempotent of QG is the sum of primitive idempotents, the last statement is a consequence of the preceding one. To prove the first statement, we observe that the result is obvious if e = G and, for e ^ G^ prove the result by induction on £. By Theorem 1.4, the case ^ = 1 is clear, so assume i > \, By Theorem 1.4 again, e — ei---e^, where each e^ is a primitive idempotent of G^. Since G — n t = i ^ n there exists an i such that e^ ^ Gi, Without loss of generality, i = i. Let f = ci" '€£^1 e Q[Gi X • • • X Gi-i], Then e = fe^ and, by Theorem 1.4, e^ = Di — D2 where Di and D2 are subgroups of G^, supp(e^) = D2 and Di ^ i^2- By the induction hypothesis,

for subgroups Hi of Hi = supp(/) and each a^ = ± 1 . Hence 2^

e = ^^a^HiDi 1=1

2^

- ^aiHiD2 i=\

2^

= ^aiHi t=i

2^

x Di - ^^a,Hi

x D2.

1=1

Clearly {Hi x Di, HiX D2 \ i = I,. ^ ^ ,2^} is a set of 2^"^^ (different) groups each contained in supp(e) = H^ x D2. Hence the result follows. D

2. Rational loop algebras of finite RA loops In this section, we give a complete description of the rational loop algebra QZ/ of a finite RA loop L. We start by recalling that any finite RA loop L is the direct product L = L2 X ^ of an indecomposable RA 2-loop L2 and an abelian group A^ by Corollary V.1.2. Since QL = QL2 ®Q Q ^ and because we now have full knowledge of QA (see Theorem 1.4), it is sufficient to determine the structure of QL in the case that L is indecomposable. For an indecomposable RA loop L, we shall show that QL is a direct sum of cyclotomic extensions of Q and Cay ley-Dickson algebras at most one of which is a division algebra. The others, being split, are isomorphic to various Zorn's vector matrix algebras. Our description will allow us to identify that primitive central idempotent which defines any Cayley-Dickson division algebra. Moreover, for each split algebra in the decomposition, we

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

183

shall give an explicit isomorphism onto the appropriate Zorn's vector matrix algebra. We begin by describing the primitive central idempotents. 2.1 Proposition. / / L is a finite RA 2-loop and L' = {l^'S}, then the primitive central idempotents of QL are L and H — {H^c^"^'^), where H is a normal subloop of L, m > 1 and either 1. the idempotent is in Q L ^ , s e H, L = {H,c) and \L/H\ = 2 ^ , or 2. the idempotent is in A(L, V), s ^ H, Z{L) = {H^ c) and \Z{L)IH\ = Furthermore

In particular, if e £ A(L,Z/') is a primitive central idempotent such that (Z(QL))e = Q, then e = H^^, where H is a subgroup of index 2 in Z(L) and s ^ H. P R O O F . Note that L/V and Z{L) are abelian 2-groups. By Corollary VI.4.8, we have QL = Q L ^ © A(L, L') and, by CoroUary VI.1.3,

Z{QL) =

QL'-^®Q[Z{L)y~^,

Thus there are two types of primitive central idempotents in QL, those belonging to Q L ^ ^ Q[L/L'] and those belonging to Q[Z{L)]^. Consider the case that e is an idempotent of the second type. Then supp(e) C Z{L) and es = —e. Hence e can be written in the form

e=

J2att^,

where T is a transversal for L' in Z{L). Clearly e / Z(L) since the coefficients of Z(L) are all positive. Thus, by Theorem 1.4, e = Hi — H2 where ^1 and H2 are subgroups of Z{L) with Hi ^ ^2? ^2 = {Hi^c^"^' ), {Hi,c) = Z{L) and \Z/H\ = 2 ^ > 1. If 5 G Lfj, then also s G H2, so sHi = Hi^ i = 1,2. It would foUow that 5e = e, a contradiction. Thus s ^ Hi and e has the required form.

184

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

On the other hand, suppose that e is a primitive central idempotent in Q/,i±£ ^ Q[LIL% Then es = e, so

ter where T is a transversal for V m L, Denoting the coset L'g by 'g and by (/?, the isomorphism QL^^ -^ QL, we have ip{e) = ^ati. If 1. If ? G ^ 2 \ ^ i , then 1 \H2\

_ \H2

while, if ? 6 i^i, then \Hi\

I//2I

\Hi\

\H2\

So e = Hi — H2', {H\^c) = L, L/H^ is cyclic of order 2^^ and H2 •ym — l I (/?i,c^ * ' ) , 2is required. In both situations. Theorem 1.4 says that {Z{QL))e = Q(^2^)For the final statement, we note that if {Z{QL))e = Q, then m = 1, Z{L)= {H,s) and e = H - {H^) = H - i | ^ ^ = H^,

D

Let K be a field of characteristic p > 0 and let L be an RA loop of finite order not divisible by p. For any primitive central idempotent e of the loop algebra A'L, Corollary VI.4.8 says that the simple ring {KL)e is either a field or a Cayley-Dickson algebra. We now describe these algebras in more detail. 2.2 Lemma. Let L = M(G, *,5fo) be a finite indecomposable RA loop with V — {1,'5}. Let K be a field of characteristic p > 0 and suppose p j^ \L\. If e G AK{L, V) is a primitive central idempotent of KL, then 1. {KG)e = {R^{xey^{yey)j a quaternion algebra, and 2. {KL)e = {R^{xey^(yey^goe), a Cayley-Dickson algebra

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

185

where G = {x^y^Z{L)) and R = K[Z{L)]e is a field. Moreover^ if (KG)e is not split (for example, if {KL)e is not split), then K does not contain a fourth root of unity. Furthermore, if K = Q, then R is a cyclotomic field and, if {QG)e is not split, then R = Q, Z{L) — {t\) X (^2) X (^3} X (it;) where t\ — s, and e — H^^, where H is a subgroup of index 2 in Z{L) not containing s. Remember that xZ{L) and yZ(L) generate G/Z{L)^ which is C2 X C2. Thus KG = S + Sx + Sy + Sxy, where S = K[Z(L)], Therefore {KG)e = R + Rxe + Rye + Rxye, where R = Se = {K[Z{L)])e. Noting that e e Z(AK{L,V)) = K[Z{L)]^, we have R = {K[Z{L)])e = {Z{KL))e, using Corollary VI.1.3; thus i? is a field. Both (xe)^ and {ye^ are in Z{L)e^ hence in i2, and (ye){xe) = yxe = sxye = —xye = —(xe)(ye) because e G KL^^. Since the elements e, xe, ye, xye are linearly independent over i2, it follows that (KG)e is the generalized quaternion algebra (/Z, (xe)^, (i/e)^). Now write KL = KG + KGu and remember that multiplication in KL is given by PROOF.

(a + bu){c + du) = (ac + god*b) + {da + bc^)u where a^b^c^d G KG. For r G {KG)e^ write r = ri + r2(xe) + ^3(2/^) + r4{xye) with the r, in R (and hence central). Then r* = ri + r2{sxe) + rs(sye) + r^^sxye) = ri — r2{xe) — rs{ye) — r^i^xye) so that the restriction of * to {KG)e is the standard involution on this quaternion algebra. It follows that {KL)e = {KG)e + {KGu)e is the Cay ley-Dickson algebra (i?,(xe)2,(ye)2,yoe). Suppose that {KG)e is not split (that is, {KG)e is a division algebra). Let a — (xe)^, /? = {y^)^- Proposition 1.3.4 says that the equation ax^ + /3y'^ = z^ has no nontrivial solutions. Since a and /? are roots of unity, so is /?/a and one of a,/3,/?/a is a square (because these elements all belong to a cyclic 2-group). Assume R contains a fourth root of unity, say i, so i^ = - 1 . If a = C^ let X = C"^ y = 0 and z = 1. If /? = C^ let x = 0, y = C~^ 2Lnd z = 1. If /3/a = C^, let x = i(^ y = 1 and z = 0. In all cases, we have a nontrivial solution to ax^ + /3y^ = 2:*^, a contradiction. Assume now that K = Q. Then R = {Q[Z{L)])e and thus, by Proposition 2.1^ R = 0(^2"*)? where ^2^ is a primitive root of unity of order 2 ^

186

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

for some m > 0. If (QG)e is not split, then m < 1 as shown above. Thus Re = Q and, by Proposition 2.1, e = H^^2 with H as described. By Corollary VI.1.3, we have Z{AiL,L'))

= Q[Z(L)]if^ = Q{hy-^[{h)

X (i3) X {w)].

By Lemma 1.2, since s is the unique element of order 2 in (ti), we have Q{^i)^T^ — Q ( 0 ^^^^ ^ ^ primitive root of unity of order equal to o(^i). So every simple component of Z{A{L^L^))^ in particular Re = Q, contains Q(^) as a subfield. Thus mi = 1, o{ti) = 2 and ^ 1 = 5 . D Recall that there are two RA loops of order 16, Mie{Qs^ 2) = M(Z?4, *, 1) and Mie(Qs) = M^Q^-, *,^i). The Wedderburn decompositions of their loop algebras over an arbitrary field (of characteristic not 2) are as foUows. 2.3 Corollary. Let K be a field of characteristic different from 2. 1. A^[Mi6(g8,2)]^8A^®3(A0 2. K[M^e{Qs)] = ^K®{K,-\,-\,-\) Let L be either of the loops in question. In both cases L/L' = C2 X C2 X C2 and Z{L) = {1,«5}. By Proposition 2.1, with H = {1} and c = s^ e = ^ ^ is the only primitive central idempotent in /S.K{L^L'). By Lemma 2.2, and with reference to Table V.4, (KL)e = (A^, 1,1,1) if L = Mi6(Q8,2) and (KL)e = (A^ - 1 , - 1 , - 1 ) if L = M^eiQs)' Thus PROOF.

A^MieCQs, 2)] ^ K[C2 x C2 x C2] 0 (A, 1,1,1) and K[Mie{Q8)] = K[C2 X C2 X C2] ® (A^ - 1 , - 1 , - 1 ) . The result now follows because A^[C2 X C2 X C2] = 8A" and (A', 1,1,1) ^ 3(A^), by Corollary 1.4.17. D Note that Z{AK{L, V)) = K[Z{L)]^ = 2{AK(G, G")) and hence, if e is a primitive central idempotent in A A ' ( G , G^)^ then e is a primitive central idempotent in AK{L,V), Recall that H(A') = ( A ' , - 1 , - 1 ) and M2(A^) = (A', 1,1). With arguments similar to those used to prove Corollary 2.3, it is easy to show 2.4 Corollary. Let K be a field of characteristic different from 2. 1. A:g8 = 4A'® H(A^)

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

187

2. KD^ ^ AK ® M2{K) We now give a description of the rational loop algebra of a finite indecomposable RA loop. 2.5 Theorem. Let L be a finite indecomposable RA loop. Then

where 1 QL^^

^ Q[L/V] ^ ®2rfllLI^'^^^^'^^' ^^^^ ^^ ^^^ number of cyclic quotients of L/ L' of order d. 2. QL^-y^ = A(L,L') is a sum of aL Cayley-Dickson algebras, with ai, the number of subgroups H in Z{L) such that Z(L)/H is cyclic and s^H. 3. Z(A(L,Z/')) ^ ®Q(^i/), where the direct sum runs over all the subgroups H described in 2 and ^H i^ ct primitive root of unity of order \Z{L)IH\. With reference to Table V.3, if L is in class C\ or £3 or £5, then all simple components 0/A(Z/,Z/') are split Cayley-Dickson algebras. If L is in C2 or £4 or Ce or C7, then all simple components of A(L^ V) but possibly one (in the case o(ti) = 2) are split Cay ley-Dickson algebras. Any component which is not split is determined by a primitive central idempotent of the form e = H^^, where H is as described below: Loop C2

r ^ 1

{1}

u {^1*2) (ii<2> X {UH) u 1 ^7 J{htj) X (^1^3) X {hw) The first statement follows from Corollary VI.4.8 and Theorem 1.4. Statements 2 and 3 follow from Corollary VI.4.8 and Proposition 2.1. To identify the simple components (QL)€ of A(L,Z/') which are not split, note that Lemma 2.2 says that e is of the form e = H^^^ where ^ is a subgroup of Z{L) of index 2 such that s ^ H and Z{L) = {h) X (^2) X (^3) X (w), with ti = s. Since \L/Z{L)\ = 8, for such H we have \L/H\ = 16. Since (QL)e = {QL)H^ ^ Q[L/H]^ is not spUt, we must PROOF.

188

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

have L/H = Mie{Qs)', by Corollary 2.3; in particular, the only element of order 2 in L/H is central. Consequently, L is not a loop of type £ i , £3 or £5 (see Table V.3) since, in these loops, the element u is noncentral and of order 2. Assuming o{ti) = 2 and that L is of type £2, £4, ^6 or £7, we now determine the subgroups H of 2{L) with s ^ H, Write Z{L) = (ti) x (^2) X (^3) X (tt;), where 0(^2)^0(^2)7o('^) > 1- For a G {^27^37^)7 define S{ti,a)=

{

tia

if o(a) > 1

1

ifo(a) = l.

Then the subgroups of index 2 in Z{L) not containing s are Hi = {t2) X (^3) X (w), H2 =

{S{h,t2))x{ts)x{w),

H3 =

{t2)x{6{ti,ts))x{w),

H4 =

{t2)x{t3)x{S{ti,w)),

Hs =

{S{tut2))x{6{tr,t3))x{w),

He =

{S{ti,t2))x{t3)x{S{tuw)),

Hj = {t2) X {6{tut3)) X {6{tuw)) and Hs = (^(^1,^2)) X {6ih,t3))

X {6{h,w)).

Because of the Classification Theorem (Theorem V.3.1) and because L/H has no noncentral element of order 2, it is straightforward to verify that H is ^ 8 - Conversely, for any indecomposable finite RA loop L of type £2? ^^4, Ce or £7 and with 0(^1) = 2, one has L/Hg = MIQ{QS). By Corollary 2.3, ( Q L ) ^ ^ ^ is not split, so the result foUows. D Let // be a finite indecomposable RA loop and let e be a primitive central idempotent in A(Z/,//'). In Theorem 2.5, we showed that {QL)e is a Cayley-Dickson algebra and described its centre R explicitly. In the case that this algebra is split (and hence Zorn's vector matrix algebra 3 ( ^ ) ) , it is possible to exhibit a surjection from QL to 3 ( ^ ) - We proceed to show how to do this.

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

By Corollary VI.4.7, Le is not associative. Since Z{Le) in t h e field (Q[Z(L)])e^

it follows t h a t Z{Le)

189

is contained

is cyclic and a 2-group since

Le is a homomorphic image of L. Thus Le is indecomposable and so, by t h e Classification Theorem, Le is of type C\ or £2- It therefore suffices t o consider just those loops which are in one of these classes. If a loop L is of type C\ or £2? then any element of QX can be written in a standard form. Specifically, and with notation as in t h e Classification Theorem, we write L = D U Du with Z{D)

= (ti).

By Lemma VI. 1.1, any

a G A(X, L^) can be written uniquely as a = 7 + 7'u with 7 , 7 ' G A ( i ) , D^), Because of Theorem 2.5, e = ^ ^ is the only primitive central idempotent of A ( X , X ' ) . Let T be a transversal for { l , ^ } in Z(D).

Then j = je

and

Y — 7'e can be written uniquely in the form 76 = (70 + 71a: + 722/ + 73X2/)e, and 7'e = (7Q + 7 I X + 73^ + 73xy)e, with each 7^ and 7^- in Q ( / i ) and t h e supports of both 7^ and 7^- contained in T . With this understanding, we call <^^ = [(70 + 1\^ + I2V + Taa^y) + (TO + l'\^ + 72^ + the normalized choice oix^y

form

of a.

l'z^V)A^

Clearly this normalized form depends on t h e

and u and on the transversal T . When we refer to "normalized

form", we mean with respect t o such particular choices. By Lemma 1.2, the map Q(^i}e -^ Q(^2^i)? denoted 7 1-^ 7-^ and defined by {t\ey

= ^2"^i ? is an isomorphism. If rrii = 1, then Q{^i)e = Q. When

this is the case, we will often omit the superscript / , thereby identifying with

je

(je)^.

In the proof of Proposition 2.7 which follows, we shall require a lemma which is also of independent interest. For any algebra R^ with 1, over a field F , and possessing an involution a \-^ a such t h a t a + a and aa are in Fl for all a, and for any scalar a, recall t h a t we have denoted by (iZ, a ) the algebra obtained by applying the Cayley-Dickson process to i2, as described in §1.3. 2.6 L e m m a . Let F be a field and let R = M2{F). map in R,

Then adj is the only involution

the identity

on scalar matrices

Let adj be the

adjoint

A ^-^ A in R which restricts

and has the property

that A -\- A ^ Fl

to and

A~A e Fl for all A £ R. PROOF.

Let A

H-^

A be an involution on R with the assumed conditions.

We wish to prove t h a t A = adj(i4) for any A ^ R. If A is a scalar m a t r i x , this is so because the given involution and the adjoint each restrict t o t h e

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

190

identity on scalar matrices. Assume A is invertible but not central. Then A A central implies that A A = A A. It follows that f{A) — 0, where f{X) = X^ - {A + 'A)X + A l . By assumption, f{X) G F[X], so f{X) is the characteristic polynomial of A and A A = {detA)I. Since A is invertible, we obtain that A = adj A, The proven cases show that R has a basis bi with bi = adj 6i, i = 1 , . . . ,4. Hence, for an arbitrary A = Y^t=i fi^i ^^ ^9 4

4

A = \ ^ 6t = 2_\ ^^J ^« — ^dj A, t=i

t=i

D 2.7 Proposition. Let L = M(Z),*,1) be an indecomposable RA loop of type C\ with (cyclic) centre generated by the element t\ of order 2^^, where D = {Z{L)^x^y) as in Table V.3. Lete = ^-=^ be the primitive central idempotent 0/A(L,LO and let F = {Z{QL))e. If m^ > I, let i = ( 6 - 0 ^ " ' " ' . Then there exists an element w £ L such that L = DUDw and (we)^ = —e. For any a 6 QZ/, write ae in its normalized form o^e = [(70 + Jix + 72T/ + jsxy) + (TO + 1\^ + l2y + 732^y)H^Then the mapping ip: {QL)e -^ 3{F) given by ae yo^ +173-^

( ^ 7 / + 7 2 ^ 7 o ^ - f ^ 7 3 ^ - ^ 7 i ^ + 72^]

[ ( - t 7 i ^ -f 7 2 ^ - 7 o ^ + ^73^ J^7(^ + 72^

70'

• 173'

if m\ > I, or

ae

70 4- 72 (71 4- 73, - 7 o + 72, 7i - 73)

(71 - 73, 7o + 72, 7i + 73) 70 - 72

if mi = 1, is an isomorphism. We have L = D U Du with u'^ = I and Z{L) = (ti). Since s is the only element of order 2 in Z{L)^ it follows that s = tf^^ . Let PROOF.

w = i

tf^^~ u

if mi > 1

xu

otherwise.

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

191

Then w'^ = 5, so (we)'^ = se = —e. Hence every element ae G (Q^)e can be written in its normalized form ^^ = [(TO + lix + 722/ + isxy) + (TO + Ti^ + 722/ + 73^y)H^By Theorem 2.5 and Lemma 2.2, {QL)e = A{L,L') = A{D,D') + /\{D,D')w is a split Cayley-Dickson algebra. Also {QD)e ^ M 2 ( Q ( 6 - i ) ) since {Z{QD))e = {Q[Z(D)])e = (Q[Z(L)])e = F^ Q ( 6 - 0 - For mi > 1, let en =

1 — Qxy e,

/l-exy\ ^12 =

/ l + 0x2/\ 2/-0a: ;:; e = —-—e

y

T:

1 + Qxy ^ e,

622 =

and ^21 = I where 0 = tf^^ let

1 + Qxy\ /l-0xy\ y + Qx 7^ y T: e= — e

e. (Note that 0^ = - e and thus {Qxy^ = e.) For mi = 1, 1 + 2/

1 — 2/

^11 = —^e

,

622 = —^e

=m<,

, .

^12 =

—77—

^21 =

—7^

.

^

y\

—77-

,

-xy + x e =

:^

e

and I- y\

fl + y\ X

— —

X + xy =

- - — € .

These are all elements oi{QD)e and it is easily verified that they are matrix units, in both cases; that is, e = en + 622 ^iid CijCki = ^jk^u for i^j^k^i £ {1,2}. In particular, these elements form a basis of {QD)e considered as a vector space over Z{{QD)e). Since {QD)€ = M2(Q(^2"^i)), these elements play the role of the classical matrix units. If Tni > 1, it is also readily verified that xe = 0ei2 — 0^21,

ye = 612 + 621

and

xye = Qeu — Qe 22

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

192

while, if m i = 1, xe = ei2 + e2i,

ye = eu - ^22

and

xye = -eu

+ €21.

It follows t h a t the mapping (p: {QD)e -> M2(Q(^2^i)) given by ae

70-^ + hs-^

iji^

+ 72-^'

•^71 + 72*^

70-^ -

ijs^

if m i > 1, and by ae

70 + 72

71 - 73

7i + 73

7o - 72

if m i = 1, is an isomorphism, where ae = (7o + 7ia^ + 722/ + 73^y)€ ^

{QD)e

is written in normalized form. Since {we^

= —e, (p{wey

adj(
= —I.

Also, by Lemma 2.6, (fi{a*e)

=

Thus (p can be extended to an isomorphism

(QL)e -^ M 2 ( Q ( 6 - i ) ) + M 2 ( Q ( 6 - i )Mwe)

= (M2(Q(6-i)),-!).

From the last part of §1.3.5, we see t h a t the m a p tp: {QL)e -^ 3(Q(^2"^i))? given by ote = [(70 + 7 i ^ + 722/ + Isxy)

+ (7o + 7(2^ + 72^/ + 7 3 ^ y ) H ^ '"^

70-^ + *73-^

(t7i^ + 72^7o^ + t73^-«7;^ + 72^)"

[ ( - 1 7 / + 7 2 ^ - 7 0 ^ + t73^»7i'^ + 7 2 ^

yo^ -

ij3^

if m i > 1, or by ae

70 + 72

(71 - 73, 7o + 72, 7J + 73)

(71 + 73, - 7 o + 72, 7i - 73)

70 - 72

D

if m i = 1, is an isomorphism.

A similar result will hold for a finite indecomposable RA loop L in the family £2? except t h a t we need not worry about the case m i = 1 because, in this case, the simple component A ( L , L') is not split, as shown in Theorem 2.5. 2.8 P r o p o s i t i o n . Let L = M ( / ) , * , ^ i ) be an indecomposable

RA loop of

type £2 '^ith (cyclic) centre generated by the element ti of order 2 ^ ^ , m i > 1, where D = {Z(L),x^y), central idempotent

as in Table V.3.

Let e

of A{L^ L') and let F — Z{{QL)e),

\-s

be the

primitive

Then there exists an

2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS

193

element w G L such that L = DUDw with {we^ = —e. Writing ae G {QL)e in normalized form, ote = [(70 + Jix + 722/ + Jsxy) + (70 + Yi^ + 722/ +

Is^v)^]^^

the mapping if): (QL)e —> 3 ( ^ ) defined by ae »-^

( - 1 ^ 7 / + ^ 7 2 ^ - 7 0 ^ + * ^ 7 3 ^ i 7 i ^ + 72^)

70^ - 1^73^

is an isomorphism. Recall that L = D U Du with u^ = t\. Since s is the only element of order 2 in 2(L)^ we have s = tj"^^ , so, setting w = t^^xu, it follows that w'^ = s and (if;e)^ = se = —e. By Theorem 2.5, we see that {QL)e = A(L,i/') is a split Cayley-Dickson algebra and that {QD)e = M2(Q(6-i)). Noting that mi > 1, let 0 = tf~\. Then 0^ = - e and {Qxy'^y = e. It is readily verified that PROOF.

1 - exy-'^ en

=

l-Qxy-^\

e ,

€22 =

/ l + 0 x y - i '\

1 + 0xy-i 2 ^' e=

y - 0a:

e

and

e2i = (

^

j ^ (^

^

je =

)xy''^

^^

-e

form a set of matrix units for {QD)e = A(/), D^). Furthermore, xe = 0ei2 - 0y^e2i,

ye = ei2 + y'^e2i and a:?/e = Qy'^eu - 02/^^22,

and the mapping (p: {QD)e —» M2(Q(^2'"i)) given by

o^e = (70 + 71^ + 722/ + l3xy)e e {QD)e

194

VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS

is an isomorphism. Similarly, and as in the proof of Proposition 2.7, the composition of (^ with the isomorphism M 2 ( Q ( 6 - i ) ) + M2(Q(6-i ))v^(^e) -^ 3 ( Q ( 6 - i ) ) (see §1.3.5) is an isomorphism i\)\ {QL)e -^ 3(Q(^2"*i)) sis required.

D

Chapter VIII

Units in Integral Loop Rings

The determination of the group of units in an integral group ring is a subject of continuing interest to many people. In any integral group ring ZG, the elements of diG, which are so obviously invertible, are called trivial units. Several of the early results about units in group rings give conditions under which certain types of units are trivial. For example, when G is abelian, it is known that all the units of finite order in ZG are trivial. In 1940, Graham Higman [Hig40] found necessary and sufficient conditions for all the units in the integral group ring of a torsion group to be trivial; later, S. D. Herman [Ber55] proved several theorems about torsion units, in particular, one for finite groups similar to Higman's. From this work, it follows that all the central units of finite order in the integral group ring of a finite group are trivial. (This was also shown independently by J. A. Cohn and D. Livingstone [CL65].) Forty years after Higman, H. Hartley and P. F. Pickel showed that if the unit group of the integral group ring of a finite nonabelian group is not trivial, then it contains a free subgroup of rank two [HP80]. In this chapter, we prove that the theorems just mentioned have natural extensions to alternative loop rings over the integers. Several of the proofs make use of two explicit constructions of units, the so-called Hass cyclic units and bicyclic units. In addition, in a loop ring ZL which is alternative but not associative, we prove that aU central units are trivial if and only if the centre of L and the quotient of L by its commutator-associator subloop are abelian groups of exponent 2, 3, 4 or 6.

195

196

VIII. UNITS IN INTEGRAL LOOP RINGS

1. Trivial t o r s i o n u n i t s We begin with a proposition which generalizes a well known property about units of finite order in group rings due to S. D. Herman [ B e r 5 5 ] . Recall t h a t an element of finite order in a loop or a ring is called a torsion element or a torsion unit. 1.1 P r o p o s i t i o n . Let a = Y^^^i^otii be a torsion ternative

unit in the integral al-

loop ring 2.L of a finite loop L. If ai ^ 0, then a = ai = ± 1 .

P R O O F . AS

in §VI.4, we denote by R the right regular representation

of the complex loop algebra CL. Thus, for x 6 iZ, R{x) is the right translation m a p CL -^ CL. This map is linear and, by Artin's Theorem (Corollary 1.1.10), / ? ( x ^ ) = (iZ(x))^ for any m > 1. Following Herman [ B e r 5 5 ] , we note t h a t li a = ^ otit G ZL satisfies a ^ = 1 for some m > 1, then the matrix for /Z(a) is similar over C to a diagonal matrix A — d i a g ( ^ i , . . . ,^n)? where n — \L\ and the ^i are m t h roots of unity. Thus the trace of i2(a) is, on the one hand, Y^=\ 6s while, on the other, it is ^ a ^ t r i 2 ( £ ) — noL\. Since 0 ^ n\oL\\ = | ^Y^-\ 61 ^ S r = i 1^1 — ^ ^^^ because OL\ is an integer, it follows t h a t all the ^i are equal to | a i | and OL\ — ± 1 . Hence /Z(a) is right multiplication by 1 or —1 and a^ = 0 if £ / 1.

D

The obvious units in an alternative integral loop ring ZL, namely, the elements ± ^ , ^ € Z/, are known as trivial units.

Hy Proposition 1.1, it

follows quickly t h a t nontrivial torsion units have no central elements in their support. 1.2 C o r o l l a r y . Let L be a finite loop such that ZL is alternative. ^ aii is a torsion unit in ZL and a^ 7^ 0 for some i G Z{L), PROOF.

The element i~^a

If a =

then a =

±i.

is a torsion unit with 1 in its support. So

Proposition 1.1 yields t h a t i~^a = ± 1 and the result foUows.

D

For abelian groups, one can actually remove the finiteness condition in Proposition 1.1. For this, we need some terminology and a lemma. A group G is said to be ordered if the elements of G can be linearly ordered with respect to a relation < in a manner which is compatible with the group multiplication. Thus any two elements of G are comparable and, for any x^y^z

^ G^ x < y implies xz < yz and zx < zy.

1. TRIVIAL TORSION UNITS

1.3 L e m m a . Every torsion-free PROOF.

197

abelian group is an ordered

group.

Let A be a torsion-free abelian group. For a subset S of A, p u t

S~^ = {s~^ \ s £ S}, Let C be the set of all multiplicatively closed subsets of A which do not contain the identity element of A.

Note t h a t C ^ ^

because {a^ | n > 0} G £ for any a G A, a 7^ 1. T h u s , by Zorn's Lemma, C contains a maximal element, say T . We claim t h a t A = T U T~^ U {1}. Suppose to the contrary t h a t there exists an element a G A, a 7^ 1, such t h a t neither a nor a~^ belongs to T, Clearly T is properly contained in the multiplicative subset 5 = T U {ta^

| ^ G r , n > 1} U {a^ | n > 1}. T h e

maximality of T implies t h a t 1 G 5*. Since A is torsion-free, it follows t h a t 1 = td^ for some t ^ T and some n > 1. Hence a~^ G T . Replacing a by a~^ in the above argument, we obtain t h a t d^ Consequently, 1 = {a^)'^{a~'^)'^

^ T for some m > 1.

G T , a contradiction.

For a,6 G A, define a < 6 if a'^b

G T.

Since /I = T U T"^ U {1}

is a disjoint union, for any a.,b ^ A precisely one of the foDowing three possibilities occurs: a~^b G T, a = 6 or ba~^ G T. Hence either a < b^ a = b or 6 < a; thus < is a linear order on A. Finally, assume a < b and c ^ A. Then a~^b G T , so {ac)~^{cb) = a~^b G T. Therefore ac < be. 1.4 P r o p o s i t i o n ( P a s s m a n ) . Let A be an abelian group. sion units of the integral group ring ZA are PROOF.

Let /i be a torsion unit in ZA.

in the set ±A.

D Then the tor-

trivial. We have to prove t h a t /i is

Replacing A by the group generated by the support of /i,

we may assume t h a t A is finitely generated. Therefore ^ = T X F is the direct product of a finite abelian group T and a free abelian group F. By Maschke's Theorem, Q T is semisimple artinian and thus

a finite direct sum of fields K^ — (Qr)e/j,, each e/^ a primitive idempotent in QT. Since ZA — (ZT)F^

the group ring of F over the ring ZT, we have

ZA C ( Q T ) F = ® ^ = i

K\F.

By Lemma 1.3, F is an ordered group, with linear order < , say. Then a unit v in the group ring KhF can be written in the form v — Yli=\

^ifi-,

with each / , G F , 0 / //^ G KH and fi < -" < fk- Write z/"^ = X ) L i ^iSi

198

VIII. UNITS IN INTEGRAL LOOP RINGS

with 0 y^ Ui e Kh, 9i G F and gi < - - < g^^ Then 1 = vv"^ = Y^J^iUJjifigj), Obviously, figi < figj for z 7^ 1 or j 7^ 1, and also figj < fkgi for i ^ k or j 7^ £. So the elements figi and fkge are uniquely represented in the set {figj \ I < i < k^l < j < £}. Since K^ has only trivial zero divisors, it follows that k = I = 1, So we have shown that any unit in KhF is trivial; that is, of the type af with a G Kh and / 6 F . In particular, any torsion unit of KhF belongs to K^* Since /i = ]C/^^/i ^^ ^ torsion unit in ZA, each fieh is a torsion unit in KhF. We obtain that /x G J^ A^/i C QT. Consequently fi eZAO QT = ZT, so /x G ±A by Proposition 1.1. D One can even extend Proposition 1.1 to arbitrary groups. 1.5 Proposition (Passman). Let a = ^g^o^gO ^^ ^ torsion unit in an integral group ring 2.G. If a\ ^ 0, then a =: ai = ± 1 . The proof is much harder than in the finite and the abelian cases. One first shows that if e is a nontrivial idempotent of the complex group ring CG then e(l) is a rational number with 0 < e(l) < 1, where e(l) is the coefficient of 1 in e. (For Qi4, A finite abelian, this follows from the results in Chapter VII.) Using this fact and working in the finite dimensional algebra C[a], with more elaborate technical arguments than in the proof of Proposition 1.1, one obtains the result. Since we will use this fact only once, we have not included the full proof. All details can be found in [Seh78] and [Pas77]. Suppose now that L — M{G^^^go) is an RA loop for some nonabelian group G with involution * and central element go. Let s denote the unique nonidentity commutator-associator of L. In §111.4, we saw that the involution could be extended to ZL by defining (x + yuY = x* + 5t/n, for x, ?/ G ZG, and that r G ZL is central if and only if r* = r. 1.6 Proposition. Let L be an RA loop with commutator-associator subloop V = {1,5}. Denote by a the image in the abelian group ring Z[L/L^] of an element a G ZL. If a is a central unit in ZL and a is trivial, then a is in the abelian group ring Z[Z{L)].

1. TRIVIAL TORSION UNITS

199

Since a = ±1 for some i ^ L and since the kernel of ZL -^ Z[L/L^] is A(L, V), we have a = ±^ + ^ for some element 6 G A{L, V), By Lemma VI.1.1, <5 = (1 - s)li for some /? G ZL. Writing /? = /?i + /?2 where supp(/3i) C Z{L) and supp(/J2) H >E(L) = 0, we have a = ±^ + (1 - s)(ii + (1 — 5)/32. Since a, /?i and 5 are central, clearly ±^ + (1 — s)fi2 is also central and hence invariant under the involution *. Note that /?2 = sfi2 because the elements in the support of (32 are not central. It follows that i is central for, otherwise, i* = si and PROOF.

±i + {l-s)P2

=

[±^+{l-s)f32r = ±se + (1 - S)sf32 = ±se

- (1 - S)f32

implying that ±£ ^ si = —2(1 — s)P2' Such an equation is not possible!, however, because the coefficients on the right are even and, on the left, odd. Thus i G Z{L)^ so (1 — 5)/?2 is also central and hence invariant under *. Now (l-^)/?2 = [(l-^)/?2r = - ( l - ^ ) / ? 2 implies (1 - s)P2 = 0, so a = db^ + (1 — s)l3\. Since i and the support of /3i are in 2{L)^ the result follows. D 1.7 Corollary. Let L be an RA loop. 1. If a is a central torsion unit in ZL, then a is trivial. 2. If e is an idempotent in ZL, then e is trivial; that is^ e = 0 or e = I. prove (1), let a be a central torsion unit in ZL. Then a is a torsion unit in the abelian group ring Z[L/L^]. By Proposition 1.4, a is a trivial unit. Proposition 1.6 implies that a is a torsion unit in Z[Z{L)]. Again using Proposition 1.4, we obtain that a is indeed trivial. To prove (2), let e be an idempotent in ZL. Then e is an idempotent in Z[L/V]. Hence P R O O F . TO

( 2 e - T ) ^ = 4 e ^ - 4 e + T = T. So 2€ — 1 is a torsion unit in the commutative group ring Z[L/L^]. By part (1), 2€ — 1 is a trivial unit and with all coefficients even, except for the coefficient of 1. Thus 2e — 1 = ± 1 , so either 1 — 6 or e belongs to (1 — s)ZL.

200

VIII. UNITS IN INTEGRAL LOOP RINGS

However, (1 ~ s)ZL contains only the zero idempotent because

{^[{\-s)ZLT=P\2^-\\-s)ZL

= {Q}.

Therefore, e = O o r e = l.

D

2. Bicyclic and Bass cyclic units Recall that for a finite subloop TV of a Moufang loop L, we write N = S n € i v ^ and note that N G ZL. (See §VI.l.) For an element g e L of order n, we write ^ rather than (g). Thus

? = l + ^ + ^' + - - + ^""'. 2.1 Definition. A bicyclic unit in an alternative loop ring ZL is an element of the form Uh^g = 1 + (1 - g)hg where g^h £ L and g has finite order. Bicyclic units were studied in several papers on generators of subgroups of finite index in the unit group of integral group rings of finite groups (see, for example, [RS91, Seh93, JL93a, JL95]). Since (1 — g)g = 0, the element Uh,g is of the type I + a with a^ = 0, a G ZL. Hence a bicyclic unit is indeed a unit, with (1 + a)~^ = 1 — a. Similarly, elements of the form 1 + gh{ I — g)y g^h £ L with g of finite order, are units. It is important to know when such units are trivial. 2.2 Lemma. Let L be a loop such that ZL is alternative. If g-th £ L and g has finite order, then the bicyclic unit Uh^g = 1 + (1 — g)h^ is trivial if and only if h~^gh = g^ for some j . If L is an RA loop, if g^h £ L do not commute and if u^^g is trivial, then (g) is normal in L. Let n be the order of ^. Since ^ = 1 + ^ + ^ ^ + - • -H-^f^'^ g^^ = ^ for any j . Thus, if h~^gh = g^^ then gh = hg^^ so gh^ = h^ and u^^g = 1. Conversely, suppose u^^g is trivial. Since this element has augmentation + 1 , we have u^^g — i for some i £ L, Thus PROOF.

1 + /i(l + y + . . . + y " - i ) = i + gh{\ + ^ + .. • + ^ " - ^ ) .

2. BICYCLIC AND BASS CYCLIC UNITS

Since h is in the support of I + h(l + g +

201

h g'^^^)^ it follows t h a t either

h = i OT h = ghg^ for some i > 0. T h e latter implies h~^g~^h

= ^% as

desired. In the case h = i^ y/e obtain

l + hig + g^ + '" + g^-') = gh{l + ^ + • • • + ^"-^) from which it follows t h a t 1 = ghg^ for some i. So /i is a power of g and h~^gh = g^ again as desired. Finally assume t h a t L is an RA loop and t h a t g and h are noncommuting elements of L such t h a t u^^g is trivial. We have seen t h a t h~^gh = g^ for some J > 1. Since L has a unique c o m m u t a t o r , 5, we also have h~^gh = sg. Hence s = g^~^ G (^), so {g) is normal in L by Corollary IV. 1.11.

D

Recall t h a t we use the symbol to denote the Euler function. 2.3 D e f i n i t i o n . Let L be a loop such t h a t ZL is alternative. Then a Bass cyclic unit is an element of ZL of the form

n where g £ L has finite order n, 1 < i < n, and gcd(i, n) = 1. T h a t a Bass cyclic unit /i, written as above, is indeed a unit can easily be verified by showing directly t h a t M-^ = ( 1 + 5 ' + • • • + P'<'-'))^(") + - ^ ^ ^ ^ ? , n where k is an integer such t h a t 1 < A: < n and n | (1 — ik). In order to give a more informative proof of this fact, we introduce more terminology and establish some elementary lemmas.

Recall t h a t a

unital subring of a ring A with unity is a subring with the same unity. 2.4 D e f i n i t i o n . A unital subring A of a Q-algebra A is called a Z-order^ or simply an order^ if it is a finitely generated Z-submodule such t h a t QA = A. Clearly an order A in a Q-algebra yl is a finitely generated additive torsion-free Z-module, so it is a free Z-module with rank equal to dimg A.

202

VIII. UNITS IN INTEGRAL LOOP RINGS

2.5 Examples. 1. If L is a finite group or an RA loop, then ZL is an order in the rational loop algebra QL and Z{ZL) is an order in Z{QL), The latter follows from Theorem III.1.3. Furthermore, if ZL is not associative, then Z{ZL) C ZL^^

© Z{ZL)^

C Z{QL),

so ZL^^®Z{ZL)^ is also an order in Z{QL) (see CoroUary VI.1.3). 2. An algebraic number field is a finite field extension of Q, this being a field necessarily of the form Q{OL) for some a G C. An element of Q(a) is an algebraic integer if it is the root of a monic polynomial with coefficients in Z. It is known that the set of algebraic integers of an algebraic number field K forms a ring O, Furthermore, O is an order in K and Mn{0) is an order in Mn(K). If a is an algebraic integer, then the subring Z[a] of Q(o;) generated by a is an order in Q(a). (See [CR88, §17].) Let H he di subloop of a Moufang loop L, Since L is an inverse property loop, as observed in §11.2.5, there is a well defined notion of the index of H in L. Recall that H has finite index if L can be covered by a finite number of left (equivalently, right) cosets of H, When this occurs, the minimal number of covering cosets is called the index of H in L. We denote this number [L: H]. The following lemma summarizes some well known facts about orders in associative algebras, but we state them here for alternative algebras. Recall that we denote by H{R) the loop of units in an alternative ring R, 2.6 Lemma. Let Ai and A2 be orders in the alternative Q-algebra A. Then the following statements hold. 1. Ai n A2 is an order in A. 2. //A2 C Ai, the index [ZY(Ai): ZY(A2)] is finite. 3. //A3 is a unital subring of A and also a finitely generated Z-module such that Ai C A3, then A3 is an order in A. 4. //A2 C Ai and /x G A2 is invertible in Ai, then fx~^ G A2. (1) and (3) are clear. (2) Since Ai is a finitely generated Z-submodule of A and because QA2 = i4, there exists a positive integer i such that M i C A2. Furthermore, PROOF.

2. BICYCLIC AND BASS CYCLIC UNITS

203

considering Ai and ^Ai as additive groups, the index [Ai: iAi] is finite. We now show t h a t the (multiplicative) index [ZY(Ai): U{A2)] is bounded by t h e additive index [Ai:

iAi].

Suppose t h a t x,t/ G ZY(Ai) are such t h a t x + M i

= y + iA\,

1 denote the common unity of A and A2. Then x~^y — 1 G ^ i

Let C A2.

Thus x~^y G A2. Similarly y~^x G A2 and therefore x~^y G ZY(A2). T h u s y G xU{K2), so xU{K2) = t/ZY(A2). (4) Observe t h a t Ai = /xAi. Hence, considering all algebras as additive groups, we see t h a t [Ai: //A2] = [/^Ai: /iA2]. Since /x is invertible in A i , it follows t h a t [Ai:/iA2]<[Ai:A2]. Since /XA2 C A2, the opposite inequality obviously holds. Hence 11K2 = A2; thus fi-^ G A2.

•

Let L be a finite RA loop, g an element of order n in L and ^n ^ primitive n t h root of unity. Let i > 1 be an integer with g c d ( i , n ) = 1. Obviously,

has an inverse in Z[^7i], namely P _ 1 ~ P _ 1 ~

^ ^^ ^

^ ^^

'

where ifc = 1 (mod n). Take ^' = 1 + y + • • • + ^^"^ G Z(^). From Theorem V n . 1 . 4 , we know t h a t

and

Set M == 0

1 Z[^^]. Then Z{g) C M and the projection of 1/ in every

component of M is a unit except when d == n, in which case the projection is i. Since g c d ( i , n ) = 1, z^(^) = 1 + ^n, for some i £ Z, So we modify u^ replacing it with

;, = ( l + 5 + ... + 5-l)^W_£p,

204

VIII. UNITS IN INTEGRAL LOOP RINGS

where ^ = ^{l+g-\

h^^~^). Then the projection of// in every component

of M is a unit, so /x is a unit in M . Since Z{g) and M are Z-orders in Q(^) and because fi G Z(^) C M , it follows from Lemma 2.6 t h a t /i is a unit in Z{g).

Hence all Bass cyclic units are indeed units in ZL, but are they

trivial? 2.7 L e m m a . Let L be a torsion

loop such that ZL is alternative

g be an element

Let i be an integer with I < i < n and

of order n in L.

gcd(z,n) = 1. If i ^ ± 1 (mod n), then the Bass cyclic ^^ is

(1+^+^2^.

..^^^-l)0(n)^l

and let

unit i4>(n) n

nontrivial. PROOF.

We prove the result by contradiction. Since the augmentation

of /x is 1, if // is trivial, then ji ^ L, Because the support of fi contains only powers of ^, it follows that /j, = g^ for some t. Let m = (n) and assume t h a t ( 1 + ^ + • • • + ^'"^ )'^ + ^ ^ ? = 9^ for some t. Multiplying by (1 - ^ ) ^ , we get (1 - g')"^ = (1 - g)'^9^, so (1)

l - m ^ ^ + ( - ) ^ 2 . + ... + ( _ i ) Y = g'-

mg'^'

+ (^)^^+'^ + • • • + ( - 1 ) - ^ ^ + - .

Since 1 < i < n, clearly n > 2, so m = (f>{n) is even. The assumption z ^ ± 1 (mod n) implies t h a t m > 2. Since the powers of g on the left side of (1) are distinct, we have g^ = I or g^'^'^ = 1. Assume first t h a t g^ = I. Since g^ ^ g^ we must have g^ — g^~^^

but

then ^^* = ^2m-2 _^ ^ ^ - 2 (^sij^ce n )( m) and thus g'^^ = g^. This says t h a t n I (2m — 4) forcing n — 2m — 4 since 2 < m < n. Thus n is even and therefore m = 0(TI) < ^ , giving a contradiction. On the other hand, if ^^+^ = 1, we must have g'^ — g^^^ — g^~'^ (since g^ ^ g~^),

b u t t h e n g'^^ = g2-2m

^ p ^ - m ^ g^ g2i _ g-2

n I (2m — 4), a contradiction.

rpj^jg 2tg3^ij^ S2^ys

D

3. Trivial u n i t s It is a classical result due to G. Higman [Hig40] t h a t all the units are trivial in certain integral group rings, specifically, in the integral group ring of an

3. TRIVIAL UNITS

205

abelian group of exponent 1, 2, 3, 4 or 6 and in the integral group ring of a hamiltonian 2-group. Recall t h a t a hamiltonian 2-group is (isomorphic to) Qs X Ey where Qs is the quaternion group of order eight and E is a, (possibly trivial) elementary abelian 2-group. We now present an elementary proof of a more general statement [ J P S 9 5 ] . 3.1 T h e o r e m . Let L be an abelian group of exponent

1, 2, 3^ 4 or 6, or a

hamiltonian

Then U{2.L) =

2-group or a hamiltonian

Moufang 2-loop.

±L,

P R O O F . We establish the result in stages. Step 1. U{ZL) = ±L implies l/(Z[L

x C2]) = ±{1 x C2).

To see this, assume C2 = {x) and suppose (a + (ix)(^ + 8x) = 1 for a,P,j,S

G ZL, This means aj+pS

= 1 and a6+/3j

- 0, so ( a + /3)(7 + ^) =

1 and ( a — /3)(7 — (5) = 1. Thus a + /? = ±^1 and OL — (i — ±^2 for some 91-^92 ^ ^- So 2 a = ±^1 ± ^2- It follows t h a t g\ — ^2 ^nd either a = 0 or /3 = 0, proving the statement. Step 2. ZY(Z[Mi6(g8)]) = ± M i 6 ( Q 8 ) and U{ZQ^)

=

±Qs.

It is sufficient to prove just the first of these two assertions since Qg C MieiQs)^

T h u s , let L = M^eiQs)

= M{Qs,^,t,)

with Qs =

(We employ the notation of §V.4.) Note t h a t Z{L)

{Z{L),x,y),

— {\^x'^] and t h a t the

nonidentity commutator-associator of L is 5 = x^. Let z/ be a unit in ZL. Since L/{x'^)

is C2 x C2 x C2, ll{Z[L/{x'^)])

is

trivial, as already shown, so, multiplying by ±g for some g in L, we may assume t h a t 1/ = 1 + (1 - x'^){ao + aix + a2y + ot^xy) + (1 - x2)(/?o + (i\x + /322/ + for some a^, A* G Z, 0 < i < 3. Write a — {\ -

X^)(Q;O

(izxy)u

+ otix + 0^22/ + «3^2/)

and /? = (1 - x2)(/3o + fiix + /32y + /^a^^y). Then z/ = 1 + a + /3n and z/* = 1 + a* + x^/3tt, so z/z/* = [(1 + a ) ( l + a*) + x2(x2/?)*/3] + [{x^fi){\ + a ) + ^ ( 1 + a)]t^.

206

VIII. UNITS IN INTEGRAL LOOP RINGS

Since x^ = 1 and ab + ba = (1 + x^)a6 for any a^b ^ {^^y^^y}^ it follows t h a t x'^/3 = —/3 and t h a t i/i/* = ( l + a ) ( l + a*) + /?/3* = 1 + 2(1 - x^){ao + al + al + ali-al

+ fi^ + /3^ + (3j + / J | ) .

Since Z(x^) has only trivial units (Step 1), we conclude t h a t

But ao + ^Q > 0, so this forces all a^ and all /?^ to be 0 except possibly ao, which could be —1. So i/ = 1 or i/ = x^ and Step 2 is complete. Step 3. If L = C4 X C4 X • • • X C4, then ZY(ZL) = We proceed by induction on the rank of L.

±L,

If the rank is 1 (so t h a t

L is cyclic of order 4), then L is a subgroup of Qg and Step 2 tells us ZY(ZL) = ±L.

Assume then t h a t rank(Z/) = n > I and t h a t the result

is true for all such groups of rank less than or equal to n — 1. We have L - A X {a) X (b) where a^ = 6^ = 1 and rank(>l) - n-2.

Let 1/ G U{2.L).

The induction hypothesis, together with the result of Step 1, tell us t h a t ZY(Z[L/(6^)]) is trivial, so (multiplying by a trivial unit) we can assume t h a t u— 1 + (1 - 62)(ao + a i a + a^a^ + aaa^ + M

+ ^^ab + /Jsa^i + ^z^i^b)

for some a^,/?^ G l-A. Now ZY(Z[L/(a'^)]) is also trivial. Applying this to i/, we obtain a\ \ OL-^ - 0, /?o + /32 = 0, /3i + /33 = 0 and QQ + a2 G { 0 , - 1 } . Similarly ZY(Z[Z//(a^6^)]) is trivial. Applying this to v^ we obtain OL\ —03 = 0, /Jo - /32 = 0, /3i - /Ja = 0 and QQ — a2 G { 0 , - 1 } . Combining the last two facts, we obtain OL\ — otz — ^^^ — ^2 — ^\ — ^'^ — ^2 — 0, and ao = 0 or — 1. This means ^' = 1 or ^/ = 6-^, so ^' is certainly trivial. Step 4. If L = C3 X C3 X • • • X C3, then U(ZV) =

±L.

Let ^3 be a primitive cube root of unity and note t h a t ± ^ 3 , i = 0 , 1 , 2 , are the only units of Z[^3]. To see this, suppose a + 6^3 is a unit in Z[^3]. Since the m a p (p: Z[^3] -^ Z[^3] defined by v^(^3) = ^ | is an automorphism, a + 6^3 is also a unit, hence so is (a + b^3){a + 6^1) = a^ + b'^ + ab^s + ab^l = a^ + b'^ - ab.

3. TRIVIAL UNITS

207

Thus a^ + b^-ab = ±1 and, in fact, +1 because o? + b'^-ab = ( a - i 6 ) ^ + | 6 ^ . Since then 3(a — b)^ + (a + 6)^^ = 4, it follows that a = 6 or a = 0 or 6 = 0. Thus a + b^s = ±^3 for some i, as claimed. Again, we proceed by induction on the rank of L and first assume that L = (a) is cyclic of order 3. Let 1/ G K(ZL). Since the augmentation of 1/ is ± 1 , multiplying by ± 1 , if necessary, we may assume that 1/ = 1 + {1 — a)(/3o + Pia) for some /?o,/3i ^ Z. The quotient ring ZL/{1 + a + a^), being isomorphic to Z[^s]^ has only ±^3 as units. In this quotient, 1/ becomes F = 1 + (1 - ^s){0o + /3i6) = 1 + /3o + /?! + (2/Ji - /3o)6It is easily seen that all possibilities for 17 = ±^^ correspond to trivial units for I/. Now assume rank(L) = n > 1 and that the result is true for all such groups of rank less than or equal to n — 1. We have L = Ax (a) x (6) where a^ = 6^ = 1 and rank(yl) = n - 2 . Let 1/ G U{ZL), The induction hypothesis tells us that ZY(Z[L/(6)]) is trivial so (multiplying, if necessary, by a trivial unit) we can assume z/ - 1 + (1 - 6)[(7o + 7i« + 120^) + (^0 + O^a + e2a?)b] for some 7i, ^t G ZA. Similarly, U(2[LI{ab)]) is trivial, so the image of v in Z[L/(a6)] must be trivial. This image is 1 + (1 - a^)[7o + 7i« + 120? + (^0 + O^a + ^2^^)^^] = 1 + To - 71 + ^1 - ^2 + (71 - 72 + ^2 - Oo)a + (72 - 70 + ^0 - ^i)a^ It follows that precisely two of the above coefficients (in TA) must equal 0. This observation leads to three cases. Case 1: 71 - 72 + ^2 - ^0 = 0 and 7 2 - 7 0 + ^ 0 - ^ 1 = 0. In this case, the induction hypothesis tells us that the image of u in Z[L/(a^6)] is trivial. This image is 1 + (1 - a)[7o + 7i« + 72^^ + (^0 + O^a + 92a})a] = 1 + 70 - 72 + ^2 - ^1 + (71 - 70 + ^0 - 02)a + (72 - 7i + ^1 - Oo)o? - 1 + ^0 - 2^1 + ^2 + (^0 + ^1 - 2^2)a + ( - 2 ^ 0 + ^1 + 02)0?,

208

VIII. UNITS IN INTEGRAL LOOP RINGS

Either the second or third coefficient must equal 0 and so the first is congruent to 1 (mod 3). It follows that (9o + <9i - 2(92 = 0 = -29^ + ^i + ^2, forcing ^Q = ^1 = ^2 stud 70 = 71 =72- Finally, we note that the image of p in Z[L/(a)] is also trivial and that this image is 1 + (1 — i)(37o + 3^o^)- It follows that 7o = ^0 = 0; we conclude that 1/ = 1. Case 2: 1 + 70 - 7i + <9i - ^2 = 0 and 71 - 72 + <92 - ^0 = 0. In this case, the image of u in Z[Ll{o?'b)] is ^0 - 2^1 + ^2 + (1 + ^0 + ^1 - 2^2)^ + ( - 2 ^ 0 + ^1 + ^ 2 ) a ^

As in Case 1, we conclude that ^0 - 2^i + ^2 = 0 = -29o + ^1 + ^2- It follows that ^0 = ^1 = ^2 stud 71 = 72 = 1 + To- The image of u in Z[L/{a)] is now 1 + (1 — 6)(2 + 370 + 3^0^)5 so we conclude that 70 = — 1, ^o = 0 and u = b. Case 3: 1 + 70 - 7i + o - ^1 = 0. Here the image of u in Z[L/(a^6)] is 1 + flo - 2^1 4- ^2 + (1 + So + ^1 - 2^2)a + ( - 1 - 200 + ^1 + 02)0^. Again, as in Case 1, 1 + ^o - 2l9i + (92 = 0 = 1 + (9o + (9i - 2^2, so 62 = Oi — I + 0Q and 72 = 7i = 1 + 7o. Finally, the image of ^ in Z[L/{a)] is 1 + (1 - 6)[2 + 370 + (2 + 3(9o)6], so we conclude that 70 ^^ ^o = - 1 and In all cases, we have shown that u is trivial, so our proof is complete.

D

G. Higman found necessary and sufficient conditions for U{ZG) to be trivial in the case of a torsion group G [Hig40]. The classical proof ultimately comes down to determining when the ring of cyclotomic integers Z[^], ^ a primitive root of unity, has no units of infinite order. The answer to this comes from the Dirichlet Unit Theorem, which we will state in Chapter XII. We have chosen, however, to present an elementary proof due to E. Jespers, M. M. Parmenter and P. F. Smith [JPS95]. 3.2 Theorem (Higman). Let L be a torsion group or a torsion RA loop. Then U{ZL) = ±L if and only if L is an abelian group of exponent 1, 2, 3^ 4 or 6, or a hamiltonian 2-group, or a hamiltonian Moufang 2-loop, In Theorem 3.1, we established the result in one direction. On the other hand, because of Lemma 2.2, if L is not an abelian group and PROOF.

3. TRIVIAL UNITS

209

not hamiltonian, there must exist a^b E L such t h a t the bicyclic unit Ua^h is nontrivial. If L is an abelian group or hamiltonian, but not of the types stated, then L contains an element g of order n which is either 5 or greater t h a n 6. In these cases, (f)(n) > 2, so a nontrivial Bass cyclic unit exists by Lemma 2.7.

D

We next settle the question, for finite loops, of when merely all the torsion units are trivial.

T h e result is due t o S. D. Berman in the case

of group rings [Ber55] and to E. G. Goodaire and M. M. Parmenter for alternative loop rings [ G P 8 6 ] . T h e elementary proof given is due to M. M. Parmenter [Par95, Theorem 3.3]. Some of the ideas used were introduced by A. WiUiamson [Wil78]. 3.3 T h e o r e m . Let L be a finite group or an RA loop.

Then all

units in ZL are trivial if and only if L is an abelian group or a 2-group or a hamiltonian PROOF.

Moufang

torsion

hamiltonian

2-loop.

If L is an abelian group, CoroUary 1.2 says t h a t all torsion units

are trivial. If L is a hamiltonian 2-group or a hamiltonian Moufang 2-loop then, by Theorem 3.1, all the units (and so, indeed, all the torsion units) of ZL are trivial. For the converse, we first prove t h a t every subloop of L is normal. For this, it is sufficient to show t h a t every cyclic group contained in L is normal. Suppose the contrary and let (g) be a cyclic group which is not normal. By Lemma 2.2, L contains a nontrivial bicyclic unit fi = 1 + (1 — g)h^' figfi-'

= (1 + (1 - g)hg)g{l

Clearly,

- (1 - g)hg) = g + {I - 2g + g^)hg.

Because U{ZL) is diassociative, fJ^gfi'^ is a torsion unit, but it is not trivial; otherwise, since h ^ (g)^ it would follow t h a t h = ghg^ for some z, so /i would be trivial by Lemma 2.2. This contradiction proves t h a t every subloop of L is indeed normal in L. Next assume L is not an abelian group and not a 2-group.

Then L

contains a subgroup of the form Qs X (a:), where x has odd prime order p and Qs = (a, 6 I a^ = 1, a^ = 6 ^ ba = a^b)

210

VIII. UNITS IN INTEGRAL LOOP RINGS

is the quaternion group of order eight. Consider t h e element y = ax of order 4p in L. Since (4p) = 2{p — 1), t h e element

1/ = (1 + y + • • • + 2/^

) ^^ ^ +

y 4p

is a Bass cyclic unit which is nontrivial by Lemma 2.7. We show t h a t h~^ub ^ u. Assume t h e contrary. Since b~^yb — y'^P'^^ and y is central, we obtain

(l + y + y'+ • • • +

y'^-'f^^-'^ = ( 1 + j,2p+l ^ ^2(2p+l) ^. . . . ^ y(2p-2)(2p+l)^2(p-l)_

Multiplying b o t h sides by (1 - y)2(p-i)(l - t/2p+i)2{p-i), we obtain (1 _ ^2p-l)2(p-l)(i _ ^2p+l)2(p-l) ^ (1 _ y)2(p-l)(l _ j/-l)2{p-l).

Hence (2 - J/2P-1 _ j^2p+l)2(p-l) = (2 - 2/ - y-l)2(P-l) and, because y^^ = y~^^^ also {2y'r> _ y _ y - i ) 2 ( p - i ) = (2 _ y - y - i ) 2 ( p - i ) . Note, however, t h a t t h e coefficient of y"^^"^ on the right hand side is 0 while t h e coefficient of y^^~^ on the left side is negative (and nonzero), a contradiction. Next we claim t h a t i/ = y'^ + (I — y'^P)a for some a G Z(2/}. To see this, let H — {y)/{y^^)

and let ^ denote the n a t u r a l image in H of an element

g e {y). Then

2p Since y = 2j9^, it follows t h a t

3. TRIVIAL UNITS

211

2p r 2(p-l)

iWi-^)'^'-'^-'+ —^ \.2p =f E »=0

'

y

2(p-l)

=f

+ —^ % ^

=f +

'

y

2p

j=i

l.[(2p-lf(P-'^-l] y +

1-(2J3-1)2(P-1)^

2p

= y

Therefore, vy~^ — 1 € A ( ( y ) , (y^^)), so t* is indeed of the form v = y^ + {l — y^^)a,

a € Z{y), as claimed.

We complete the proof by showing t h a t uhu~^ is a nontrivial unit. For this, write v = ^'ij/ + ^'2 and u~^ = uj\y+u>2^ ^i^'^i € Z(2/^) for i = 1,2. Since 2/^ is central, ^-2^ = ^^"2 and, since yb = 6y^''+^ and u = j/'^ + (1 — y^'^)a, it follows t h a t v^y = 1 — y^^a\y

for some a i G Z(y^). Thus

(z/it/)6 = (1 - y ^ ^ K y f t = (1 - y2^)ai62/''+^ = 2/^^(1 - y^^K^^/ = - ( 1 - y'^^)aiby

= - 6 ( 1 - y^^)aiy

=

-buiy.

Hence ubv

= b{-uiyujiy

- U\yu2 + ^2^\y + ^2^2)

= b{-uiujiy'^

- v\UJ2y + i^2^iy + ^2^2)-

But (^it/ + ^2)(^iy + ^2) = 1 implies i/\Uiy'^-\-U2(J^2 = 1 ^.nd ^ia;2 + ^2^i = 0. Consequently i/bu~^ = 6 ( - l + i/2(-^2 + ^2^\y

+ ^2^12/ + ^2^2)

= 6 [ - l + 2(1/20^2 + ^2^12/)]; therefore, 6 G supp(i/6i/~^). Since vbu~^ has augmentation 1, if ubu~^ is trivial unit, then ubu'^

= 6, contradicting what we showed earlier.

D

212

VIII. UNITS IN INTEGRAL LOOP RINGS

4. Trivial central u n i t s In this section, we give necessary and sufficient conditions for all the central units in an alternative (not associative) loop ring to be trivial. This question has been settled for group rings by J. Ritter and S. K. Sehgal [ R S 9 0 ] and for alternative loop rings t h a t are not associative by E. G. Goodaire and M. M. Parmenter [ G P 8 6 ] . First we introduce the notion of local finiteness. As with groups, we say t h a t a loop L is locally finite if any finitely generated subloop of L is finite. 4.1 L e m m a . Let T denote the set of torsion Then T is a locally finite normal then T is

subloop of L.

elements

of an RA loop L.

If L is finitely generated^

finite.

P R O O F . TO

prove that T is a subloop, it is sufficient to show t h a t T is

closed under inverses and products, by Corollary II.2.3. Clearly T is closed under inverses. Also, if a and b are elements of T which commute, then certainly ab € T. On the other hand, if a and 6 do not commute, then ab is again of finite order since

{

a^6^

if n = 0 or 1

(mod 4)

5a^6^

if n = 2 or 3

(mod 4)

where s is the unique nonidentity commutator-associator of L. So T is a subloop. Because of Corollary IV. 1.11, to prove the normality of T , it is sufficient to show t h a t for ^ G T and £ G L, t'^ti

has the same order as t.

Since the loop generated by t and / is a group (Moufang's Theorem), this fact is obvious. Suppose t h a t K is a finitely generated subloop of T . If A' is commutative, then it is a group (see Corollary IV. 1.3) and, since it is generated by a finite number of torsion elements, it has finite order. If K is not commutative, then it contains V — { 1 , ^ } . Since s is both a unique nonidentity commutator and associator in L, KjV group of the abelian group LfV, so is K.

is a finitely generated torsion sub-

Hence KIV

is finite. Since V is finite,

Finally, note t h a t s ^ T since 5^ = 1. If L is finitely generated,

then T / L ' is finitely generated as a subgroup of the abelian group L/L\ before, we see t h a t T is finite.

As D

5. FREE SUBGROUPS

4 . 2 T h e o r e m . Suppose L is a torsion

Moufang

native loop ring which is not associative.

213

loop and 2.L is an

Then the central units in ZL are

trivial if and only if all units in 7.[Z{L)] and Z[L/V] and only if both Z{L) PROOF.

and LjV

are trivial; that is, if

are abelian groups of exponents

If all units of Z[Z(L)]

alter-

and Z[L/V]

2, 4 or 6.

are trivial, then Proposi-

tion 1.6 implies at once t h a t the central units of ZL are trivial. Conversely, assume t h a t all central units of ZL are trivial. Clearly t h e units in Z[Z{L)]

are trivial. Because L is torsion, replacing L by a finitely

generated RA subloop containing the support of a central unit, we may assume t h a t L is finite, by Lemma 4.1. Therefore, as noted in §2.5, and ZL^

0 Z{ZL)^

are both Z-orders in Z{QL) Z{ZL)

By Lemma 2.6, U{Z{ZL)) U{Z{ZL))

C ZL^^

0

LIL'

and

Z{ZL)^.

is of finite index in U{ZL^^®

is trivial and hence finite, so also is U{ZL^^

Therefore the unit loop of Z[L/L'] = ZL^^

Z{ZL)

Z{ZL)^).

Since

0 Z{ZL)^)

finite.

is finite as well; in particular,

is finite. Since the torsion units of the integral group ring of a finite

abelian group are trivial (CoroUary 1.2), all the units in Z[L/L'] are trivial.

D In concluding this section, we draw attention to the fact t h a t J. Ritter and S. K. Sehgal have proven that the central units of the integral group ring of a finite group G are trivial if and only if, for each g £ G and each j relatively prime to |G|, g^ is conjugate to g or g"^ [RS90]. This result holds for alternative loop rings in general and an elementary proof in the not associative case can be deduced from Theorem 4.2. The proof given by Ritter and Sehgal for the integral group ring of a finite group G ultimately relies on the Dirichlet Unit Theorem and on showing t h a t the character fields of the absolutely irreducible representations of G must be either rational or quadratic imaginary extensions of Q.

5. Free s u b g r o u p s In this section, we show t h a t for most finite loops L with ZL alternative, the unit loop U{ZL) contains a noncyclic free group.

VIII. UNITS IN INTEGRAL LOOP RINGS

214

We begin with the construction of free subgroups of SL(2, C), the special linear group of degree two over C. Although the result is stated with more generality t h a n Theorem 14.2.1 in [ K M 7 9 ] , the proof is almost identical. 5.1 T h e o r e m . Let ci and C2 be complex Define elements

numbers

with \ci\ = \c2\ > 2.

t\2 = ^12(^1) and ^21 = ^2i(<^2) ^^ SL(2, C) by

tuici) =

1

ci

0 1

and

^21(^2) =

1

0

C2

1

Then ti2 o,nd ^21 generate a free subgroup o / S L ( 2 , C ) ; that is, there are no nontrivial

relations

PROOF.

on these two

elements.

Let A — ^12(^1) ctnd B = ^2i(<^2)- Let W be an alternating

product of nonzero powers of A and B. We have to show t h a t W ^ I^ the identity matrix. Of course, this fact is obvious if W is simply a nonzero power of A or B. So we assume henceforth t h a t W contains nonzero powers of both A and B. If W begins with a power of 5 , then conjugate W by this power of 5 , and consider instead the resulting product, which begins with a power of A. Thus we may assume t h a t

where C is A or B and each a^ ^ 0. For each i = 1 , . . . , r , let Ri denote the first row of the matrix A^^ B^^ • • • C^'. R2k = R2k-\B''^^

If R2k-\

— (^2A:-i -> ^2A;)? then

= {x2k-{-UX2k)

R2k^\ = /22A:^''"^"^' =

{x2k-^l,X2k-^2),

where X2k-\-\ = X2k-\ + C2a2kX2k and X2A:-f2 = X2k + cia2k-\-iX2k-\-i' The last two equations can be combined in the single equation Xi^2 = Xi +

mai^iXi^i,

1 < i < r — 1, where m G C and |m| = |ci| = |c2| > 2. To prove the result, it is sufficient to show t h a t the number \xi\ increases as i increases from 1 to r + 1. Obviously 1 = | x i | < | a i | | c i | = |x2|. The inductive step k t + 2 | > |m||ai4-i||x,^.i| - |x,| > 2\xi^i\ yields the result.

- \xi\ = \xi^i\ + (|x,-+i| -

\xi\). D

5. FREE SUBGROUPS

Let r ( 2 ) be the principal

215

congruence group of level two; t h a t is, the set

of matrices in SL(2, Z) of the type

with a^b^c^d

G Z.

1 + 4a

26

2c

l + 4d

Denote by / the identity m a t r i x of SL(2,Z).

PSL(2, Z) = SL(2, Z ) / { / , —/} is the projective

Then

special linear group of degree

two over the integers. It is easily verified t h a t r ( 2 ) is the kernel of the n a t u r a l epimorphism P S L ( 2 , Z ) -^ PSL(2,Z2). For more information on principal congruence groups, we refer the reader to [ N e w 7 2 ] . 5.2 C o r o l l a r y , The group T{2) is a free subgroup o/SL(2, Z) and by the matrices PROOF.

generated

^12(2) and ^21(2).

Let A = ^12(2) and B = ^21(2). T h a t A and B generate a

free group follows from Theorem 5.1. Clearly (A^B)

C r ( 2 ) . To prove the

reverse inclusion, let X = [^21 xH ] ^ r ( 2 ) . We have to show t h a t X can be decomposed as a product of powers of A and B. Consider first the case |xi2| > |a;ii|. Noting t h a t xn ^ 0, we can write ^12 + k i l l = v{2xu) with v^r £Z

+ r

and 0 < r < |2a:ii|. Note t h a t r / 0 since X12 is even and x n

is odd. It follows t h a t Xii

—2x111; + a:i2

Xii

X21

- 2 x 2 i t ; + X22

2^21

r-

\xu\

-2x2it^ + X22

with \x[2\ < \x[i\ = | x i i | . Secondly, consider the case | x i i | > |xi2|. If X12 = 0, then x n = X22 = 1 and so X G ( 5 ) C ( ^ , B). On the other hand, if X12 ^ 0, write xu + k i 2 | = 'w;(2xi2) + r with ii;,r G Z and 0 < r < 2|xi2|. Let

X" = XB-"^ = Then X ' ' =

11 •*'12 L-*'21 ^ 2 2 J

x i i •- 2it;xi2 X21 — 2wx22

X12 2:22

with \x'li\ < [x'l^l = |xi2|.

r - |xi2|

X12

Xii — 2WX22 2:22

216

VIII. UNITS IN INTEGRAL LOOP RINGS

Applying the previous steps several times, if necessary, it follows that X is indeed a product of powers of A and B. D Let L be a finite RA loop which is not hamiltonian and let g and h be elements of L such that the cyclic group (g) is not normal in the group (^,/i); in particular, by Corollary IV. 1.11, L' = {l,'^} ^ (g). By Lemma 2.2, the bicyclic unit u^^g is not trivial. 5.3 Theorem. Let L be a finite RA loop which is not hamiltonian. Let g^h ^ L be such that {g) is not normal in the group (g^h). Then the group generated by the units Uh,g = 1 + (1 - g)hg

and

u'^g = 1 + gh{l - g)

is a free group of rank two. Let s be the unique nonidentity commutator-associator of L. Clearly s ^ (g). Let G be the group generated by g and h and recall that G/Z{G) ^ C2 X C2. Any group homomorphism (f:G^>^H determines a ring homomorphism (f: ZG -^ ZH and it is easily verified that
^i^h^g) = l + k{\-

(f(g))ip{h)(f{g)

= (1 + (1 - ^{g)){h)<^)y

= {u^ik)M9))'

and, similarly,

¥ ' « , , ) = (1 + ^Mh){l

- fig))' = Kiki^^g))'.

So, to prove the result, we may replace G by any homomorphic image ^{G) in which {(f{g)) is not normal in {(p(g)^(f{h)); in particular, since s ^ (^r^), we may replace G by G/{g^) and assume that g has order 2. Since (g) is not normal in G, the idempotent ^ = -^ is not central in QG. Hence, there exists a primitive central idempotent e in QG^^ so that the idempotent - ^ e is not central in (QG)e, which is a quaternion algebra, by Lemma VIL2.2. Note that - ^ e is a zero divisor in {QG)e because ^^ = 1, so {QG)e is split and hence isomorphic to a 2 x 2 matrix ring over a field

5. FREE SUBGROUPS

217

F , by Corollary 1.4.17. By Theorem VII. 1.4, F — Q(
^21 = V ^ ^ ^ ,

^22 - V ^ .

It is easily verified that these elements form a set of matrix units in (QG)e, so we can represent the elements of (QG')e as 2 x 2 matrices with respect to these matrix units. To do so, we write /? G (QG)e as /? = S K ^ 7<2 EaPEjj. Hence and

ge

he =

0 a 1 0

for some a G Q(0* I^ follows that a 0 0 a Since h^ is a central torsion unit and the only torsion units in Q ( 0 ^^e powers of ^ or the additive inverses of such elements, we see that a = ±^^ for some i > 0. Therefore Uh.ge

=

[1 0 0

1

+

"0 0 0 2

0 1

0

2 0 0 0

and, similarly. %g'

1 ±4f 0 1

By Theorem 5.1, the elements Uh,,ge and u^ e generate a free group; therefore Uh.,g and w^ generate a free group in H{Z{g,h)). D 5.4 Corollary. Let G be a finite group such that G/Z{G) S C2 X C2. If Ufi^g = 1 + (1 — g)hg is a nontrivial unit, then u^^g and u^ = 1 + 'gh{l — g) generate a free subgroup ofU{ZG). Since GIZ{G) = C2 x C2, the group G has the LC property and a unique nontrivial commutator (Proposition III.3.6). Hence G can be embedded in an RA loop (Theorem IV.3.1). The result now follows at once from Theorem 5.3. D PROOF.

VIII. UNITS IN INTEGRAL LOOP RINGS

218

5.5 Remark. Z. S. Marciniak and S. K. Sehgal have shown that if g and h are elements of an arbitrary group G with g of finite order and if the bicyclic unit Uh^g G ZG is nontrivial, then u^^g and 1 +^/i~^(l — g~^) generate a free group [MS]. To construct free groups in integral loop rings of hamiltonian Moufang loops, we need the following application of a more general result of J. Tits [Tit72] on the construction of free subgroups in linear groups. (See also the proof of Lemma 5.3 in [Seh93].) 5.6 Proposition. Let ^p denote a primitive pth root of unity. Then there exists a positive integer n such that the subgroup o/GL2(C) generated by

[ 1 [-ip

n

^p' 1 _

and

" 1 - ^pi 0

0 l + ^pi

is free of rank two. Recall that for a finite subloop TV of a Moufang loop L, we write N — m\ ^neN ^ ^ Q^* Also, for g e L of finite order n, we write ^ rather than (g). Thus g= 1 ( 1 + ^ + ^ 2 ^ . . . + ^ - - ! ) . 5.7 Theorem. Let L be a finite hamiltonian is not a power of 2. Thus there exist a^b^c quaternion group of order 8 and c is a central p. Let fii and ^2 denote the following elements

Moufang loop whose order £ L such that (a,6) is the element of odd prime order of QL:

//i = 1 + (1 — a)(l - Z)ac ;X2 = 1 + ( 1 - S ) ( 1 - ? ) 6 C . Let m be the order of the unit group of the finite ring Z[^4p]/2pZ[^4p], where ^4p is a primitive Apth root of unity. Then^ for some positive integral multiple n ofm, the group generated by ji^ and ^'2 is a noncydie free group contained inUiZL). PROOF.

Let G - {a,b,c) = Q^x Cp^ where Qs = {a,b\a^

^ b'^.a'^ = l,ba = aH)

5. FREE SUBGROUPS

219

and Cp = (c) is cyclic of order p. Then e = i ( l - a 2 ) [ l - l ( l + c + - . . + cP-i)] = ( l - a 2 ) ( l - c ) is a central idempotent in the rational group algebra QG. Because of Corollary VII.2.4, QQs has only one noncommutative simple component, namely, H(Q). By Lemma VIL2.2, the corresponding primitive central idempotent is ^ , so ( Q ^ s ) ^ = H(Q). Because of Lemma VIL1.2,

where ^p is a primitive pth root of unity. Thus, (QG')e ^ QQs'-^

®Q QCp[l - i ( l + c + • • • +

CP-^)]

^ H(Q) ®Q QC^p) ^ H(Q(^p)). Consequently, (CG)e ^ C ®Q (QG)e ^ H(C ®Q QC^p)) = ©H(C), as shown in §1.5.12. By Theorem 1.3.4, H(C) is split and hence isomorphic to M2(C), by Corollary 1.4.17. Thus (CG)e ^ ®M2{C). Let / be a primitive central idempotent of (CG)€, Then {CG)f = M2(C). Let X = fa^ y = fb and z = fc. Noting that a^e = —e and hence a? f = —f^ii follows that x{iy) = {iy)x^,

x^ = /

and

{iyf

= /.

Hence, in (CG)/, the elements x and iy generate a group isomorphic to 2)4, the dihedral group of order eight. It follows that ^11 = U^+iy)

and

E22 = U^ - iy)

are noncentral idempotents of (CG)/. Together with ^12 = 1(1 + iy)xl{l

- iy)

and

E21 = - ^ ( 1 - iy)xl{l

+ iy),

it is easily verified that these four elements form a set of matrix units of (CG)/ = M2(C). The representations of a:, y and z as matrices with respect to these matrix units are 0 1 ' -1 0

' -i ,

y =

0

0 1 i \

anc

z =

^p

0

0

ep

VIII. UNITS IN INTEGRAL LOOP RINGS

220

Hence the matrix representations of / / i / and /X2/ are 1

Ml/ =

^p

and

/i2/ =

1 - ^pi

0

I-(pi

0

By Proposition 5.6, there exists a positive integer k such that the group generated by (//i/)^ and (/X2/)^ is free of rank two. Since the group iif^if)^^^i/^2f)^^) is a homomorphic image of the group ( M I ' ^ , / X 2 ^ ) , it remains to show that /xf^ and /X2^ are units in ZG. By symmetry, it is sufficient to prove this for/xf'". Because of Theorem VII. 1.4, e = ( l - a 2 ) ( l - 2 ) is also a primitive idempotent of Q(ac) with (Q(ac))e ^ Q(e4p) = Q(t^p), where ^4p is a 47?th root of unity. Under these isomorphisms, the element ace corresponds to i^p, thus (Z{ac))e = Z[i^p]. Clearly, 1

(i + i^p)(i-»ep) = i + ep' = £^'2 - 1 ' Sp

and the latter has inverse ep -1 1

^'^ tp' - 1 - = l + e p ^ + . . . + e^^(^-^)GZ[iep],

where g' is a positive integer such that 2q = I (mod p). Hence 1 + i(p is invertible in Z[i^p], so (1 + ac)e is invertible in (Z(ac))e and thus also in (ZG)e. It follows that /Xi = 1 + ecic = ( 1 — e)-|-e(l-|- ac) is invertible in the order (ZG)e © (ZG)(1 — e). The definition of m shows that (1 + i^p)^ G 1 + 2pZ[i(p]. Consequently, /x^^^ = (1 - e) + e(l + 2j9a) for some a 6 Z{ac). Hence //^^ = 1 + 2pae 6 ZG. Finally, Lemma 2.6 (part 4) yields that /x^^ is a unit of ZG. D Since RA loops are not commutative, the foDowing corollary is the loop analogue of the Hartley-Pickel result for group rings [HP8O]. 5.8 Corollary. Let L be a finite RA loop. ThenU{ZL) only ifU{ZL) contains a free group of rank two.

is nontrivial if and

5. F R E E SUBGROUPS

221

Clearly, iiU(ZL) contains a free subgroup of finite index, then U(ZL) is not trivial. For the converse. Theorem 3.2 implies that L is not a hamiltonian 2-loop. The result then follows from Theorems 5.3 and 5.7. D PROOF.

Chapter IX

Isomorphisms of Integral Alternative Loop Rings

In this chapter also, most group rings and loop rings will be integral; that is, the coefficient ring wiU generally be the ring Z of rational integers, and our results specifically apply to alternative loop rings which are not associative. Our objectives are three-fold. First we settle the "isomorphism problem" for RA loops: we show that a torsion RA loop is determined up to isomorphism by its integral loop ring. Secondly, we show that if L is an RA loop, then every normalized automorphism of ZL is the composition of an inner automorphism of the rational loop algebra QL and an automorphism of L. Finally, we establish the validity for integral loop rings of RA loops of three conjectures about group rings made by H. J. Zassenhaus in the 1960s. For instance, we show that every torsion unit a € ZL is of the form ±7.^^ (71"^071)72 for some a ^ L and units 71,72 in QL. Most of this chapter is based upon [GM88, GM89, GM96b]. Our consideration of normalized automorphisms and the Zassenhaus conjectures requires knowing that every automorphism of a composition algebra is inner. This is a theorem of N. Jacobson, discussion of which forms the basis of Section 2. Let i2 be a commutative, associative ring with unity and of characteristic different from 2. Let L = M{G, *?5'o) be an RA loop. Thus G is a nonabelian group with an involution * and go is a central element of G. The centres of G and L are the same and //' = G' = {1,6}, where s is necessarily central of order 2. Recall that the involution * extends to RL in such a way that a G RL is central if and only if a* = a. Let e: RL -^ Rhe the augmentation map. If TV is a normal subloop of //, AR(L^N) denotes the kernel of the natural homomorphism €j\j: RL -^ 223

224

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

R[L/N], (We often write A(L,iV) when R is clear from the context.) The identity r = e{r) + {r — €{r)) for r G RL implies that RL = R-\- A(iy), where A(L) = A(L,L) = ker€. Since A{L^N) is the ideal of RL generated by all elements of the form n - 1, n G TV, it follows that A{L, N) = RL(A(N)) = A(N) + A(L)A(7V). From this it is easy to see that (1)

[^{L,N)f

C A{L)A{N)

C A(L,7V).

Since TV* = TV, we have also

(2)

[A(L, TV)]* C A(L, TV)

and

[A(TV)]* C A(TV).

Throughout this chapter, we rely heavily on the fact that a central torsion unit in an integral alternative loop ring is trivial (see Corollary VIIL1.7). We also make extensive use of the following two lemmas, the second one a result of A. Whitcomb which we state here for integral alternative loop rings. 0.1 Lemma. Let L be an RA loop or a group with V = { l , ^ } .

Then

L n ( l + Az(L)Az(L')) = { l } . Since Az{L') is the ideal of ZV generated by 5 — 1, if £ G L n (1 + Az{L)Az{V)), then i - \ ^ r{s - \) for some r G Az{L). Thus {(. - \){s + 1) = 0 and so ^ + 5^ = 5 + 1. It follows that i G { l , ^ } . Now Az{L) is the Z-linear span of {p — 1 | p G Z/} because of the identity h{g - 1) = {kg - \) - {h- 1). In particular, Az{L') = Z{s - 1). Since ^ - 1 G Az{L)Az{L'), we have PROOF.

(3)

£_i = ^a,(y-l)(6-l),

a, G Z .

geL

Letting T be a transversal for L' in L (with 1 G T) and writing each g ^ L in the form g = ti\ t £T^ i^ £ L', the map g ^-^ t' extends to a Z-linear map r\ZL-^ZL'. Now apply r to (3). Smc^T{g-\){s-\) = r{tt's-tt'-s+l) = t's-e - s ^ \ = {e - l){s - 1), which is 0 or (5 - 1)2 = 2(5 - 1), we obtain an equation of the form t — \ = 2a{s — 1) for some integer a. Since the coefficients on the right are even and those on the left are odd, we obtain £ = 1, as claimed. D 0.2 Lemma (Whitcomb). Let L be an RA loop or a group and let TV be a normal subloop of L. If x G ZL and x = t (mod A(L, TV)) for some i ^ L,

1. THE ISOMORPHISM THEOREM

225

then there exists an element £o £ L such that x = io (mod A{L)A{N)). N = L', then IQ is unique. PROOF.

If

With e the augmentation map ZL -^ Z, we have X = i+ ^

an{n - 1),

= £+Y^{{an

an G ZL

- e(an))(n - 1) + e{an){n - 1)}

neN

= £+J2<^n){n-l)

(mod A(L)A(7V))

neN

since a^ — ^(<^n) G A(Z/). For ni,n2 € iV, observe that (ni-l) + (n2-l) = (nms - 1) - (ni - l)(n2 - 1) - (^1/12 - 1)

(mod A(L)A(A^)).

Hence, applying this to nj and 712 with nin2 = 1, we obtain that —(n—l) = {n~^ — 1) (mod A{L)A{N)) for all n ^ N. Since €(a) is an integer, we obtain, for some UQ £ N^ x = i+{no-l)

= ino-{i-

l)(no - 1) = irio

(mod A(L)A(A^)).

With £0 = £^0? we obtain an element of the required type. If A^ = L', uniqueness of io is an immediate consequence of Lemma 0.1. D 1. T h e i s o m o r p h i s m t h e o r e m The isomorphism problem for group rings, posed by Graham Higman in his 1940 thesis, asks if a group is determined by its group ring; that is, given a group G and a ring /2, does RG = RH for some group H imply that G = /T? It is well known that the strongest context in which to study this question is that of integral group rings because if ZG = ZH for groups G and H^ then RG = RH for every commutative and associative ring R with unity since RG = R®z ZG. G. Higman himself proved that if G is a finite abelian group and H is another group such that ZG = ZH^ then G = H. Later, A. Whitcomb proved that finite metabelian groups are determined by their integral group rings [Whi68]. (A group G is metabelian if G has an abelian normal subgroup N with an abelian quotient G/N; equivalently, if both G' and G/G^

226

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

are abelian.) An interesting result of R. Sandling states t h a t a finite group which can be realized as the group of units of a ring is determined by its integral group ring [ S a n 7 4 ] . If G and H are finite, the hypothesis ZG = 2.H implies t h a t G and H have the same table of characters. Since finite permutation groups are determined by their complex character tables [ J K 8 1 , Theorem 2.6.1], it follows t h a t Sn is also determined by its integral group ring. K. W. Roggenkamp and L. Scott showed t h a t finite nilpotent groups are determined by their integral group rings, a result which also follows from a recent result of A. Weiss (see Section 4). The isomorphism problem over fields was first considered by S. Perils and G. L. Walker who proved t h a t if G is a finite abelian group and H is any group with QG = QH, then G = H [ P W 5 0 ] . A similar result was obtained by W. Deskins who proved t h a t if G is a finite abelian p-group, if F is a field of characteristic p > 0 and if ^ is a group such t h a t FG = FH ^ then G = H [Des56]. At this point, we draw the attention of the optimistic reader to a striking counterexample of E. Dade who exhibited two metacyclic groups G and H which are not isomorphic but are such t h a t FG =. FH for all fields F [ D a d T l ] . (A group G is metacyclic that both A^ and GjN

if G has a normal subgroup A^ such

are cyclic.)

Finally, we should mention t h a t it has been a long-standing conjecture in this area t h a t if G and H are p-groups and F^ is the field of p elements, then Fj,G ^ F^H implies G ^ H (see, for example, [ Z M 7 5 ] ) . R. Sandling has shown this to be the case when G is a central-elementary-by-abelian p-group [ S a n 8 9 ] . For a detailed history of the isomorphism problem, the reader is referred to the survey of R. Sandling [ S a n 8 5 ] . T h e purpose of this section is to show t h a t the isomorphism problem over Z has a positive solution for torsion RA loops.

1.1 T h e o r e m . Let L\ and L he RA loops with L\ torsion morphic

and TL\

iso-

to 2.L. Then Li = L.

PROOF.

normalized

Let 0: ZL\ in

-^ ZL be an isomorphism.

We assume t h a t 6 is

the sense t h a t it preserves augmentation. There is no loss

of generality in so doing since, if p: ZLi -^ ZL is any isomorphism, then

1. THE ISOMORPHISM THEOREM

227

ep{tj = ±1 for any ^ G Z/i, so the map p: ZL\ -^ ZL defined by p(i) = [€p(£)]~^p(£) extends linearly to a normalized isomorphism ZLi -^ ZL. Denote by x the image in Z[LlV] of an element x ^ ZL under the natural homomorphism e^i: ZL -^ Z[L/L^], Let £ E L-^, Then 9{i) is a torsion unit whose image 6(1) in the abelian group ring Z[L/L^] is a torsion unit and hence trivial, by Corollary VIII. 1.4. Since its augmentation is + 1, 0(i) = t^ for some t £ L. Since kereit = A(i/,iy'), we have 0(£) = t (mod A(Z/,L')) and so, by Whitcomb's Lemma, e{i) = i,

(mod A(L)A(L'))

for some unique ip G L, Thus p: i \-^ ip is di well defined map Li —^ L. We complete the proof by showing that p is, in fact, an isomorphism. For this, note first that if e{l>^) = hp

(mod A(L)A(LO)

and ^(£2) - ^2p

(mod A(L)A(LO),

then

e{i^i^) 3Z e{t,)e{t2) = iiphp (mod

A(L)A(L')).

By uniqueness, ^(^1^2) = hp^2p = p{(^\)p{('2)'> so /> is a homomorphism. Suppose p{t) — 1. Then (4)

e{i)

= 1

(mod

A(L)A(L0),

so e(e'^) = 9{if = 1 (mod A(L)A(LO). Since i'^ is central in Lu 0{e^) is a central torsion unit in ZL, and it is a trivial unit. Since its augmentation is l,e(P) = a e Z{L). Since a-\ G A(L)A(L'), Lemma 0.1 shows that a = 1; that is, 9{iy' = 1. Consider again equation (4). Since A(L)A(L') is closed under the involution * on ZL, we have 9{iy = 1 (mod A(L)A(L')) and hence 9{i)9{iy = 1 (mod A(L)A(LO). Now 9{t)9{ty is a central torsion unit in ZL, and of augmentation 1, so it is in L. Another application of Lemma 0.1 shows that 9(i)9{iy = 1. Comparison with 9(i)'^ — 1 shows that 9(tj — 9{iy, Hence 9{JC) is central. As a central torsion unit of augmentation 1, 9{Ji) G L, hence 9(J[) — 1, by (4). Since 9 is one-to-one, i — 1, so /> is one-to-one.

228

IX. ISOMORPHISMS O F INTEGRAL ALTERNATIVE LOOP RINGS

Suppose t G Z{Li),

Then 6(i) is a central torsion unit in ZL, hence

trivial and, because it has augmentation 1, in Z{L). p(i)

= e{i).

Thus p{Z{Li))

C Z{L),

Also, if a G Z{L),

central, torsion and of augmentation 1, hence in Z{L\). just observed, p{6~^{a)) Z{L).

— 0{6~^{a))

By definition of p, then ^ " ^ ( a ) is

By what we have

— a. It is clear then t h a t p{Z{L\))

=

Since /9 is a one-to-one m a p from one RA loop to another, and

since an RA loop is generated by its centre and any three elements which do not associate (see §IV.3) it is immediate t h a t p is onto and hence an isomorphism.

D

2. I n n e r a u t o m o r p h i s m s of a l t e r n a t i v e a l g e b r a s Let A be an alternative ring with unity and, for x G ^ , let R{x) and

L{x)

denote the usual right and left translation maps respectively. Thus aR{x) = ax

and

aL{x) = xa

for a ^ A. For consistency with other parts of this book, whenever these maps arise, all maps will be written on the right. If a , x G v4 with x invertible, we showed in §11.5 t h a t a, x and x~^ generate an associative subring of A. It follows t h a t R(x) and L{x) are invertible maps with inverses and L{x~^)

R{x~^)

respectively.

2.1 D e f i n i t i o n . An inner automorphism

of A is an automorphism which

is in the group generated by {/2(x), L{x) \ x ^ A invertible}. If A is associative, any automorphism 0 which is inner in the above sense, is of the form 6{a) = uav for some invertible elements u^v £ A, Since 6(1) = l^ u = v~^ and so 9 is inner in the associative sense. We wish to characterize the inner automorphisms of an alternative ring in a different way. To do so, we introduce in A three maps which we have previously defined on a Moufang loop. For invertible elements x , y G ^4, T{x) = R{x,y)=

R{x)L{x)-^ R{x)R{y)R{xy)-'

L{x,y)=L{x)L{y)L(yx)-\

2. INNER AUTOMORPHISMS OF ALTERNATIVE ALGEBRAS

229

2.2 Proposition. Let Aut(A) and Inn(A) denote, respectively, the group of automorphisms and the group of inner automorphisms of an alternative ring A. Let H be the group generated by {T(x), i2(x,y), L(a:,y)} for all invertible elements x and y in A. Then Inn(A) = Aut{A) 0 H. PROOF.

We present a proof suggested by D. A. Robinson. Let G - {R(x), L{x) \x e A invertible)

so that, by definition, Inn(A) = Aut(A) fl G. Since 77 is a subgroup of G, it is clear that Aut(A) f) H C Inn(^). Thus it suffices to prove that Inn(^) C H. For this, we consider the set

s = {0eG\ee

HR{\6)]

and show that SG C S. To see why, let 0 E S^ let x be an invertible element in A and let r = 6R(x). Then 0 = pR{\0) for some y9 G H^ so T = 0R{x) = pR{\0)R{x)

= pR{l0,x)R{l0'x)

=

pR{l0,x)R{\T);

hence r £ S. Similarly, with r = 0L{x) and 0 = pR{l0)^ p £ H^ we have r = 0L{x) =

pR{l0)L{x)

= pT{10)L(l0)L{x), =

since R{10) = T{10)L{10)

pT{l0)L{l0,x)L{x'l0)

= pT{10)L{l0,x)T-\x

' \0)R{x ' 10)

= pT{10)L{l0,x)T-\x

. 10)/?(lr);

hence r G 5. Since G is generated by all R{x) and L(x)^ it is clear that SG C 5, as asserted. Since the identity map id^ on A is in H^ id^ = id^ R{1 id^i) belongs to 5, so G = S. Finally, if ^ G Inn(/l), then 0 e G = S, so 0 e HR{\0) = H because \0 = 1. This completes the proof. D Suppose now that A is a composition algebra over a field F of characteristic different from 2. We refer the reader to §1.4 for a general discussion of composition algebras and, in particular, to the proof of Hurwitz's Theorem, Theorem 1.4.11. We denote by q{a) and a i-^ a, respectively, the multiplicative quadratic norm and involution on A] thus q{a)\ = aa for a e A. Writing A = Fl ® (Fl)-^ and a = a l -f a' with a e F and

230

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

o! e (Fl)-^,

recall t h a t a = al - a\

a^ = —aa G Fl.

If a G (Fl)-^,

then a ^ F l , but

Conversely, if a = al + a^ ^ Fl and a^ e Fl^ then a' ^ Q

and a? = Q^l + 2aa' + a'^ = ( a ^ l - a V ) + 2aa' shows t h a t a = 0. Thus ( F l ) ^ = {a G A I a ^ F l , a 2 G F l } . It follows t h a t (5)

e{Fl)^

for any F-automorphism 6{a\)

C (Fl)^

9 oi A ( t h a t is, any automorphism of A satisfying

= al for a G F ) . Let a = a l + a' with a G F and a' G ( F l ) - ^ and let

e be an F-automorphism of A, Then e{a) = al + 0{a') with e(a') G ( F l ) - ^ , so '0(a) = a l - d{a') = 9{a)\ thus, e{a) = e{a) for any a ^ A. Furthermore

(6)

q{a) = qiOia))

for any a e A because q{a)l = 6{q{a)l)

= 6{aa) = 6{a)9{a) —

q{9{a))\.

T h e primary aim of this section is to prove a theorem of N. Jacobson showing t h a t any F-automorphism of a composition algebra with centre F is inner. To do this, we follow the general pattern of proof in [ J a c 5 8 ] . An F-automorphism r of yl is a reflection if r'^ = i d ^ , the identity map on A, Let J5 be a subalgebra of A of dimension ^ dim A and on which the restriction of q is nondegenerate. Thus A = B ® B^ and B^

= Bi^ with

q{i) = —a ^ 0. Moreover, multiplication in A is defined by (7)

(x + yi){z + w(,) = (xz + awy) + {wx + y'z)i

for x,t/ G B,

Define r: A -^ Ahy

r^^ = i d ^ , T\ ^ = — i d ^ ± . Clearly r

is one-to-one and onto, and r'^ = id^. Moreover, it follows directly from (7) t h a t r is a homomorphism; thus r is a reflection. We call such r the reflection

in B, In fact, every reflection r is reflection in some subspace of

dimension | dim A and on which the restriction of r is nondegenerate, as we proceed to show. Suppose t h a t r is any reflection of A, The equation a= \{a + T{a)) + \{a -

T{a))

2. INNER AUTOMORPHISMS OF ALTERNATIVE ALGEBRAS

231

shows that A = A^ © A_ is the direct sum of subspaces where A^ = {a e A \ T{a) = a}

and

A- = {a e A \ T{a) = - a } .

Clearly T{A^) C A ^ and r(A_) C A_. Let / be the bilinear form associated with q. Thus / ( x , y)l = (q(x + y)-

q{x) - q(y))l = xy + yx

for any x,y e A. Let x e A^ and y G A-. Since r fixes F l , (8)

xy + yx = T{xy + yx) = T{x)T{y) + r(2/)r(x) = -xy - yx

so that f{x,y) = 0. It follows that / is nondegenerate on A^ and also that A- = (yl_|_)-^. Again, from the proof of Theorem L4.11, we see that for some nonzero i G ^ | , A^i C y l | = A-. Since >l = A^ © >1_ and because dimA^i = dimi4_|-, we have A^ = A^l. In particular, dim A4. = ^ dim >1 and r is the reflection in A^. 2.3 Proposition. Every reflection of a quaternion or Cay ley-Dickson algebra over a field of characteristic different from 2 is inner. Let A he a, quaternion or Cayley-Dickson algebra over a field F of characteristic diflFerent from 2. Denote by q and / , respectively, the norm on A and its associated bilinear form. Suppose r is a reflection of A in a subalgebra 5 , where A = B ® B^, B = A^ and B^ = A^. If A is quaternion, then B = Fl + Fi for some i G (^l)"*" with q{i) ^ 0. (Thus i is invertible.) For any a G B^^ we have 0 — f(i^a^ — ia + ai — —ia — ai^ so aT(i) = i~^ai — —a — ar. Thus T{i) and r agree on B^, They also agree on 5 , since B = A^ is commutative. Thus r = T(i) is inner. If yl is a Cayley-Dickson algebra, then B = Fl + Fi + Fj + Fij is quaternion, where i and j are invertible and ij + ji = 0. In view of (7), for any a G B^^ we have aL{j^i) = {ij)~^{i -jo) = {ij)~^(ji * ct.) = —a = ar^ so r and L{j,i) agree on 5-^-. They also agree on B because B = A^ is associative, so r = L{j^ i) is inner. D PROOF.

2.4 Theorem (Jacobson [Jac58]). Over a field F of characteristic different from 2, any F-automorphism of a quaternion or Cayley-Dickson algebra is the product of reflections and hence inner.

232

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS PROOF.

Let A be a quaternion or Cayley-Dickson algebra over F and

let 6 be an F-automorphism of A, Let q{a) — aa be the quadratic form on A^ where a \-^ a denotes the involution on A^ and let / be the bilinear form associated with q. Thus f{x^y)\

= xy + yx for x^y £ A. Let i G

be such t h a t q{i) ^ 0. Then 0 / V = i — 6{i). Since 6{F\)^

i^ = - n G F l .

C {F\)^

(Fl)-^

Set u = i + (9(i) and

by (5), each of u and v is an element

of (Fl)-'-. This implies u = —u and v = —v.

Since i^ G F l , we have

i^ =z 6(P)^ thus uv + vu = 0, so f(u^v)

= 0. Since q{0{i)) = g(i), by (6),

we have q{u) — q{i) + q{0{i)) + f(i^0(i))

— 2q{i) + f(i^6{i))

q{v) = 2q{i) - f{i,e{i)).

and, similarly,

Thus either q{u) ^^ 0 or q{v) ^ 0.

Assume first t h a t q{u) ^ 0. If A is quaternion, let B — F\ -\- Fu and let r be the reflection in B. Note t h a t T{U) = u and T{V) = —v because v G B-^. Now the restriction of / to (Fl)^

is nondegenerate by Corollary L4.6 and dim(Fl)-'- = 7 by

Proposition L4.4. If A is Cayley-Dickson, let Bi be the subspace of which is orthogonal to u^ v and uv.

(Fl)^

Since u^ v^ uv span a subspace of

(Fl)-*- of dimension at most 3, it follows again from Proposition 1.4.4 t h a t d i r n d l > 4. Since dim(Fl)-'- = 7, the maximum dimension of a totally isotropic subspace is 3, by Corollary 1.4.3, so there exists an element j G 5 i with q{j) /

0. Since f{j^u)

= 0, the subalgebra B2 generated by j and

u is quaternion.

Let r be the reflection in B2.

f{l,v)

= f{u,v)

= f{j,v)

= 0 and f{ju,v)

Note t h a t v G B2 since

= f{vuj)

= -f{vuj)

= 0.

(See (1.32).) Again we note t h a t T{U) — u and T(V) = —v. Thus, whether A is quaternion or Cayley-Dickson, we have i + e{i) = u = T{U) = T{i) +

T(0(i))

i - 0(i) = v= -T{V)

+ r((9(0),

and = -T{i)

and hence T(0(i)) = i. Now assume t h a t q{v) 7^ 0. As above, there exists a reflection r^ such t h a t Ti{v) = V and TI(U) = —u. Thus T\6{i) = —i. Since q{i) 7^ 0, i is part of a basis provided by the Cayley-Dickson process; hence yl = S3 © B^ for some B^ with i G ^ 3 - . With T2 the reflection in ^ 3 , we have T2r\d{i) = i.

3. AUTOMORPHISMS OF ALTERNATIVE LOOP ALGEBRAS

233

T h u s , whether q(u) 7^ 0 or q{v) / 0, there exists a reflection or a product of reflections r such t h a t T9(i) = i. Replacing 6 by r^, it suflRces to prove t h a t 0 is the product of reflections in the case t h a t 0{i) ~ i. So assume 6{i) = i and let j G ( ^ 1 + Fi)^ Then ij + ji = 0, so also iO(j) + 0{j)i 0{j) e (Fl + Fi)^,

Let X = j+e{j)

be any element with q{j) ^ 0.

= 0 (because 0{i) = i) and hence

and y = j-e(j).

Then x,y e (Fl +

Fi)^,

f(^x^y) = 0 and, as before, q{x) 7^ 0 or q(y) 7^ 0. Suppose A is quaternion. If q{x) 7^ 0, F l + Fx is nondegenerate and we let r be the reflection in Fl + Fx.

Then T(X) = x and T{y) = —y, so

T0{j) = j . Also T6{i) = T{i) = —i, so T9 is the reflection in F l + Fj.

If

q(^y^ 7^ 0, then F l + Fy is nondegenerate and we let r be the reflection in F l + Fy.

This time we have TO(J) = —j. Since again T9(i) = —i, we see

t h a t T6 is the reflection in F l + Fij.

This completes the case t h a t A is

quaternion. Suppose t h a t >1 is a Cayley-Dickson algebra. Then the subalgebra generated by i and x, if q(x) 7^ 0, or by i and y^ if q(y) 7^ 0, is quaternion. Let r be the reflection in this subalgebra. Thus T6{i) = i and T6{J) — ±j. TO(J)

If

= —J, choose £ with ^(^) 7^ 0 orthogonal to the quaternion algebra

generated by i and j and let ri be the reflection in the quaternion subalgebra generated by i and i. Then T\T6{i) = i and

TIT9(J)

= j . Replacing 0

by r r i ^ , we can assume 6{i) = i and 0{j) — j . Now choose ^ = ( F l + Fi + Fj + Fij)^ 3indz = i-e{i).

Theni,e{i),z,te

with q{i) 7^ 0. Set w = i + 9(1)

(Fl + Fi+Fj+Fij)^

^^nd f{w, z) = 0. If

q{w) 7^ 0, let T be the reflection in the quaternion subalgebra generated by i and w and, otherwise, the reflection in the quaternion subalgebra generated by i and z. Then T6(i) = z, r0(j') = —j and

T6((.)

= ± ^ , so

T6

is a reflection

in a quaternion subalgebra containing i. This completes the proof.

D

3. A u t o m o r p h i s m s of a l t e r n a t i v e l o o p algebras Naturally related to the isomorphism problem, there is another question, which we first discuss within the context of group rings.

W h a t are the

automorphisms of an integral group ring ZG? Any automorphism of the group G can be extended linearly to an automorphism of ZG. Also, if 7 is an invertible element in the rational group algebra QG such t h a t j~^gj

G ZG

234

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

for all ^ G G, then the map (f^: 2.G -^ ZG given by (p^{g) = J~^gj is also an automorphism of ZG [Seh93, Lemma 34.4]. It has been conjectured that all normalized automorphisms of ZG are compositions of automorphisms of these two types. 3.1 Conjecture ( A u t ) . If 0 is a normalized automorphism of ZG, then there exists an automorphism p of G and a unit 7 in QG and such that % ) == 7 " V ( ^ ) 7 for aU ^ e G. S. K. Sehgal has established this conjecture for finite groups which are nilpotent of class two [Seh69]. G. Peterson confirmed the conjecture for finite symmetric groups Sn [Pet76] and extended his results to some classes of metacyclic groups [Pet77]. The conjecture has also been verified by A. Giambruno, S. K. Sehgal and A. Valenti for the wreath product ^4 i ^n of a finite abelian group A with Sn [GSV91]. It has also been verified for P I Sn, where P is a finite p-group for some prime p. The result for odd p is due to A. Giambruno and S. K. Sehgal [GS92] and, for p = 2, to M. M. Parmenter and S. K. Sehgal. In general, however, the problem remains open. It is unresolved for alternating groups, for instance, though it holds for ^5 [LP89] and, in fact, for An li n < 8 [Gia]. In this section, we prove that, if L is a finite RA loop, then any normalized automorphism of ZL is the composition of an inner automorphism of QL and an automorphism of L, 3.2 Theorem. Let L be a finite RA loop and let 9 be a normalized automorphism ofZL. Then there exists an inner automorphism ip of the rational loop algebra QL and an automorphism p of L such that 6 = ip 0 p. Our first observation is that 9(s) = s where, as always, s denotes the unique nonidentity commutator-associator of L, To see this, let k and £ be any elements of L which do not commute. Then PROOF.

(9)

ki - ik = {I - s)ke

and it follows that the (ring) commutator of any two elements of ZL is contained in (1 — s)ZL. Also, applying 6 to each side of (9), we obtain (l-9{s))0{ke) = 9{k)9{i)-9{t)9{k) G {\-s)ZL, Multiplying by I + 5 gives (1 + 5)(1 - 9{s))9{ki) = 0. Since both 1 + 5 and 1 - 9{s) are central and

3. AUTOMORPHISMS OF ALTERNATIVE LOOP ALGEBRAS

235

0{k£) is invertible, we obtain (1 + 5)(1 — 0(s)) = 0. Hence (10)

l + s = e{s) + s0{s).

As a central torsion unit, 6{s) G L (Corollary VIII.1.7), so each side of (10) is a sum of loop elements. Since 0(s) ^ 1, necessarily 9{s) = s. As in the proof of Theorem 1.1, there exists an automorphism p of L defined by p(£) = ^p, where i^ is the unique element of L satisfying e{i)

= ip

(mod

A(L)A(L0).

We denote also by p the linear extension of this map to ZL and note that this extension is a ring automorphism of ZL. We now claim that 6{i) is central if and only if p{i) is central and, in this case, 9{i) = p{i). For this, we first note that, if 6{i) is central, then 6{i) G L (as a central torsion unit of augmentation 1), so p{t) — t^ — 6(i) (by uniqueness) is central. On the other hand, suppose that p{£) =: £p is a central element of L. Then 0{i) = i^ (mod A{L)A{L')) implies e{e)^ = ij (mod A{L)A(L')). But 6{i)^ = 9{(?) is a central torsion unit (because i'^ is a central torsion unit) and of augmentation 1. By uniqueness,

(11)

e{if^tl.

Now d{iy = £; (mod A{L)A{L')) since A(L)A(L') is closed under the involution * on ZL and, since ip is central, i* = i^. Thus d(ty = ip (mod A(L)A(L')) and 9{i)9{ey = i] (mod A{L)A{L')). Since rr'^ is central for any r G ZL, we see that 9{i)9{iy is a central torsion unit of ZL, and of augmentation 1. By a now familiar argument, 9{i)9{iy = £^. Comparison with (11) shows that 9{i) = 9{iy^ so 9{(.) is central. As a central torsion unit of augmentation 1, 9{i) G L and, by uniqueness, 9(i) = £p. This estabUshes our claim. As one consequence, observe that p(s) — s. Since 9{i) = p{£) (mod A(L)A(L')) and A(L') = ZL'(1 - s), we have (12)

^i + s)9{i) =

{\^s)p{t)

for any C £ L. Recall that the centre of a loop ring RL is spanned by Z{L) and {£ + si \ i £ L} (Corollary III.1.5). The claim above shows that for £ G 2 ( L ) , 9{£) rr p{£), while (12) shows that 9{{1 + s)£) = p{{l + s)£), since p and 9 fix s. The point is that 9p'~^ fixes elementwise the centre of ZL and.

236

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

more importantly, the centre of the rational loop algebra QL,

This loop

algebra is the direct sum of simple alternative algebras A^, which are either fields or Cayley-Dickson algebras (see Theorem VI.4.8). Since the centre of QL is fixed by Op~^^ the restriction of this m a p to any commutative component is the identity and to any component which is a Cayley-Dickson algebra an inner automorphism, by Theorem 2.4. Define ipi'. QL -> QL by

M^j) where ai G Ai.

Op-^{ai)

i = j

aj

i 7^ j

It is easy to see t h a t tpi is an inner automorphism of QL

for each i and hence t h a t tp = Yii^i is an inner automorphism such t h a t ip == Op~^ ^ which gives 6 = tp o p dis desired.

D

4. S o m e c o n j e c t u r e s of H. J. Z a s s e n h a u s We denote the loop of units in the integral alternative loop ring ZL by U{ZL) and the loop of normalized

units by U\{7.L)\ t h a t is,

U^{ZL) = {aeUiZL)

I 6(a) = 1}

where e: ZL -^ Z \s the augmentation m a p . By TU{ZL)

and TU\{ZL)

we

denote, respectively, the loops of torsion units and normalized torsion units

of ZL. If G is an abelian group, then we know t h a t the units of finite order in ZG are trivial. (See Chapter VIII.) When G is not abelian, an obvious way to exhibit new torsion units in ZG is to compute conjugates of the form ± 7 ~ ^ ^ 7 , with g ^ G and 7 G U{ZG),

In fact, one could take 7 from the

rational group algebra QG, provided that 7~^6'7 C ZG. In the mid 1960s, H . J . Zassenhaus conjectured t h a t all torsion units in ZG can be constructed in this way. More precisely, he formulated 4,1 C o n j e c t u r e ( Z C l ) . Let a G ZG be a normalized unit of finite order. Then there exists an invertible 7 G QG and an element a ^ G such t h a t 7 ~ ^ a 7 = a. If a G ZY(ZG), 7 G U{QG) and 7 ~ ^ a 7 = a G C , we say t h a t a is rationally conjugate conjugate

to a. More generally, a subgroup H of ZY(ZG) is

to a subgroup K of G if j~^Hj

— K for some 7 G U{QG).

rationally There

4. SOME CONJECTURES O F H. J. ZASSENHAUS

237

are two stronger versions of Z C l dealing with subgroups of normalized units. 4.2 Conjecture (ZC2). Let ^ be a subgroup of normalized units in ZG such that 1^1 = \G\. Then H is rationally conjugate to G. 4.3 Conjecture (ZC3). Let H be any finite subgroup of normalized units in ZG. Then H is rationally conjugate to a subgroup of G. The reason for restricting attention to subgroups of normalized units should be apparent since, for /^ G ^ , 7 G U{QG) and j'^hj

G G, then

1 = €(7-1/^7) = e{h). Clearly Z C 3 implies t h e first two conjectures and a positive answer to Z C 2 implies a solution to the isomorphism problem. It is also easy to verify t h a t Z C 2 implies the conjecture A u t described in §3.1 and t h a t a positive answer to both A u t and the isomorphism problem implies Z C 2 . The Zassenhaus conjectures have been established for various kinds of groups. All of them have long been known to be true for finite nilpotent groups of class two [Seh69].

T h e most far reaching result to date is a

theorem due to A. Weiss [Wei91] (see also [Seh93]) which says that Z C 3 holds for the group ring of any finite nilpotent group. Z C l was proved for metacyclic groups of the form G = (a) XI ( x ) , where gcd(o(a),o(x)) = 1, by C. Polcino Milies, J. Ritter and S. K. Sehgal [ M R S 8 6 ] . A. Valenti has shown t h a t Z C 3 holds for a group of the form G = (a) Xl X^ where gcd(o(a), | X | ) = 1 and X is abelian (see also S. 0 . Juriaans [Jur94] ). T h e book by S. K. Sehgal [Seh93] contains an exposition of most of the known results on this subject. In another direction, it should be mentioned t h a t Z C l was proved for 5*4 by N. A. Fernandes [Fer87], for A4 by I. S. Luthar and I. B. S. Passi [LP89] and, for ^ 5 , by I. S. Luthar and P. T r a m a [LT]. On a negative note, K. W . Roggenkamp and L. Scott found a metabelian group of order 2^ • 3 • 5 • 7 which is a counterexample to Z C 2 [Rog91] and later L. Klinger found another counterexample with a group of the same order, but using different methods [ K l i 9 l ] . The goal in this section is to show t h a t , with minor modifications, all three conjectures hold for alternative loop rings which are not associative.

238

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

4 . 4 P r o p o s i t i o n . Let L he an RA loop with centre Z{L). torsion

unit in ZL.

Then there exists a unique a £ L such that a = a

(mod A ( L ) A ( L ' ) ) . Moreover, PROOF.

Let a be a

a^ = a^ G ^ ( L ) .

Since the image a of a in the abelian group ring Z[L/L^] is

a torsion unit, a is trivial.

Thus a = ±t (mod A(Z/,L')) for some t G

L and, by Whitcomb's Lemma, there is a unique a £ L such t h a t a = ±a (mod A{L)A{V)).

Thus a^ = a^ (mod A{L)A{V)),

(mod A ( L ) A ( L O ) . Write a ' ^ a ^ = i + S(l-s),6

so a-^a^

e ZL,supp{6)

= 1

C T , where

T is a transversal for { l , ^ } in L and 1 G T . It follows t h a t 1 or 5 is in the support of a~^a^, hence a~^a^ G // by Corollary VIII.1.2. Now Lemma 0.1 shows t h a t a^ = a^.

D

If // is a finite RA loop, the alternative loop algebra CL of L over the complex numbers is the direct sum of simple alternative algebras which are either fields or Cayley-Dickson algebras. (See Chapter VI.) In a CayleyDickson algebra, every element is either invertible or a zero divisor because a is invertible if and only if aa /

0, where a \-^ a denotes the canonical

involution. Thus every element of CL is either invertible or a zero divisor. An argument of C. P. Milies and S. K. Sehgal can therefore be adapted in the alternative case and gives us a key lemma. 4.5 L e m m a . [MS84] Let k C K be infinite whose loop algebra over k is semisimple fi zn 7 ~ ^ a 7 for some invertible invertible

fields.

Let L be a finite loop

and alternative.

element 7 G KL,

If ct^P G kL and

then (3 = 7 ~ ^ a 7 for

some

element 7 G kL.

PROOF.

Consider the equation ax — x(i which, relative to the basis

L (of either vector space kL or A'L), can be expressed in matrix form as x M = 0, where M is the matrix of the linear transformation L{a) — R{l3) and X is the coordinate vector of x. Note that M is an n x n matrix, n = |Z/|, with entries in /c, and singular since xM — 0 has a solution x corresponding to an invertible element x G KL.

Let x i , . . . ,Xi be a basis for the nuUspace

of M in k'^ and let Xi G kL correspond to x^. We prove t h a t some A;-linear combination of these vectors corresponds to an invertible element of kL. If this is not so, then, for any A i , . . . ,At G A;, the element x = Yl^i^i zero divisor in kL; equivalently, the matrix of R{x) is singular.

^^ ^

Consider

4. SOME CONJECTURES OF H. J. ZASSENHAUS

239

the determinant det Z KR{xi)

= / ( A i , . . . , At) G A:[Ai,... , A,].

Since k is infinite and we are assuming t h a t / ( A i , . . . ,At) = 0 for all Xi G k, the polynomial / must be identically zero. But this implies t h a t / ( c i , . . . , Ct) = 0 for all Ci G K, which is not true. Thus / is not identically zero, so some solution of x M = 0 in Z;;^ corresponds to an invertible element in kL.

n

Let a and /3 be elements of ZL. In what follows, it will be convenient t o say t h a t a and /3 have a common

conjugate

in QL (or CL) provided there

exist 7i and 72 in QL (or CL) such t h a t 7 f ^ a 7 i = 12^(^12' T h e following corollary is immediate. 4.6 C o r o l l a r y . If a and fi are elements of the integral alternative ZL which have a common conjugate

conjugate

loop ring

in CL, then a and (3 have a

common

in QL.

We now present an analogue of Z C l for alternative loop rings which are not associative. Clearly our result reduces immediately to Z C l in the associative case. 4 . 7 T h e o r e m . Let L be a finite

RA loop and suppose a is a

torsion unit in ILL. Then there exists an element QL such that J2^{j^^aji)j2 PROOF.

normalized

a E L and units 71,72 in

= CL-

By Proposition 4.4, there exists a unique a ^ L such t h a t a = a

(mod A ( i / ) A ( L ' ) ) and a^ — a}. As in the proof of Theorem 3.2, o. is central if and only if a is central. If this is the case, then a = a by Lemma 0.1 and there is nothing to prove. Therefore, we assume henceforth t h a t neither a nor a is central. By Corollary VIII. 1.2, a has no central elements in its support, so a* = ^o;. Since also a* = ^a, it follows t h a t aa-^-ota = (aoL-\-adY is central (Corollary III.4.3) and, since OL — a <^ A(Z/)A(L') and A ( L ' ) = Z(l - 5), t h a t a ( l + 5) = a{\ + s). We will show t h a t a and a have a common conjugate in CL. Because of Corollary 4.6, it will follow t h a t a and a have a common conjugate in QL, which is the assertion of the theorem. Write CL — A\®"

•© An, where the

Ai are simple alternative algebras. Denoting by oti and a^, respectively, the

240

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

components of a and a in A^-, it is sufficient to show t h a t a^- and a^ have a common conjugate in Ai. Suppose Ai is commutative and hence a field. By Proposition VI.4.6, the projection Si of s in Ai is the unity of Ai. Since (1 + s)a = (1 + s)a^ we have 2ai = 2ai^ so a^ = a^-; thus a^ and ai have a common conjugate. Suppose t h a t Ai is a noncommutative component; thus (by Proposition VI.4.6) Si ^ I; hence Si = ~1. In this case, since a is not central, ai is not central, by Corollary VI.4.7. Since a^a^ + ct^a^- is central in Ai^ it follows t h a t ai is not central in Ai (because ai is a unit). For convenience, we now drop the subscripts on a^ and a^. Let B be the (associative) subalgebra of Ai generated by a and a. Since the centre of a simple finite dimensional algebra over C is C, and since a^ — a} and aa-\-aa

are central, we can write

a^ = o? — } ^ z^ ^ and aa-\-aa — // with A, /i G C. Every element in 5 is a linear combination of 1, a, a and aa. Set j \ — | ( l - | - a / A ) and J2 — \{\ — ajX). Then j \ and /2 are orthogonal idempotents different from 0 and 1 such t h a t / i + /2 - 1 and a = A(/i - h \ Thus B = {h + f2)B{h Defining fij — fiafj

and

/2i=/2a+—/2.

and / 2 ^ / 2 each have dimension 1 (with bases f\ and

/2 respectively) and, for i / j , t h a t d\m{fiBfj) in which case fiBfj

has basis fij.

< 1, with equality if fij ^ 0,

Moreover, 2

/?2 = / l i = 0 ,

UBf,.

we have, by straightforward computation, t h a t

f\2 = f\a-—fi It follows t h a t f\Bf\

+ /2) - E

/,2/2i = (A^ - £ ^ ) / ,

2

and

/2i/,2 = ( A 2 - i ^ ) / 2 .

If A^ — ij?IAX^ / 0, replacing /12 and /21 with /12 = c~Vi2 and /21 = c"V2i 2

respectively, where c G C is a root of X'^ — {X^ — j^)

= 0, it is apparent

t h a t { / i , / 2 , / i 2 5 / 2 1 } form a set of matrix units and hence t h a t B = M2(C). Since the minimal polynomial of both a and a over C is X'^ — A^, each element is similar to [ Q J \ ] , so a and a have a common conjugate in B and hence in Ai. If A^ - /i2/4A2 = 0, then fi/2X^ = ± 1 . If fi/2X'^ = 1, defining p = ( l / 2 A ) ( a -h a ) , it can be checked directly t h a t p^ = I and p~^ap

= a.

Thus a is a common conjugate of a and a in ^ i . If ///2A^ = — 1 , define g = ( l / 2 A ) ( a — a ) and check t h a t q'^ = I and q~^aq

= —a. Since a is

4. SOME CONJECTURES OF H. J. ZASSENHAUS

241

not central, for some h in the projection of CL onto Ai^ we must have h~^ah =z sa = —a, so it follows here too that a and a have a common conjugate in Ai, namely —a. D We now turn our attention to the other two conjectures of H. J. Zassenhaus, ZC2 and ZC3. The following proposition generalizes in a completely straightforward manner a known result for group rings. 4.8 Proposition. Let L be a group or an RA loop. Then any set of normalized torsion units in ZL which forms a loop, Li, is linearly independent overZ. In particular, if L is finite, then \Li\ < \L\, P R O O F . If X)r=i ^«7« — ^' ^^^^ ^« ^ ^ ^^^ 7i? • • • ^ Tn distinct normalized torsion units contained in a loop Li, then we can write n

(13)

ail = - 5 ] a , ( 7 r S i ) . t=2

Each 7f ^7t is a torsion unit and different from 1, hence, when expressed as a linear combination of loop elements, the coefficient of 1 is 0, by Proposition VIII.1.1. Comparing coefficients of 1 on each side of (13), we obtain ai = 0. Similarly, all a^ = 0. D Recall that an RA loop L has the LC property; that is, if x,y G L and xy = yx, then one of x, i/, xy must be central. 4.9 Lemma. If LQ is a subloop of an RA loop and LQ is not commutative, then Z{Lo) C Z{L), Let a G Z{LQ) and suppose that a ^ Z{L). Let x^y e LQ be such that xy ^ yx; thus, none of x, y, xy is in the centre of L. Since ax = xa but neither a nor x is in 2 ( L ) , it must be that ax is central in L and, similarly, ay and a{xy) are central. But then PROOF.

a{xy) = a{{a~^ • ax){a~^ • ay)} = a~^{ax)(ay) impfies that a~^ is central in L, contradicting a ^ Z{L),

D

4.10 Remark. Suppose that L is a torsion RA loop with V = {1,^5}. If a is any normafized torsion unit in ZL, then, as noted Proposition 4.4, there is a unique element p(a) G L such that a = p(a) (mod A(L)A{V)) and

242

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

cP' — p{oif'. Noting the uniqueness of pia) as an element of L, it is clear that /9(a) = a if a G //. As in the proof of Theorem 3.2, a G TZYi(ZL) is central if and only if p{o) is central and, in this case, P(OL) — a. In particular, p(^s) — s. Suppose a i and a2 are normalized torsion units in ZL such that aia2 is also a torsion unit. Then ai = />(<^i) (mod A(L)A(L')) and a2 = pi(^2) (mod A(L)A(L')), so aia2 = p{ai)p{a2) (mod A(L)A(L')). On the other hand, aia2 = p(tti0^2)? so p(aia2) = p{ai)p{a2) by uniqueness. It follows that the restriction of p to any subloop of normalized torsion units is a homomorphism; moreover, this restriction is one-to-one because \{ p{a) = 1, then a = 1 because 1 is central. We remark, in passing, that these comments essentially provide another proof of the isomorphism theorem for alternative integral loop rings of torsion RA loops. 4.11 Lemma. Let L be a finite RA loop with L' = {1,^} and let H be a finite subloop ofU\{TL). Let a^/3^j be elements of H. If the loop commutator (a,/?) is not I, then (a,/3) = s and, if the loop associator (a,/3^j) is not I, then (a,/?,7) = s. If H is not commutative, then Z{H) C Z[L). Let p: H —^ L he the homomorphism discussed above. Since p{a^f5) = (/>(a),/9(/3)) and p is one-to-one, (a,/3) / 1 implies p{a^(i) ^ 1; hence p{a^f3) == 5 is central. As noted, it follows that (a,/?) = p(a^/3) = s, A similar argument establishes the second statement of the lemma. If H is not commutative and a G Z(H)^ then p(a) is central in the subloop of L which is the image of p. By Lemma 4.9, p{a) is central in L, so a = p{a) G 2 ( 1 ) . D PROOF.

4.12 Lemma. Let H be a finite subloop of U\{7-L) containing s and let p: H -^ L be the homomorphism described in ^4-^0, Then H* = H and p commutes with *. If a G ^ is central, then it is trivial, so a* = a G fl^. If a £ H \s not central, then it has no central elements in its support, by Corollary VIII.1.2, so a* = 5a G H. In any event, ^ * C H, so 7?* = H. Let a E H, If a is central in ZL, then a is trivial, so p{a) = a and p(a*) = p{a) = a = a* = />(<^)*- If (^ is not central, then a* = sa and also p(a) is not central. So again p{a*) = p{sa) = p{s)p(a) = sp{a) = />(«)*• Thus p commutes with *. D PROOF.

4. SOME CONJECTURES OF H. J. ZASSENHAUS

243

For alternative loop rings which are not associative, the second and third conjectures of Zassenhaus take the following form. 4.13 Theorem. Let L be a finite RA loop. If H is a finite subloop of normalized units in ZL, then H is isomorphic to a subloop of L, Moreover, 1. if H is an abelian group, then there exists an inner map tj) ofQL such that tp{H) C L and the restriction of ip to QH is a monomorphism; 2. ifH is not associative, then there exists an inner automorphism ip of the rational loop algebra QL such that i^{H) C L; and 3. if H is a group, then there exists an inner automorphism ip of the complex loop algebra QL such that ip{H) C L. Let /T be a finite subloop of Ui{ZL), Since HZ{L) is also a torsion subloop of Ui{ZL)^ we may assume Z[L) C H^ so, if H is not commutative, we may also assume that 2{H) — Z[L)^ by Lemma 4.11. Let p: H -^ L he the homomorphism described in §4.10 and set Hp = p(H). Since p fixes Z{L)^ our assumption about H implies that Z[L) C Hp and, if H is not commutative, that Z[Hp) = Z{L). If H is an abelian group, then Hp is an abelian group contained in L. If H is central, H = Hp C L and there is nothing to prove. Thus we assume that H is not central and select h £ H and i E L such that (h^i) ^ 1. Since h is not central, hp = p(h) is an element of Hp which is not central in L. If x is any other element in Hp^ since xhp = hpX^ either hp or x or hpX is central since L has property LC. (See §IV.2.) In the latter case, x = x'^{hpx)~^hp is a central multiple of hp. It follows that Hp = Z{L) U Z{L)hp and that H — Z{L) U Z{L)h. By Theorem 4.7, there exists an inner map ^0 on QL such that ip(h) = hp — p{h). Clearly -0 fixes central elements of L, thus ip{H) = p{H) C L. This proves statement (1). Next we shall show that there exists an inner automorphism V^, of QL if H is not associative and of CL if ^ is a nonabelian group, such that PROOF.

(14)

i^{a) = p{a)

for dX\ a^ H. Assume H is not associative. Expressing QL as the sum of fields and Cayley-Dickson algebras, it is sufficient to show that in each simple component A = (QL)e of QL, e a primitive central idempotent, there exists an

244

IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS

inner m a p 'tpA such t h a t (14) is valid in t h a t component (for all a G ^ ) . Let TT: Q L -^ Ahe

the natural projection. Since, for any a £ H^ a = p(a)

(mod A ( L ) A ( L O ) and A ( L ' ) = Z ( l - 5), we have ( 1 + 5)a = {l + s)p{a).

If

A is commutative, then 7r{s) = e, the unity of A, by Proposition VI.4.6, so 7r(a) = 7r{p{a)) and we can take ipA ^o be the identity m a p on A. It remains only t o consider components which are Cayley-Dickson algebras. Assume henceforth t h a t A is such a component; it follows t h a t 7r(s) = — e, again from Proposition VI.4.6. We know t h a t Z{L)

C H and t h a t H = H^ =

p{H)

is a subloop of L which is not associative. Since L is generated by

Z{L)

and any three elements which do not associate (Theorem IV.3.1), Hp = L. Thus H is isomorphic to L. Since the Q-subalgebra of QL spanned by H is isomorphic to Q ^ , by Proposition 4.8, clearly p extends to an isomorphism Q ^ -^ QL, t h a t is, to an automorphism of QL. Since p fixes 2{L)

and

hence, in particular, p{l — s) = 1 — s^ we see t h a t p defines an automorphism oi A which fixes its centre. (See Corollary III.1.5.) By Theorem 2.4 (Jacobson's Theorem), this automorphism of A is necessarily inner. This establishes statement (2), the case t h a t H is not associative. Assume t h a t H is a, nonabelian group. Expressing CL as the sum of fields (each isomorphic to C) and Cayley-Dickson algebras with centre C (each isomorphic to Zorn's vector matrix algebra over C), as before, it is sufficient to show t h a t , in each simple component A = {CL)e of CL, e a primitive central idempotent, there exists an inner map ipA such t h a t (14) is valid in t h a t component (for aD a G ^ ) . If A is commutative, we can again take tl)A to be the identity map on A, so assume A is not commutative. Again, 2{H)

= Z{L)C

H ^ Hp = p{H)C

L,so

L = Hp[jHpU

= M{Hp,^,u^)

for

some element u ^ L. By Corollary VI. 1.3, e is in CH and it is a primitive central idempotent of this algebra. know t h a t (CH)e

Also, because of Lemma VII.2.2, we

is a split quaternion algebra and hence isomorphic to

M2(C). (See §1.3.3 and Theorem 1.4.16.) Since p fixes central elements. Corollary VI. 1.3 shows t h a t p defines a C-automorphism from {CH)e {CHp)e = M2(C) C (CL)e = A.

Let ^ 1 = {CH)e.

to

By Theorem VI.4.5,

the m a p n: CL —> CL defined by n(6) = 66* restricts to a nondegenerate quadratic multiplicative form on A and on Ai. As in the proof of Hurwitz's Theorem (Theorem 1.4.11), there exists ir e Ai, t h a t A = Ai QA\i\

el = - n ( ^ i ) ^ 0, such

with the usual Cayley-Dickson multiplication (see §1.3).

4. SOME CONJECTURES OF H. J. ZASSENHAUS

245

Replacing ii by ( \ / n ( ^ ) ~ ^ ^ i , we may assume i\ = —\, In a similar way, A = A2® ^2^2 with A2 = {CHp)e and 1^ = -I, Finally note that A* C Ai for i = 1,2 and p commutes with *, by Lemma 4.12. It is straightforward now to verify that the map A = Ai Q Ai£i ^^ ^2 © >l2^2 = ^ defined by a + bii-^ p{a) + p{b)i2 is an automorphism V^ of ^ which extends p and is inner, by Jacobson's Theorem. Finally, the proof of (3) in the case that H is an abehan group is similar to the foregoing, by essentially applying the methods of Hurwitz's Theorem twice. We leave this to the reader. D

Chapter X

Isomorphisms of Commutative Group Algebras

In Chapter IX it was shown that the isomorphism problem has a positive answer for integral loop rings of torsion RA loops. At this point, our primary interest is the isomorphism problem for loop algebras of finite RA loops. Thus, for finite RA loops Li and L2, and for a field F , we investigate whether the existence of an F-algebra isomorphism between FLi and FL2 implies that Li is isomorphic to Z/2. Since abelian groups play such an important role in the structure of RA loops, we first need to study this question for group algebras of finite abelian groups. The present chapter is devoted to that goal; in Chapter XI, we apply our results to loop algebras. Since familiarity with tensor products of fields is crucial to our study, we start this chapter by establishing some results in this area. In the second section, the isomorphism problem for semisimple abelian group algebras FG is investigated. An affirmative answer is given for rational group algebras and, in general, a reduction is given to the case where G is a p-group for some prime p. These results are consequences of the decomposition of the algebra FG as a direct sum of cyclotomic fields. In Section 3, we direct our thoughts to the case where the characteristic of F divides the order of the group G. Such group algebras are called modular. It is shown that the isomorphism problem in this situation can be reduced to the semisimple case and to the case that G is a p-group. In the fourth and last section, we determine when two fields F and K are equivalent on a class S of finite abelian groups; that is, we find conditions which imply that FG and FH are F-isomorphic if and only if KG and KH are A'-isomorphic for all G, J? € S,

247

248

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

1. Some results on tensor products of fields In this section, we discuss the structure of a tensor product of the form E ®F K^ where E and K are fields both of which contain F as a subfield. We are interested mainly in the case where the fields involved are cyclotomic extensions of F , that is, extensions by a primitive root of unity. We begin by summarizing the essential ingredients of Galois theory, referring the reader to [Jac74, Chapter IV] and [Jac64] for details. A field extension E/F is said to be finite if the dimension [E: F] of JS, considered as a vector space over F , is finite. The extension E/F is algebraic if every element of F is a root of a polynomial with coefficients in F . A polynomial in F[X] is separable if it has no multiple roots. An algebraic extension E/F is separable if the minimal polynomial over F of each element of E is separable. A finite extension of F is normal if it is the splitting field of a polynomial in F[X], that is, the smallest subfield of the algebraic closure of F containing all the roots of some polynomial. A finite extension E/F is Galois if it is both normal and separable. An F-automorphism of E is an automorphism of E which fixes all the elements of F . The Galois group of the extension EjF^ denoted G a l ( F / F ) , is the set of all F-automorphisms of F . A finite extension E/F is Galois if and only if [ F : F] = I G a l ( F / F ) | and the set of elements of E which are fixed by aU the automorphisms of G a l ( F / F ) is precisely F . It is easy to see that if EjF is Galois and K is any field containing F such that both E and F are contained in a common field fi, then the composite field K{E) (defined in fi) is Galois over K. If not explicitly specified, the field fi will always be clear from the context. As in Chapter VII, we shall need to work with roots of unity. We recall the main definitions in the general case. So let F be an arbitrary field. An element ^ G F which is a root of the polynomial X"^ — 1 is called an nth root of unity \ if ^ has order n in the multiplicative group of F , then ^ is said to be a primitive nth root of unity. For n > 1, a primitive nth root of unity over F means a primitive nth root of unity in a splitting field of X'^ — 1 over F . Unless specified otherwise, such an element is consistently denoted ^n- If char F does not divide n, a primitive nth root of unity over F always exists and F{^n)lF is a Galois extension called the nth cyclotomic extension

1. SOME RESULTS ON TENSOR PRODUCTS OF FIELDS

of F,

249

The elements i\^ I < i < n^ g c d ( i , n ) = 1, constitute precisely the

primitive n t h roots of unity in

F{^n)'

Let

l
Since any a 6 G a l ( i ^ ( ^ n ) / ^ ) maps ^^ to a primitive n t h root of unity, it follows t h a t * n ( - ^ ) is invariant under each a and therefore $ n ( - ^ ) has coefficients in F, We call $ n ( ^ ) the cyclotomic

polynomial

of order n and

note t h a t $ n ( - ^ ) is a polynomial of degree <^(n) where, as always (f) denotes the Euler function. Recall t h a t the prime field of a field F is Q if char F = 0 and the field Fp of p elements if char F = p ^ 0. 1.1 L e m m a . Let F be a field containing

a subfield P. Let ^ be a

root of unity over P whose order is not a multiple o/char P, Then is isomorphic PROOF.

to a direct sum of[P{^):

P]/[F{^):

F] copies of

primitive F®pP(^)

F{^).

Let n be the order of ^ and let / G P[X] be the minimal (monic)

polynomial of ^ over P. Then / is irreducible and P{^) ^ P[X]/(f),

Thus

(See §1.5.11.) Let / = /1/2 • * • /t be the decomposition of / as a product of irreducible (monic) polynomials in F [ X ] . The assumption on the characteristic of P implies that the roots of X ^ — 1 are distinct. Thus / is separable and hence f- zji f- if i ^ j . Using the Chinese Remainder Theorem, we can write

\h) Now F[X]l{fi)

\Jt)

= F(0i)^ where Oi is a root of fi and thus also a root

of / , for 1 < i < t.

Since the roots of / in P{()

are all primitive n t h

roots of unity over P , each 0i is a primitive n t h root of unity over P. Thus F{ei) ^ F ( 0 . It follows t h a t

(1)

F®pP{0 =

FiO®'-@F{0"-

>/ t summands

'

250

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

Since [F ®p P{^): of (1) is t[F{^):

F] = [P{0 • ^ ] ^ ^ ^ the dimension of the right hand side

F], we have t = [P(^):

P]/[F{0'

F] as required.

D

1.2 L e m m a . Letp be a prime and let F be a field of characteristic from p. If the polynomial in

X^ — a has no root in F, then X^ — a is

different irreducible

F[X]. PROOF.

Write X^ - a = f{X)g{X),

where f,g

e F[X] and / is irre-

ducible. Working over the splitting field of / , we have f{X)

= (X-

a i ) ( X -a2)'"{X-

am)

for some integer m < p. Since each of the roots of / is also a root of X ^ — a, we have a^^ = a and, with C,i = a^^ai^ it is clear t h a t (f = 1 for 1 < i < m. Let b = aia2 '' -am- Then b £ F and 6^ = ^^1^2 '' 'O^ = CL^- If m < p^ we can find integers r and s such t h a t rp + sm = 1, so a = a^'^a'^^. Since ^5771 _ f^sp^ ^^ hdi^we a = {b^d^y X ^ — a has no root in F.

with b^a^ G F , contradicting the fact t h a t

Consequently m = p and X^ — a = f{X)

irreducible in F [ X ] .

is D

1.3 C o r o l l a r y . Let p be prime and let F be a field of characteristic ent from p which contains a primitive

pth root of unity.

differ-

Let ^ be a pHh root

of unity over F. Then [ F ( ^ ) : F] is a power of p. PROOF.

Consider the following sequence of field extensions: F = F{e'~')

C F{e'~')

C . . . C F{e)

C F(0.

Since F contains a primitive pth root of unity, F contains all pth roots of unity.

By Lemma 1.2, it is easy to see t h a t for each j , j = l , . . . z , the

dimension

[F{^P^

) : F{^^^)] is either 1 or p. The result follows.

1.4 L e m m a . Let ^ be a primitive acteristic

D

rth root of unity over a field F of char-

relatively prime to r. Then [F{^): F] is a divisor of (t>{r).

PROOF.

Let a G G a l ( F ( ^ ) / F ) .

Since (7(^) is also a root of unity of

order r , ( J ( ^ ) = ^* for some rational integer t relatively prime to r. It is easy to see t h a t the mapping a »-^ ^ is a monomorphism from G a l ( F ( ^ ) / F ) into the multiplicative group Z* of invertible elements of the ring of integers modulo r. Since Z* has order (r), we see t h a t | G a l ( F ( ^ ) / F ) | is a divisor of (r). Since F ( 0 is Galois over F , [ F ( 0 : F] = | G a l ( F ( 0 / F ) | and the result follows.

D

1. SOME RESULTS ON TENSOR PRODUCTS OF FIELDS

251

1.5 Lemma. Let E and N be subfields of a field il and suppose E is a finite Galois extension of a field F. Then [E{N): N] = [E: ED N]. Let B = { e i , . . . ,6^} be a basis of E over E O N, Since B generates E{N) over iV, we have only to show that it is also linearly independent. Assume, by way of contradiction, that B is linearly dependent over N and let { e i , . . . ,6^^}, m < n, be the smallest subset which is still linearly dependent over N, Then there exist elements a i , . . . , a ^ in TV, not all 0, such that ^ ^ 1 aiei = 0. Without loss of generality, we may assume that ai = 1. Let a G Gal(E(N)/E), Since a is an F-automorphism and since N/F is normal, we have cr(ai) £ N for 1 < i < m. Moreover, a{ai) = ai (since PROOF.

aj = 1), so m

^{ai

- cr(a,))e, = 0.

t=2

The choice of m implies that a^ — (T(ai) = 0 for z = 2 , . . . ,m. This shows that each coefficient a^ is fixed under any element a G G3l(E(N)/E). So a, G ^ n TV for 1 < i < m, a contradiction, since iB is a basis of E over EON, D 1.6 Corollary. Let E and N be subfields of a field SI and suppose E is a finite Galois extension of a field F. Then E(N) = E ®EnN ^ • It is easy to see that e ® n y-^ en defines a homomorphism from E ®EnN N onto E{N), Also [E ®EnN N: N] = [E: En iV], so, by Lemma 1.5, E{N) and E^EHNN have the same dimension as vector spaces over A^. The result follows. D PROOF.

1.7 Theorem. Let p be a prime and let F be a field of characteristic different from p which contains a primitive pth root of unity. Let ^ps and ^r be roots of unity over F of orders p^ and r, respectively^ with gcd(p, (r)) = l = gcd(p,r). Let E = F{^ps), Then E ®F H^r) = E{^r) and [E{^r)' E] = [F{^r):F], Let h = [F{^r) n F{^ps): F]. Then /i is a divisor of [F(^^): F] and, since gcd(p, r) = 1, also a divisor of (r), by Lemma 1.4. On the other PROOF.

252

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

hand, h is also a divisor of [F(^ps): F] which, by Corollary 1.3, is a power of p. Since gcd(;>,^(r)) = 1, we see that h = I, Thus F{^r) H F{^ps) = F. From Corollary 1.6, one obtains

and, by Lemma 1.5, [^(^r): E] = [F{Cr): F].

D

2. Semisimple abelian group algebras In this section, we survey some of the more important results concerning the isomorphism problem for semisimple group algebras FG over finite abelian groups G; so F is a field whose characteristic does not divide \G\. Throughout, an isomorphism between group algebras FG and FH means an F-isomorphism. For group algebras over fields, the isomorphism problem was first formulated by T. M. Thrall at the Michigan Algebra Conference in the summer of 1947. The first results on this subject appeared in a well known paper by S. Perils and G. L. Walker [PW50]. We begin our study with the first theorem of that work, a description of the Wedderburn decomposition of a commutative group algebra, though we shall give a different proof. Recall that in Chapter VII, we solved this problem for rational group algebras. 2.1 Theorem (Perils—Walker [PW50]). Let G be a finite abelian group of order n and let F be a field whose characteristic does not divide n. Then

where a^ = ndl[F{^d)'' F]^ n^ denotes the number of elements of order d in G and ^d denotes a primitive dth root of unity over F. We proceed by induction on n. Assume first that G = (a) is cyclic and consider the map tp: F[X] -^ FG given by if^{f) = /(tt)- It is easy to see that -^ is a ring epimorphism. Hence, PROOF.

ker ip Since F[X] is a principal ideal domain, kerV^ = (/o) is the ideal generated by the monic polynomial /o of least degree such that fo{a) = 0.

2. SEMISIMPLE ABELIAN GROUP ALGEBRAS

253

Clearly a^ = 1 impUes X^ - 1 € kerV^. Also, if f{X) = E L o ^ t ^ ' is a polynomial of degree r < n^ then / ( a ) = Si=o ^i^^ T^ ^ because the elements { l , a , a ^ , . . . ,a^} of FG are linearly independent over F, Thus kerV^ = (X^ — 1) and, therefore, FG =

F[X] (X"-l)'

We know that X " — 1 = JJ \ $d(X), the product of all cyclotomic polynomials $d(X) in F[X]. For each d dividing n, let $d(X) = n":^i fd,{X) be the decomposition of $d(X) as the product of irreducible monic polynomials in F[X]. Since char F )f n, the polynomial X " — 1 is separable, hence the polynomials fj- are relatively prime. By the Chinese Remainder Theorem,

where (dt denotes a root of /^^, 1 < i < adFor a fixed rf, all the elements Q^ are primitive roots of unity of order d; therefore, all the fields F(Ccf,), 1 ^ ^ ^ ^d? ^.re isomorphic, so

where ^d is a primitive root of unity of order d and adF(^d) denotes the direct sum of ad copies of F(^^). Since degfd^ = [F{^d)- F]^ all the polynomials fd^ have the same degree. Since ^d{X) is a polynomial of degree (d),

{d) =

ai[FiU):F].

Since G is a cyclic group of order n, for each divisor d of n, the number rid of elements of order d in G is precisely (d); hence.

This completes the case that G is cyclic. Now assume that the result holds for all abelian groups of order less than n. If G is cyclic, we have already shown that the proposition is valid. Otherwise, G = Gi X H^ where H is cyclic and |Gi| = rii < n. By the induction hypothesis, FGi = 0 i ad^F{^di), where a^^ = ndJ[F{^di)' F] aj

Til

and Ud^ denotes the number of elements of order di in Gi. Hence FG = F[Gi xH] = {FGr)H ^ 0 . ,

ad,F{U,)H.

254

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

Applying the first part of the proof to the group algebras F{^cii)H of the cyclic group H ^ we obtain

where ad^ = «d2/[-^(^''i'^rf2)- ^i^di)] and nd2 denotes the number of elements of order d2 in H. Let d = lcm(di,d2). Then F{^di,U2) = F{U), so

flrf = Y^C'diO.d^, the sum taken over all pairs di, ^2 such that lcm(di,d2) = d. Since [FiU): F] = [FiUr,^d,): F{U,)][F(Ur)• F], it foUows that

ad[Fi^d): F] = ^

a^. a,jF(^d,, ^^J : Fi^.Wi^d,):

d\^d2

F] = J^ ^d.Ud,. d\,d2

Since each element g ^ G — G\ x H can be written uniquely in the form g = g^h^ with gi G Gi and h e H^ and because o(^) = lcm(o(yi),o(/i)), it is clear that ^ ^ ^ '^di'^d2 — ^cf, the number of elements of order d in G, Hence _ and the result follows.

n^ D

Our proof of the result of Perils and Walker concerning the isomorphism problem for rational group algebras will require the following fact. 2.2 Lemma. Let S,di (^f^d ^d2 ^^ primitive roots of unity of order di, ^2, respectively J with di > ^2- Then Q(^di) = Q{^d2) if cirid only if di = ^2 or di = 2^2 with ^2 odd. Since [Q(^d): Q] = (l>{d) for any primitive dth root of unity over Q and since (j){di) = (f>{d2) implies di = ^2 or di = 2^2, the result is clear. D PROOF.

2.3 Theorem (Perlis-Walker [PW50]). LetG be a finite abelian group. If H is any other group such that QG = QH^ then G = H.

2. SEMISIMPLE ABELIAN GROUP ALGEBRAS

255

We follow the argument of [Seh78, Theorem III.2.12]. Set n = |G|. If QG = QH^ then H is also a finite abelian group of order n so, by Theorem 2.1, PROOF.

where ^d denotes a primitive dth. root of unity, ^d = 77^77-7—7^5

^d =

and n^, rrid denote, respectively, the numbers of elements of order cJ in G and in H, So ad (respectively bd) equals the number of cyclic subgroups of order d in G (respectively H), Since Q(^d,) — Q if and only if d^ = 1 or 2, it follows that ai + a2 = &1+62. Since ai = 61 is always the case, we also obtain a2 = 62- For a 2^th root of unity, r > 2, we have by Lemma 2.2 that Q(^2^) = Q{^d) if ^^^ only if d = 2^. Thus the isomorphism of QG and Q ^ impUes that 02^ = 62'"• On the other hand, if p is an odd prime. Lemma 2.2 implies that if Q(^pr) = Q(^ci), then either d = p^ or d = 2p^. Therefore apr + a2pr = bpr + b2pr. Now a cyclic group of order 2p^ can be written uniquely as the direct product of a cyclic subgroup of order 2 and a cyclic subgroup of order p^. Hence a2pr = a2apr^ so our equality now reads apr (I + a2) = 6pr(l + ^2)- Thus apr := bpr.

We have shown that, for any power p'^ of a prime dividing n, both G and H have the same number of elements of order p'^. This implies that the p-primary components of G and H are isomorphic for all p^ so G = H. D We now turn our attention to the isomorphism problem for finite abelian groups over other fields. If G is a finite abelian group. Theorem 2.1 shows that CG is a direct sum of \G\ copies of C. Hence CG = C ^ whenever G and H are two abelian groups of the same order. This shows quite clearly that the answer to the isomorphism problem for group algebras over fields depends critically on the field of coefficients under consideration. In order to give further results, we shall reduce the study of the isomorphisms of group algebras of finite abelian groups to the study of isomorphisms among the group algebras of their corresponding p-primary components. This was also done in [PW50], but there is an oversight in the arguments given there. Independent corrections to this gap appeared in

256

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

papers by R. Bautista [Bau67] and D. E. Cohen [Coh68]. Our proof will follow the latter, but first, we need a lemma. 2.4 Lemma. Letp be a prime and let K be a field of characteristic different from p. For n > 1, denote by ^pn a primitive p'^th root of unity over K, If [A^(^pn+i): /ir(^pn)] = p and n>lifp = 2, then [/ir(^pn+r+i): K{^pn-\-r] = p for every positive integer r. It will suffice to prove that [A'(^pn+2): A^(^pn+i)] = p since the full statement will then follow by induction. Clearly we may assume K{^pn) = K and thus [A"(^pn+i): A"] = p. Let a i = (^pn+2)P and a2 = (ai)P. Then a i is a primitive p^'^^ih root of unity and a2 is a primitive p^th root of unity. So a2 € A^, a i ^ K and a i G A^(^pn+i). Since 02 is not the pt\\ power of an element in A', it follows from Lemma 1.2 that f{X) = X^ - OL2 is irreducible in K{X\. By Corollary 1.3, {K{ipn^i)\ A^C^^n+i] is 1 or j9, so it is sufficient to show that ^pn+2 ^ A'( 1, then 0^2 is a 2^th root of unity if and only if —a2 is. Hence, X^ + a2 is irreducible in PROOF.

2. SEMISIMPLE ABELIAN GROUP ALGEBRAS

257

K[X] if and only if X^ — a2 is irreducible in A^[X]; however, —a2 = A;^, so A; is a root of X^ + a2? giving again a contradiction. D 2.5 Theorem. Let G and H he abelian groups of the same order n and, for each rational prime p dividing n, denote by Gp and Hp the sets of pprimary components of G and of H respectively. Let F be a field whose characteristic does not divide n. Then FG = FH (as F-algebras) if and only if FGp = FHp for all primes p dividing n. PROOF.

If G = Gp^ x Gp^ x • • • x Gp^, then, by Lemma VII.0.1, we have FG ^ FGp, ® FGp, ® • • • ® FGp,,

so the conditions of the theorem are clearly sufficient. To prove the converse, we will show that FG determines FGp for each prime divisor p of n; that is, that FG = FH for some other group H implies FGp = FHp. Write G = Go X Gp X Gi, where Go is the product of all the g^-primary subgroups of G with q < p and Gi is the product of all the g'-primary subgroups with q > p. Then FG = FGo ® FGp ® FG\. We prove the result by induction on the number of prime factors of |Gi| (so, for the first step in the induction process, Gi will be trivial). It is sufficient to establish two facts. Fact 1. Let ^ be a primitive root of unity over F of order p and let K = F ( 0 . Then KGp determines FGp, If A^ = F , that is, if F contains a primitive pth root of unity, this is obvious. So assume A' 7^ F; in particular, p is odd. Using Theorem 2.1, write ^Gp = ®,>o ««^(6)

and

KGp ^ e . > , 6,F(6),

where ^i denotes a primitive root of unity of order p\ By Lemma VII.0.2, KGp ^ K ®F FGp ^ 0 . > o a,F{0 ®F H^t) and, for i > 1, F{^) ® F{(i) ^ [A^: F]F{^i), by Lemma 1.1. Then, by Lemma 2.4, we see that 61 = ao + ai[A^: F], where ao = 1, and bi = ai[K: F] for z > 1. Clearly the coefficients bi determine the coefficients a^ for alH > 1 and KGp determines FGp,

258

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

Fact 2. Let \Go\ = T^ let 9^ he Si primitive rth root of unity and let E = K{9r). Then EG determines KGp. (Note in the sequel that we may replace r by any multiple which is a product of primes q with q < p.) With notation as in Fact 1, write KGp = 0^>i biF{^i). As shown in Corollary 1.3, [A^(^^): A'] = p'^ for some m. We denote by K^ the cyclotomic extension of K whose dimension over K is precisely p'^. Then we can also write KGp in the form KGp = 0^>o CiKi for some integers Q . Again, we have EGp = 0^>o CiE®K Ki. Because of the choice of Go, we have gcd((/>(r),p) = I = gcd(r,p) so, using Theorem 1.7, each Ei = E®K Ki is a field and EGp = 0^>o ^iEi. Theorem 2.1 shows that EGQ = TE^ SO, by Lemma VILO.l, EG ^ EGo ®E EGp ®E EGi ^ vE ®E ( © , > O c^^^) ®E

EGL

Since we are assuming that FG\ is determined by FG, we have that EGi ~ E ®F FG\ is determined by EG. Assume that a tensor product of the form Ej ® EG\ contains Sji summands isomorphic to Ei. Clearly Sji = Q\i j > i and Sii > 1, because EG\ contains at least one summand isomorphic to E. Thus Ei®E EG\ will again contain at least one summand isomorphic to Ei. Let fii be the number of summands in EG which are isomorphic to Ei. Then I5i = r{ciSii + Y^j^iCjSji)^ so we can determine the integers Ci inductively from this formula. Fact 2, and hence the result, follow. D

3. Modular group algebras of abelian groups In this section, we consider isomorphisms of a group algebra FG, where F is a field of prime characteristic p and G is a finite abelian group whose order is divisible by p. This subject was first studied by W. E. Deskins in [Des56]. We begin by showing that a finite abelian p-group is determined by its group algebra over any field of characteristic p. We introduce some notation. For a given group G, define

G"" = {x^\xe

G}.

If G is an abelian group, then G^ is a subgroup and it follows from the decomposition theorems for finite abelian groups that two abelian groups G and H are isomorphic if and only if G^ ^ H^ and G/G^ ^ H/m.

3. MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS

259

For a commutative algebra A over a field F of characteristic p > 0, we denote by A^ the set of all elements a'^, a £ A, Clearly this is an algebra over the field F^. As noted at the beginning of the proof of Theorem IX. 1.1, if FG = FH as F-algebras, there exists a normalized isomorphism ip: FG -^ FH; that is, an isomorphism FG -^ FH such that €('tp(g)) = 1 for any g ^ G. 3.1 Theorem. Let F be a field of characteristic p > 0 and let G and H be finite abelian p-groups. Then FG = FH (as F-algebras) if and only if G^ H, Let FG = FH. Then 1^1 = \H\ = p"" for some n and we prove the theorem by induction on n. If n = 1, both G and H are cyclic groups of order p and hence isomorphic. So assume that the result holds for p-groups whose order is less than p'^. Certainly FG ^ FH implies that (FGy ^ (FHy and, since (FG^ = F^GP, we obtain F^G^ ^ F^H^. Since F^ is a field and charF^ = p, our induction hypothesis implies that G^ = H^. On the other hand, if xp: FG -^ FH denotes any normalized isomorphism, then ip{g^ — 1) = i^igy - 1 e A{H,HP), Thus V ^ ( A ( G , G P ) ) C A{H,HP), Working with ip~^ we obtain the reverse inclusion, so 'tp(A{G^G^)) = A{H^ H^), Hence '^ induces an isomorphism of the respective quotient rings and we have PROOF.

F[G/G^] ^ FG/A{G,G^)

^

FH/A{H,HP)

^

F[HIH^].

Since | G / G P | < |G|, the induction hypothesis yields that GJG'P = H/H^ and thus, as noted at the beginning of this section, G = H, The converse is trivial. D To study the general abelian case, notice that if G is an abelian group and p is a prime dividing |G|, then G = P X A^ where P is a p-group and yl is a group such that p jf \A\. We shall show that if Gi = Pi X Ai and G2 = P2 X A2 are two abelian groups whose orders are divisible by p and which are decomposed as above, then an isomorphism FGi = FG2 splits into two others; namely, FPi = FP2 and FAi = FA2, We follow the proof of [PM78]. 3.2 Lemma. Let F be a field of prime characteristic p and letG = PxA be a finite abelian group, where P is a p-group andp](\A\. Thenl + A(G,P)

260

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

is a p-group of exponent p'^ for some integer m > 0 and U{FG) PROOF.

= (1 + A ( G , P)) X

U{FA),

Clearly the set 1 + A ( G , P) is closed under multiplication so, to

prove t h a t it is a subgroup, it suffices to show t h a t its elements are invertible with inverses contained in 1 + A ( G , P ) . For this, let u G A ( G , P ) . Since A ( G , P ) is nilpotent, by Proposition VI.2.6, there exists a positive integer t such t h a t A(G', P)^ = 0. Let n be an integer such t h a t p^ > t.

Then

(1 + u)^" = 1 + u'P'' = 1, so 1 + t^ is an invertible element of 1 + A ( G , P ) , of order a divisor of p ^ . Hence 1 + A ( G , P ) is a p-group of exponent p ^ for some m > 0. An element in U{FA)

has in its support no elements of the form xa

with 1 / a: G P and a G >1, while every nontrivial element in 1 + A ( G , P ) necessarily contains at least one such element in its support.

Therefore

ZY(F/I)n(l + A ( G , P ) ) = { 1 } . To prove t h a t U{FG)

is actually the product of 1 + A ( G , P ) and

consider an arbitrary element a G U{FG). momorphism FG -^ F[G/P] u = a-^a

Denote by ep the natural ho-

=. FA and let a i = €p{a).

as an element of P C , we have QI G U{FG)

U{FA),

Thinking of a\

and ep{a'[^a)

— 1.

Hence

- 1 G A ( G , P ) and a = (1 + u)a^ G (1 + A ( G , P ) ) X U{FA),

desired.

as D

3.3 T h e o r e m . Let F be a field of prime characteristic

p and, for i — 1,2,

let Gi = Pi X Ai be finite abelian groups, where Pi is a p-group and Ai is a group such that p )( \Ai\. thus Pi ^ P2) and FAi ^ PROOF.

Then FGi = FG2 if and only if FPi = FP2

(and

FA2.

Assume t h a t FGi

= FG2 and let ip: FGi

-> FG2 be a nor-

malized isomorphism. By Lemma 3.2, 1 + A ( G 2 , P 2 ) has exponent p ^ for some m > 0 and thus a^

e U{FA2)

for every element a G ZY(PG2)-

Since p )( \A\\^ the group homomorphism A\ -^ A\ defined by a H-> a^"^ is one-to-one and onto. Thus, for every a G ^ 1 , there exists b ^ Ai such t h a t a = b^ . Since iP{a - I) = ^{b^^ - 1) = ij(by^

- 1

4. THE EQUIVALENCE PROBLEM

and since '0(6)^" G U{FA2),

it foUows t h a t

V^(6)P"

261

- 1 G A(G2, A2). This

shows t h a t ' 0 ( A ( G i , A i ) ) — A(G27^2) ^ind therefore ^^ induces an isomorphism of the respective quotient rings. Thus

^ FG2IHG2IA2)

^ F[G2lA2\

^

On the other hand, if a G 1 + A ( G i , P i ) , then V^(a) G U{FG2)\ Lemma 3.2, '0(a) = xt/ with x G 1 + A(G2,P2) and y G U{FA2).

FP2. thus, by Also,

by Lemma 3.2, there exists n > 0 such that a^" = 1. Hence xl^^aY"^ — xv""yv"" - i^ ^Q j.p'' - yv"" -

\^ Writing y - ^aeA2

^«^' ^^ ^ ^ ' ^ ^ ^^^^

yP" _ ^ ^ ^ ^ Ta a^" = 1. Since ;? J^ |yl2|, it is easy to see t h a t y^" = 1 if and only if y = 1. Thus jp{a) = x G 1 + A(G2, F2). Hence V^(A(Gi, P i ) ) = A(G2,P2) and, similarly, P ^ i ^ P ^ 2 . The converse is an immediate consequence of Lemma VILO.l.

D

4. T h e e q u i v a l e n c e p r o b l e m In the previous sections, we saw t h a t the rational group algebra of a finite abelian group G determines G in the sense t h a t , if H is another finite abelian group such t h a t QG = Q/T, then G = H.

Similarly, we showed t h a t the

modular group algebra of an abelian p-group determines the group. In this section, we carry this investigation further, developing tools which allow us to decide which other fields have similar properties. We begin by introducing a new concept. 4.1 D e f i n i t i o n . Let S denote a family of finite groups. We say t h a t two fields F , A^ are equivalent

on S if, for any G^ H £ S^ it is the case t h a t

FG ^ FH if and only if KG ^

KH.

This definition is due to E. Spiegel [Spi75], who first studied the equivalence problem in the class of finite abelian groups and who later, with A. Trojan [ S T 7 6 ] , extended this study to the class of finite 2-groups. In the next chapter, we shall apply these results to the study of the isomorphism problem for loop algebras of finite RA loops. We first extend Theorem 2.5.

262

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

4.2 Theorem. Let G and H be abelian groups of the same order n and, for each rational prime p dividing n, denote by Gp and Hp the p-primary components ofG and of H respectively. Let F be any field. Then FG = FH if and only if FGp = FHp for all p dividing n. In the case c h a r F jf \G\^ this is precisely Theorem 2.5, so assume c h a r F = q | |G|. Write G = Gq X A^ where Gq is the g-primary component of G and q )f \A\^ and write H = Hq x B^ where Hq is the ^-primary component of H and q )f \B\. By Theorem 3.3, FG = FH if and only if FGq ^ FHq and FA ^ FB. The result then follows from Theorem 2.5. D PROOF.

4.3 Definition. Let K be a field and p a rational prime. Let ^pn denote a primitive p'^th root of unity over K and set 7/c,p(n)-[/i'(^p.+i):A'(V)]The p-sequence of K is the sequence {7A",p(^)}n=i,2,...These sequences can be neatly characterized. 4.4 Theorem. Let p be a prime and let K be a field of characteristic different from p. If p is odd, then the p-sequence of K has one of the forms 1,1,1,... 1,1,1,... , l,p,p,... p,p,p,... while 2-sequences have similar forms except that the first terms can be arbitrarily I or 2. PROOF.

This follows at once from Corollary 1.3 and Lemma 2.4.

D

We now exhibit two concrete situations which show that, in the case p = 2, the arbitrary choices of 1 or 2 in the first term of a 2-sequence can, indeed, occur. 4.5 Examples.

(i) Let K = R, the field of real numbers. Then [/v(62):A'(6)] = [R(0:R] = [C:R] = 2,

while

[/ai2n+0:/i'(6n)] = [C:C] = l

4. THE EQUIVALENCE PROBLEM

263

for all n > 1. Thus the 2-sequence of K is 2 , 1 , 1 , — (ii) Let K = Q{V2).

Then

[Ki^2^): and, since ^2^ = ^{l

^ - ( 6 ) ] = [Q{V2,i):

Q(x/2)] = 2

+ i)^ we have

It is now easy to see t h a t the 2-sequence of A^ is 2 , 1 , 2 , 2 , Given a finite abelian group G of order p'^ and a field K of characteristic different from p , we have, by Theorem 2 . 1 ,

where ^p, denotes a primitive root of unity of order p^ and a^ = '^i/vi^ rii the number of elements of order p^ in G and Vi = [K(^px): K]. Notice t h a t ao = 1 and a^ = 0 if G contains no element of order p\ Associated with the field Ii\ a sequence {Q;n}n=o,i,2,..o can be defined as follows: set QQ = 1 and, for n > 0, let a^-f 1 be the least integer r such t h a t Vr > '^a„ when this integer exists; otherwise, set a^+i = (^n-\-2 = - - - = oc. With this definition, we have A'(^pan) = A'(^pan+i) = ••• = A'(^ a„^.i-i) if an-fi 7^ 00, while if a^^ 7^ 00 and a^i+i = 00, then A'(^pan) = ^^(^p^) for /j >

a^.

We define another sequence of numbers associated with the group algeb r a KG.

Set bm = <

dam + ^ « m - h l H

^- « « m + l - l

0

if « m /

OO

if a ^ = 00.

Then

I
264

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

then KG = KH if and only if bi = 6^ for all i = 1 , . . . ,n; equivalently, if and only if the numbers of elements of order at most p^' in G and in H are equal. 4.6 Theorem. Letp he a prime and let K and F be fields of characteristic different from p. Then K and F are equivalent on the class S of all finite abelian p-groups if and only if they have the same p-sequence. PROOF.

Suppose K and F have the same p-sequence and, for ^ > 1, set VK{s) = [K{^^sy.K]

and

VF{S) = [F{^j,s): F].

Then 5-1

VK{S) = VK{\)

J J 7A>(0

5-1

^nd

VF[S) = VF{\)

1=1

J ^ 7F,P(01=1

Hence, for 5 > 1, VK{S) — \VF{S)^ where A = 't'A"(l)/'^F(l)Let G and H be abelian groups of order p^ •> n > 1, and suppose that FG ^ FH. We have to show that KG ^ KH. Write FG ^ ®:=ob^Fi^,.)

and

FH ^

etoKnCp^)-

Then KG ^K®

e;^^, A6,A'(^pO

and

KH ^ A' 0 0 : ^ , , Xb[K{^^.).

Since FG = FH implies bi = 6' for all i = 1 , . . . , n, we have Xbi = Xb[ for all z; thus KG ^ KH. To complete the proof, we show that if the p-sequences of F and K differ, then there exist finite abelian groups G and H such that KG =. KH but FG^ FH. Assume that 7 F , P ( ^ ) 7^ lK,p{'^) foi* some positive integer n. First, consider the case where p is odd, or p = 2 with n > 1. Because of Theorem 4.4, we can assume, without loss of generality, that 7 F , P ( ^ ) = V a-nd 7 A > ( ^ ) — 1Let G — Cpn+i and H - Cj,n x Cj). Then KG = KH since G and H have the same number of elements of order at most p'^ for r < n^ but FG ^ FH since G and H do not have the same number of elements of order at most

4. THE EQUIVALENCE PROBLEM

265

Now consider the case where p = 2, n = 1 and assume t h a t we have two fields F and K whose respective p-sequences are 2,1,1,...

and

1,1,1,... .

Let i denote a 4th root of unity and ^ an 8th root of unity. T h e hypotheses on the p-sequences imply t h a t [F{i): F ( —1)] = 2 and [K(i): so i e K but i i

F,

Since [ F ( 0 : F{i)\

F ( 0 = F{i) and A^(0 = K{i).

A^ —1)] = 1,

= [ A ^ O - ^^XO] = 1, we have

Let G - Cg and 77 = C4 x C2. Applying

Theorem 2.1, it is easy to see t h a t FG^2F®

3F{i)

and

KG ^ 8A^

FH ^AF®

2F(i)

and

KH ^

while

Hence, KG = KH

8K.

but FG ^ FH^ so F and K are not equivalent over

s.

a In the next chapter, we shall be interested especially in finite abelian

2-groups. In order to work with such groups, we introduce here some isomorphism invariants of a field. 4 . 7 D e f i n i t i o n . Let K be a field. We define 1 ind2(A) = {n

if

00

0{K) =

t{K)

if7A',2(l) = 2 7A',2(^)

= 2 and

7A',2(^

if 7 A ' , 2 ( ^ ) = 1 for n >

- 1) = 1 for n > 2

1,

1

if x^ + 1 = 0 is solvable in K

0

if x^^ + 1 = 0 is not solvable in A',

1

if x^ + y-^ = — 1 is solvable in K

0

if x'^ + y^ = — 1 is not solvable in K.

With these definitions, Theorem 4.6 readily gives the following. 4.8 T h e o r e m . Let K and F be fields of characteristic

different

from 2.

Then K and F are equivalent on the class S of all finite abelian 2-groups if and only ifO(K)

= 0(F)

and md2(K)

= ind2(F).

266

X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS

As mentioned earlier, E. Spiegel and A. Trojan extended this study to the class of all finite 2-groups [ST76]. We quote their result here since it will be interesting to compare with its extension to 2-loops, which we shall give in the next chapter. We omit the proof because we shall not need the result directly in our work. 4.9 Theorem. Let K and F be fields. Then K and F are equivalent in the class S of all finite 2-groups if and only ifO{K) = 0(F), ind2(A^) = md2{F) andt{K) = t(F),

Chapter XI

Isomorphisms of Loop Algebras of Finite R A Loops

In this chapter, we turn our attention to the isomorphism problem for alternative loop algebras over fields. Given a finite RA loop L and a field F such that c h a r F )( |L|, we saw in Chapter VI that the loop algebra FL is a direct sum of ideals which are simple rings. We shall refer to this situation as the semisimple case. On the other hand, if char F | |Zy|, then such a decomposition does not exist; we call this the modular case. Since the techniques involved in the study of each of these cases are quite different, we deal with them separately. As in the previous chapter, all isomorphisms between loop algebras over the same field F are F-algebra isomorphisms. 1. S e m i s i m p l e l o o p algebras We begin with an easy lemma. 1.1 L e m m a . [dBM95a, Lemma 2.5] Let L\ and L2 be RA loops and let F be any field such that FLi ^ FL2. Then FlL^/L'^] = ^[1211'^] and A{LuL[)^A(L2.V^). Let ei^f : FLi -^ F[Li/L\] be the natural epimorphism. By Lemma VI.1.1, ker 6^/ = A(//i, L[) = F//i(l —5), where L' = {1, ^s}. Denote by [FLi^ FLi] the left ideal of FLi generated by all elements of the form a/3 -Pa with a,/3 G FL^. We claim that [FLuFLi] = A{Li,L[). In fact, given two elements ^,m 6 Li which do not commute, we have im-mi = im{l-s) G FLi{l-s); thus [FLuFLi] C FLi(i-s). On the other hand, if i and m are any two elements of Li which do not commute, PROOF.

267

268

XI. ISOMORPHISMS OF LOOP ALGEBRAS

then 1 — 5 = 1 — (i^m)

= m~^i~^(im

— mi) G [FLi^ ^ ^ i ] ; thus t h e opposite

inclusion also follows. Now given an isomorphism ip: FL\ i^{[FLi,FLi])

-^ FL2 we have ' 0 ( A ( L i , L^)) =

= [FL2, FL2] = A(L2, L'2) and thus A{Lu

L[) ^ A ( L 2 , V^)-

This shows t h a t 1/^ also induces an isomorphism ip of the corresponding factor rings, so F [ L i / L ' i ] ^ F L i / A ( L i , L ; ) S FL^I^iL^,

L'^) ^ FiL^/L'^].

D

If L is an RA loop with unique nonidentity commutator-associator s and if F is any field of characteristic different from 2, then, by Lemma VI. 1.2, FL = FL^^

® FL^

^ F[LIL']

© A ( L , V),

Thus Lemma 1.1 implies the following. 1.2 T h e o r e m . Let L\ and L2 be RA loops and let F be any field of characteristic

different from 2.

F[L2lL'^]

and A{Lu

Then FL\

= FL2 if and only if F[L\IL\]

=.

L\) ^ A(L2, L'^).

In the semisimple case, this result is not surprising since it is essentially a consequence of Corollary VI.4.8. Indeed, for a finite RA loop L and a field F such that char F j( |Z/|, we know that F[L/L']

is the sum of all

the simple components of FL which are both commutative and associative while A ( L , L ' ) is the sum of those simple components which are neither commutative nor associative (these are actually Cayley-Dickson algebras). Accordingly, it is apparent t h a t the structure of A(L,Z/') will be of fundamental importance for the study of the isomorphism problem.

We

first study the centre of this ideal. 1.3 P r o p o s i t i o n . Let L be a finite RA 2-loop and let F be a field such that c h a r F 7^ 2.

Write Z{L)

i — 1 , 2 , . . . , r. Assume ZiAf{L,

L'))

= {ti) x ••• x (tr),

that V C {t\). ^

where {t,) ^ €2^^

Then

^ ^ ^ J ^ L ^ - ^ F C ^ n . )[C2n2 X • • • X C a n . ] .

P R O O F . Because of Corollary VI.1.3, 2 ( A H L , L ' ) ) - F[Z{L)Y-^

= [F{t,y-^][{t,)

X . . . X (f.)]-

for

1. SEMISIMPLE LOOP ALGEBRAS

269

From Theorem X.2.1, we know t h a t ni

^<">"®in?^^«where Vi is the number of elements of order 2^ in C2«i. Thus

On the other hand, because of Lemma VI. 1.2, F{h)

= F{hy-^

® ^ ( ^ i ) ^ = mi)/L']

® AFiih),

L')

and

Comparing equations, we have

and the result follows.

D

Recall t h a t any finite RA loop L can be written L — M X A d.s the direct product of an RA 2-loop M and an abehan group A of odd order (Corollary V.1.2). We show that isomorphisms of semisimple alternative loop algebras "split" into two separate isomorphisms. 1.4 T h e o r e m . [ d B M 9 5 b , Theorem 2.5] Let L\ and L2 be finite RA loops and let F be a field whose characteristic

does not divide the order of either

of these loops. For i — 1,2^ write Li — Mi x Ai, where Mi is an RA and Ai is an abelian group of odd order. FMi

^ FM2 and FAi ^ PROOF.

F[Li/L\]

Then FL\

=. FL2 if and only if

FA2.

Suppose first t h a t FLi

= FL2.

By Theorem 1.2, we have

^ F[L2/L'^]] t h a t is, F [ ( M i / M O X A,] ^ F[{M2lM'2)

x A2].

As these are commutative group algebras, Theorem X.4.2 yields t h a t F[MilM[]

2-loop

^ F[M2lM!^]

and

FA^ ^

FA2.

270

XI. ISOMORPHISMS OF LOOP ALGEBRAS

In view of Theorem 1.2, in order to prove that FMi = FM2 as well, it will suffice to show that A(Mi,M{) ~ A(M2,M^). By Theorem 1.2, A(Z/i,i/2) ~ A(Z/2,i/2)- Moreover, for i = 1,2, denoting by Si the unique nonidentity commutator-associator of LJ, we have A(L,-, L[) = FLi^

^ FMi^

® FAi ^ A(M,-, M/) ® FAi

since Si G M^ Thus A(Mi, MO ® F ^ i ^ A(Li, L[) ^ A ( L 2 , 4 ) = ^ ( ^ 2 , M^) ® FA2. By Theorem X.2.1, FAi ^ nF ® ®d'^dF{U) = FA2, where ^d is a primitive root of unity of odd order d and d runs over the set of divisors of I All such that F{U) ^ F, For i = 1,2, A(L„ L\) ^ nA{M,, M[) ® 0 , m , ( A ( M „ M[) ® F{^d))^ By Corollary VI.1.3, Z ( A ( M , - , M ; ) ) = F [ 2 ( M , ) ] i ^ . Thus, again using Theorem X.2.1, 2(A(M,-,M/)) ^ 0^/^(^a,), where the ^f^^ are primitive roots of unity of order 2^K Consequently,

z(A(M., Ml) ® F{u)) = ®, n^a,) ® n^ci)Since (d,2"-') = 1, Theorem X.1.7 yields Fi^a,)® F{U) = FiUMa,)

= Fi^a.d),

where ^ajd is a primitive root of unity of order 2^^d. We claim that this field is never isomorphic to a field of the form F(^a^)' In fact, assume that FUa.d) = F{^a,)- Then FiU) C F{^a.) and so n ^ a . ) ®F F{U) ^ F{^,,d) = F{^a,){U) = F{U). However, as F{^d) / F^ the tensor product F{^a,) ®F f'i^d) has dimension at least two over the field F{^a^)'> ^ contradiction. Hence the centre of the algebra A(Mi, M-) is a direct sum of fields which are all different from those appearing in the decomposition of the centre of A(Mi, M!-) ® F{(d)- Since nA{MuM{)

® ® ^ md(A(Mi, M{) ® FiU)) ^ nA(M2, M^) ® ® ^ md(A(M2, M^) ® F{U))

2. RATIONAL LOOP ALGEBRAS

271

and because A(M^-,M/) is a sum of Cayley-Dickson algebras over fields of the form F{^aj) while Z(A{M-M-) ® F{^d)) contains no such direct summands, it follows that nA{Mi,M[) ^ nA{M2,M!^). Hence A(Mi,M{) ^ A(M2,M2), as desired. The converse is straightforward. D Theorem 1.4 reduces the isomorphism problem for semisimple alternative loop algebras to the study of the problem for group algebras of finite abelian groups, which we discussed in Chapter X, and the study of isomorphic loop algebras of RA 2-loops over fields of characteristic different from 2. This last question will be discussed in two stages. First, we shall study loop algebras over the field of rational numbers; then we shall see how to use the results obtained in this case to give general theorems over arbitrary fields.

2. Rational loop algebras In this section, we consider the isomorphism problem for finite RA loops L over the field of rational numbers. We shall see that the results depend on whether or not the unique nonidentity commutator-associator of L is a square in the centre of L. For this reason, we introduce some terminology due to L. G. X. de Barros [dB93a]. 2.1 Definition. An RA loop L is of type I if the unique nonidentity commutator-associator 5 of L is a square in Z{L); that is, if there exists a G 2(L) such that a^ = s. If there exists no such element, the loop is said to be of type 11, If L is a finite RA loop with L' = { l , ^ } , it follows from Theorem IV.3.1 and Theorem V.1.8 that the centre of L can be written in the form Z(L) = {h) X (^2) X (^3) X A, where o(^,) = 2"^*, mi > 1, m2,m3 > 0, 5 6 {^1) and A is an abelian group. It is clear that L is of type I if and only if mi > 1 and hence of type II if and only if Z{L) = V x H for some (abelian) group H. The isomorphism problem for loops of type I was studied by G. Leal and C. Polcino Milies in [LM93] and their work was completed with the study of loops of type II by L. G. X. de Barros [dB93a].

272

XI. ISOMORPHISMS OF LOOP ALGEBRAS

2.2 T h e o r e m . Let L be a finite RA loop of type I and let M be

another

loop. Then QL ^ QM if and only if L/L' PROOF.

^ M/M'

and Z{QL)

^

Z(QM).

Assume first t h a t QL ^ Q M . Then, clearly, Z{QL)

^

Z{QM),

Also, Lemma 1.1 shows t h a t A ( L , L') ^ A ( M , M ' ) and Q[L/V] so L/L'

^ M/M'

^ Q[M/M'],

by Theorem X.2.3.

By Corollary VI.4.8, QL ^ Q[L/L'] 0 A ( L , V) and Q M ^ Q [ M / M ' ] © A ( M , M ' ) where A(L,LO = A i © . - . © A n and A(M,

M')^Bi®'"®Bm

are the direct sums of Cayley-Dickson algebras Ai and 5 j , respectively. Comparing respective centres, we obtain n = m and, after possible reordering, Z(Ai)

= Z{Bi)

foTi=

1,.. .,n.

Let L' = { l , ' ^ } . Since L is of type I, there exists a G Z(L) a2 = s. Then a ^

such t h a t

G Z ( A ( L , L ' ) ) = 2 ( A ( M , M ' ) ) and ( c ^ ^ ) ^ = - 1 ^ ,

so the projection of this element onto each simple component is a central element whose square is —1. It follows t h a t the centres of each Ai and each Bi are not Q and so, in view of Lemma V n . 2 . 2 , these simple components must be split Cayley-Dickson algebras and hence unique over their centres (Theorem L4.16). So A^ ^ B^{oT i = I,,.,

, n a n d Q L ^ Q M as desired.

D

This result shows that finite RA loops of type I need not be determined by their rational loop algebras. To give a specific example, it is useful to get some information about the quotient L/L\

We adopt the notation of

Theorems V. 1.7 and V.1.8. 2.3 P r o p o s i t i o n . Let L be a finite indecomposable

RA loop.

Write

Z{L) = (^i) X (^2) X (^3) X (w) ^ C2- X C26 X C2C X C2d, with a> I, b,c,d>

0, and V C (^i). If L G £1 U £ 3 U £5 U £ 7 , then L/L

= 0 2 0 - 1 X 0 2 6 + 1 X C2C+1 X C2d-\-i]

otherwise L/L

= 0 2 0 X 026+1 X 02C+1 X 0 2^*

2. RATIONAL LOOP ALGEBRAS

273

Our proof relies on the classification of indecomposable RA loops given in Theorem V.3.1. For an element x G L, we denote by x its image in the quotient L/ V, \i L eCi U £ 3 U £ 5 , then PROOF.

LjL' = (?7) X (x) X (y) X (u) = C2a-i x C26+1 x C2C+1 x C2. If L G £2, we have L/L' = (y) X (uT) X (Ff) ^ C2a x C2 x C2, where i^i = uy^j ~^ and Xi = xy^j If L G >C4, we have

~^.

L/L' = {x) X (y) X (tZT) = €2^ x C26+1 x C2, where i^i = ux/j L/L'

~ \ while, if L G /^e? then = ( i l ) X ( x ) X (y)

=

C 2 a X C26+1 X C2C+1.

If // G £7, we have L/L' = (?7) X (x) X (y) X (tU) = C2a-i X C26+1 x C2C+1 x C2d+i. The result follows.

D

When a = I (that is, ti = s) and L G £2 U £4 U Ce^ then d = 0, so a = d + 1 and we have the following. 2.4 Corollary. Let L be a finite indecomposable RA loop of type II and write 2{L) = L' x C26 X C2C x ^2^ with b,c,d> 0. Then L/V = C26+1 X C2C+1 X C 2 d + i •

2.5 Example. We now give an example which illustrates that finite RA loops of type I need not be determined by their rational loop algebras. Let L G £3 be such that Z{L) = C4 X C2 and let L2 G £2 be such that Z(L2) = C4. Set M = L2 X C2. Clearly LfM. On the other hand, we use Theorem 2.2 to show that QL ^ QM. By Proposition 2.3, LIV ^ C2 X C4 X C2 X C2 = M/M', so we have only to establish that Z{QL) and Z{QM) are isomorphic. We have Z(QL)SQ[L/L']®Z(A(L,I'))

274

XI. ISOMORPHISMS OF LOOP ALGEBRAS

with Z{A{L,V)) Now

= Q[Z{L)]^

and Q[L/L'] ^ Q[M/M'] ^ 12Q © 6Q(0.

Q[Z(L)] ^ Q[C4 X C2] - 4Q ® 2Q(0. Since

Q[Z{L)]^Q[Z{Lr-±^eQ[Z{Lr-^ and I 1 + 5 rv.

we see that Q[Z{L)]^ ^ 2Q{i). So Z(QL) ^ 12Q + 8Q(0. Similar computations show that Z(QM) ^ 12Q + 8Q(i) = Z{QL), as desired. We now turn our attention to RA loops of type II. By contrast, these indeed do turn out to be determined by their rational loop algebras. We begin with a refinement of Corollary VI.4.8. 2.6 Lemma. Let L be a finite RA 2-loop of type II and let N = Then

L/Z(Ly.

where A(A^, A^') is a direct sum of Cay ley-Dickson algebras each with centre Q and A is a direct sum of split Cay ley-Dickson algebras each with centre different from Q. Let L' = U,s}. Since L is of type II, Z{L) = L' x H with H ^ C2"2 X • • • X C2nt. We know that PROOF.

Z{A(L, L')) ^ Q[Z{L)Y-^

^ [{QVy-=^]H ^ QH,

Also, Q ^ ^ QC2n2 ® ••• ® QC2nt and, by Theorem VII.L4, QC2n. ^ 01^=0 Q ( 6 0 = Q ® Q ® ^7=2^(^2^^ where ^2* denotes a root of unity of order 2\ Hence Z{A{L,L'))^

Q®---®Q

05,

2^~^ summands

where <S is a sum of fields of the form Q(^2»)? ^ ^ 2. Thus, by Corollary VI.4.8 and Lemma VIL2.2,

A{L,L')^^t\'

MQ) ® A,

2. RATIONAL LOOP ALGEBRAS

275

where each Ai(Q) is a Cayley-Dickson algebra over Q and A is a> sum of split Cayley-Dickson algebras over Q(^20? ^ > 2. Now N = LlZ{Ly is an RA 2-loop of order 2*+^ and Z{N) has exponent 2. Furthermore (as above)

2*~1 summands

Thus A{N,N') QL ^ QLZ{Ly

^ 0 ^ 1 " / Ai{Q). Furthermore, using Lemma VI.1.2, ® Qi.(l - . i f ^ ^ ) ^ QL/A{L, Z{Lf)

^ Q7V © A ( l , Z{Lf) thus A{L, Z(Ly)

® A(Z, Z ( Z f )

^ Q[N/N'] ® A(JV, N') © A ( I , ^(L)^),

= !F Q A, where T is the direct sum of fields. Hence QL ^ Q[N/N'] © JT © A{N, N') © A,

so Q[N/N']®T = Q[L/L'] is the sum of the simple commutative, associative components of QL and the result follows. D 2.7 Theorem. Let L and M be finite RA 2-loops of type 11. Let N and P denote the quotient loops L/Z{L)^ and M/Z{My, respectively. Then QL ^ QM if and only if L/L' ^ M/M', Z{L) ^ Z{M) and A{N, N') ^ A(P,P')Suppose L and M are RA 2-loops of type II such that QL = QM. By Theorem 1.2, Q[L/L'] ^ Q[M/M'] and A(L,L') ^ A ( M , M ' ) , so Z{A{L,L')) ^ Z{A{M,M')). By Theorem X.2.3, L/L' ^ M/M'. Writing Z{L) — L' X H and Z{M) = M' x K\ where H and K are abelian 2-groups, we have PROOF.

Z(A(L, L')) ^ Q[Z{L)]{1 - V) ^ [{QL'){\ - L')]H ^ QH and Z ( A ( M , M ' ) ) = Q[Z(M)](1 - M') ^ [(QM')(1 - M')]K ^ QK. Theorem X.2.3 shows that H = A', which implies Z{L) ~ Z{M). Since Lemma 2.6 shows that A(7V, N') and A(P, P') are the sums of the simple components of QL and QM which are Cayley-Dickson algebras with centre Q, it is also clear that A(7V, N') ^ A(F, P').

276

XI. ISOMORPHISMS OF LOOP ALGEBRAS

Conversely, assume t h a t L/V

^ M/M\

Z{L)

^ Z(M)

and A(7V, TV') ^

A ( F , P O - By Lemma 2.6, QL^Q[L/L']®A{N,N')®A and Q M ^ Q [ M / M ' ] ® A ( P , P ' ) ® i? where ^ and 13 are the sums of the simple components of QL and Q M , respectively, which are split Cayley-Dickson algebras with centres not Q. The first two summands in each of the above decompositions are pairwise isomorphic. Furthermore,

and hence

= Q[Z(L)]^,

Z{A(N,N')QA)

where L' = { l , ^ } , so Z{A)

is

the sum of the components of Q [ 2 ( Z / ) ] ^ ^ which are not Q, and similarly for Z[B).

Since V is a direct factor of Z{L)

and M' is a direct factor of

>Z(M), we may assume t h a t the isomorphism Z[L) M', It follows t h a t Z{A)

= Z{B),

-^ Z{M)

maps L' to

hence the result.

D

In what follows, we shall require a description of the rational loop algebras of finite indecomposable RA loops with centres of exponent 2. According to the Classification Theorem (Theorem V.3.1), there are (up to isomorphism) seven such loops, which we describe below. If Z{L)

= C2, then |L| =

Mi6(Q8?2) or I2 =

MIQ{QS)'

16 and L is isomorphic to either /]

=

T h e corresponding rational loop algebras

were described in Corollary VII.2.3: Q/i = 8 Q ® 3 ( Q ) , Q/2 = 8 Q © ( Q , - 1 , - 1 , - 1 ) . H e r e 3 ( Q ) and (Q, - 1 , - 1 , - 1 ) denote, respectively, the split Cayley-Dickson algebra over Q and the division algebra of Cayley numbers over the rationals. If Z{L)

= C2 X C2, then |Z/| = 32 and there are two such loops (see

Table V.4): /3 = M ( 1 6 r 2 C i , * , l ) G > C 3

2. RATIONAL LOOP ALGEBRAS

277

and h = M(16r2C2,*,/i) G A , where 16r2Ci = (a:, ^,^1,^2 I ^^ = ^1 = ^2 — 1^2/^ — ^29^2 2ind ^i = (a:,i/) central) and 16r2C2 = (a:,y,^1,^2 Ml == ^2 — l?^'^ = ^1^2/^ = ^25^2 ^nd ^1 = (x^y)

central).

Since Is and I4 are of type II, using Corollary 2.4, we see t h a t Is/I^

—

C2 X C2 X C4 ^ 14/14- Since QC2 ^ Q ® Q and QC4 ^ Q ® Q 0 Q ( 0 , we have Q[/3//^] = Q[/4//4] = 8Q ® 4 Q ( 0 . According to Theorem VII.2.5, all simple components of A ( / 3 , / 3 ) are split Cayley-Dickson algebras, while all but one of the simple components of A ( / 4 , / 4 ) are also split. Computing dimensions, we see t h a t both A ( / 3 , / 3 ) and A ( / 4 , / 4 ) have precisely two components, so

Q/3^8Q®4Q(0®23(Q) and Q/4 ^ 8Q ® 4Q(0 ® 3(Q) ® (Q, - 1 , - 1 , - 1 ) . If Z{L)

= C2 X C2 X C2, then |L| = 64 and, according to Table V.4,

there are two indecomposable RA loops of this order, both constructed from the same indecomposable group 32r2A: = {x,y,h,t2,t3

\ x^ = t2,y^ = h.tj

= tj = ^ = l , ( x , y ) = ^1),

with ^1,^2,^3 central. The corresponding loops are /5 = M ( 3 2 r 2 A : , * , l ) G £ 5 and

Ie =

M{32T2k,^,h)eCe.

278

XI. ISOMORPHISMS OF LOOP ALGEBRAS

Once again, using Corollary 2.4, we see that h/I's — C2 X C4 X C4 = h/I^ so, as before, we see that Q[hir^] = 8Q 0 12Q(i) ^ Qi/e/^]- Applying Theorem VII.2.5, we get Q/5^8Q®12Q(0e43(Q) and Q/e ^ 8Q e 12Q(i) ® 33(Q) 0 (Q, - 1 , - 1 , - 1 ) . FinaUy, if Z{L) = C2 X C2 x C2 x C2, then L ^ I7 = M{32T2k x C'2, *,'";) G Cj^ where w ^ \ generates the factor C2. Here, hll'-j = C4 X C4 X C4 and Q[/7//7] = 8Q 0 28Q(z), so Theorem VII.2.5 gives Q/7 ^ 8Q © 28Q(0 0 73(Q) 0 (Q, - 1 , - 1 , - 1 ) . 2.8 Lemma. Let L be a finite RA 2'loop of type 11. Write L = Li x H, where L\ is an indecomposable RA 2'loop and H is an abelian group. Set N = LlZ{Lf, Ni = L i / 2 ( L i ) 2 and p = rank{H/H^) = rank{H). Then A{N,N')^2f'A{NuN[). Observe that A^i is an indecomposable RA loop with centre of exponent 2 and that rank(2:(A^i)) = rank(2:(Li)). Since N = N^ x{H/H'^), PROOF.

A{N,N')

= QN 1 -25 = Q A ^ i ^ ®Q Q[H/H^] ^ AiNuN{)

where N' = N[ = {\,s}.

So the result follows.

®Q Q[H/H% D

2.9 Lemma. Let L and M be finite RA 2-loops of type IL Write L — L\X E and M — M\ x K, where L\ and M\ are indecomposable RA loops and H and K are abelian groups. If QL = QM, then rank(/r) = rank(A^) andL^IZ{Lif^ L2lZ{L2f. Let N = L/Z{Lf, P = MlZ{Mf, N^ = L^lZ^L^f and Pi = Mi/Z(Mi)2. Set p = rank(//) = T^iik{H/H^) and r = rank(/l) = rank(A7A'2). If QL ^ QM, then, by Theorem 2.7, L/V ^ M/M', Z{L) ^ Z{M) and A(7V,7V') ^ A ( P , P ' ) . so, by Lemma 2.8, 2^A{NuN[) ^ 2^A{Pi,P{). Since Ni and Pi are indecomposable RA loops with centres of exponent 2, both loops are either in 7^ = {/2,/4,/e,/?} or in I = {/i,/3,/5}, where PROOF.

2. RATIONAL LOOP ALGEBRAS

279

/^, 1 = 1 , . . . , 7 denote the seven indecomposable RA loops described above, according to whether or not (Q,—1,—1,—1) appears as a direct summand in the decompositions of QL and QM. If Ni and Pi are in H, then p = r since (Q,—1,—1,—1) appears only once as a direct summand in the decompositions of Q/2, Q/4, Q/e and Q/7. On the other hand, if Ni and Pi are both in J , then we shall show that the assumption p ^ r leads to a contradiction. For this, it is sufficient to consider three cases. Case 1: TVj ^ h and Pi ^ I3. In this case, rank(Z(i/i)) = rank(>Z(iVi)) = 1 and rank(Z(Afi)) = rank(Z(Pi)) = 2. Hence Z{Mi) ^ C2 X C^b with 6 > 0. Since Z{Li) ^ C2, Z{L) ^ Z{Li) X H ^ Z(Mi) X K ^ Z{M), we have C2 x 7^ ^ C2 x C26 X K, which implies H = C26 X K. Therefore L ^ Li x C26 X K and L/L' ^ Li/L[ X C26 X K. By CoroUary 2.4, Li/L[ ^ C2 X C2 x C2 and Mi/M{ ^ C2 X C2 X C26+1. Since L/L' ^ M / M ' ^ (Mi/M{) x A^ it follows that C2 X C2 X C2 X C26 X K = C2 X C2 X C26+1 x A% which implies that 6 = 0, a contradiction. Case 2: A^i ^ A and Pi ^ /s. Here rank(2(Li)) = 1 and rank(2(Mi)) = 3. Hence 2 ( M i ) ^ C2 X C26 X C2C with 6 , 0 0. Since Z{Li) ^ C2, 2 ( L ) ^ Z(Li) x 7/, Z ( M ) ^ 2 ( M i ) X A^ and Z{L) ^ Z{M), we have C2 X ^ ^ C2 X C26 X C2C X A^ which implies H = C26 X C2C X K. Thus L = Li x C26 x C2C X A^ and L/L' ^ Li/L[ X C26 X C2C X K. By Corollary 2.4, L I / L ; ^ C2 X C2 X C2 and Mi/M{ ^ C2 x C26+1 x C2C+1. Since L/L' ^ M / M ' , it foUows that C2 X C2 X C2 X C26 X C2C X K = C2 X C26+1 X C2C+1 X A^ which implies 6 = c = 0, a contradiction. Case 3: A^i ^ /a and Pi ^ /g. In this final case, rank(2(Li)) — 2 and rank(Z(Mi)) = 3. Hence Z{Li) ^ C2 X C2r and 2 ( M i ) ^ C2 x C26 x C2C, with r,b,c > 0. As before, we have C2 X €2^ X H = C2 X C26 X €2^ X K, If r = 6, then H ^C2cX A^ so L ^ Li X C2C X K. Thus L/L' ^ C2 x C2 x C2r+i x C2C x K and M / M ' ^ C2 X C2r+i x C2C+1 X A'. Since 1/1' ^ M/M\ we conclude that c = 0, a contradiction. The case r = c is similar, so assume

280

XL ISOMORPHISMS OF LOOP ALGEBRAS

r ^ b and r ^ c. Since €2^ X H = C26 X C2C X K^ there exist subgroups Hi and A'l of H and K, respectively, such that H = C26 X C2C X ^ 1 , K ^ C2r X Ki and i7i ^ R\. Then L ^ Li x C26 x C2C x Hi and M ^ Ml X C2r X i7i, hence L/V ^ C2 x C2 x C2r+i x C26 x C2C x ZTi and M/M' ^ C2 X C26+1 X C2C+1 x C2r x i^i. Since L/V ^ M/M', we obtain r = 6 = c = 0 , a contradiction. Finally, p = r implies that A{Ni,N{) = A(Pi,P{) and, since Ni and Fi are indecomposable RA loops with centres of exponent 2, it foUows from the discussion prior to Lemma 2.8 that Ni = Pi. D 2.10 L e m m a . Let L and M be finite RA 2-loops of type II such that QL = QM. Write L = Li x H and M = Mi x K, with Li and Mi indecomposable RA loops and H and K abelian groups. If H = K, then Li = Mi. Clearly, Z{L) ^ Z{Li) x H, Z{M) ^ Z{Mi) x K and, by Theorem 2.7, Z{L) ^ Z{M). Hence, if 77 ^ A^ then Z{Li) ^ Z{Mi). By Lemma 2.9, Q[Li/Z{Li)'^] ^ Q[Mi/Z{Mi)^], so these algebras either both admit (Q,—1,—1,—1) as a direct summand or both do not. Because of Lemma 2.6, the same is true for QLi and QMi. Since Z{Li) and Z{Mi) have the same rank, and because of Theorem VII.2.5, it follows that Li and Ml are in the same family Cj described in Table V.3. Furthermore, since Z{Li) ^ 2 ( M i ) , we have Li^ Mi. D PROOF.

We now proceed with the main result of this section. 2.11 T h e o r e m . Let L and M be finite RA loops of type II. Then QL = QM if and only if L = M. One implication is obvious. For the converse, assume QL = QM. By Theorem 1.4 and Theorem X.2.3, we may assume that L and M are finite RA 2-loops of type II. Write L = Li x H and M = Mi x K, where Li and Mi are indecomposable RA loops and H and K are abelian groups. Let N = L/Z{L)\ P ^ M/Z{M)\ Ni ^ LilZ{Lif and A = MilZ{Mif. By Lemma 2.9 and Theorem 2.7, LjV ^ M/M\ Z{L) ^ 2 ( M ) , A{N,N') ^ A ( P , P ' ) and Ni ^ Pi. Because of Lemma 2.10, we now need only prove that H = K. We do this by analyzing all possibilities for A^i and Pi. Recall that A^i = Pi is one of the indecomposable loops li described in the discussion prior to Lemma 2.8. PROOF.

2. RATIONAL LOOP ALGEBRAS

281

If Ni ^ Pi is isomorphic to either h or h, then 2 ( L i ) ^ C2 = So Z{L) ^ Z{M)

readily implies H ^

2:(Mi),

K,

Suppose Ni ~ Pi is isomorphic to either I3 or I4. Then r a n k ( 2 ( L i ) ) = r a n k ( 2 : ( M i ) ) = 2.

Write Z{Li)

^ C2 X Cs^ and Z{Mi)

^

C2 X C^b,

r > 0,6 > 0. By CoroUary 2.4, L I / L ; ^ C2 X C2 X C2r+i and M i / M {

^

C2 X C2 X C26+1. Since 2 ( L ) ^ 2 : ( M ) , we obtain C2r x H ^ C^^ x K,

If

r ^ b^we can find subgroups Hi and A^i of H and /iT, respectively, such t h a t H ^C2bX

Hi, K ^ C2r X Ki and Hi ^ Ki.

M ^ MixC2rxHi.

Then L ^ Li x C26 x i7i and

Since L/L' ^ {Li/L[)xH

smd M/M' ^

{Mi/M[)xR\

we have L/V

^ C2 X C2 X C2r+i X C26 X Hi

and M/M' and hence LjV

^ C2 X C2 X C26+1 X C2r X Hi,

^ MjM',

a contradiction.

Hence r = b and therefore

7/ ^ A\ Suppose A^i = Pi is isomorphic to either Is or IQ, Then r a n k ( Z ( L i ) ) = r a n k ( 2 ( M i ) ) = 3. Write Z{Li)

^ C2 X ^2^ X C2^ and Z{Mi)

^ C2 X C26 x

C2C, r, 5,c > 0. Hence, again by Corollary 2.4, Z/l/Z/i = C2 X (72r+l X C25+I and Mi/M[

^ C2 X C26+1 X C2C+1.

From 2:(L) = 2 ( M ) , we conclude t h a t C2r X C2' X H ^ A'. If {b,c}

/

C26 x C2C x

{^,'5}, simple arguments (as in the previous case) lead to

contradictions in aD possible cases, so H = K. If Ni = Pi is isomorphic to /y, we write Z{Li) and Z{Mi)

= C2 X C2r X €2^ X C29

= C2 X C26 X C2C X C2d, with r,s,q,b,c,d> Z/l/Li = C2r-|-1 X C2.'+l X C2qr+1

and Mi/M[

= C26+1 X C2C+1 X C2d+i.

0. Then

282

XL ISOMORPHISMS OF LOOP ALGEBRAS

Since Z(L) = Z{M), we obtain C2r X C23 xC2
c{K) = I

1

\i x'^ -{- y^ -{- z^ -\- w'^ = -\ is solvable in A'

0

if x^ + t/"^ + 2:-^ + tt;-^ = — 1 is not solvable in K

As shown in Theorem 1.3.4, c{K) = 0 if and only if the Cay ley-Dickson algebra (A', —1, —1, —1) is a division algebra. Hence, by Corollary VII.2.3, K[Mie{Q)] ^ A^[Mi6(g,2)] if and only if c{K) - 1. 3.3 Proposition. Let p he a prime and let Fp denote the field with p elements. Then t{Fp) = 1 and thus also c{Fp) = 1. P R O O F . We have to show that —1 is a sum of two squares in Fp. This is obvious for p = 2^ so assume that p is odd. Let ^i and 5*2 be the subsets

3. THE EQUIVALENCE PROBLEM

283

of Fp defined by 5l = { o ^ l ^ 2 ^ . . . , ( 2 ^ ) ' } and 52 = { - 0 ^ - 1 , - 1 2 - 1 , - 2 2 - l , . . . , - ( H f i ) 2 - l } . Since x'^ = t/^ (mod p) implies p\ {x — y)oTp\ (x + y), both 5*1 and 5*2 have precisely ^ ^ elements. Since Fp has only p elements, there exist x^ G 5*1 and —y^^ — 1 G 5*2 so that x'^ = —y*^ — I. This proves the result. D 3.4 Proposition. Let L he a finite RA 2'loop and K a field such that char/I / 2. (i) / / L is of type /, then all simple components of AK(L^ L') are split algebras. (ii) If L is of type II and A is a simple component of AK{L, L') which is not splits then A = {Ii\ —1, —1, —1). Write Z{L) = {ti) x (^2) x ••• x (tr), where (ti) = C2n., i = 1,2,..., r a n d V C {ti). By Proposition 1.3 and Theorem X.2.1, PROOF.

oni-l

2 ( A K ( L , 1 0 ) = f 7 7 7 - T - ^ ^ ^ ^ ( ^ 2 n i )[C2n. X . . . X C2ne] ^ e ^ ^ i

lU,

where A\ = A^(^2"^») ^^^ ^2"^» is a primitive 2"^»th root of unity, i = 1 , . . . , r. By Corollary VI.4.8, AK{L, L') = ®Ui ^^^ where each Ai is a CayleyDickson algebra over its centre Ki, Furthermore, because of Lemma VII.2.2, if the centre of Ai contains a 2^th root of unity with n > 1, then Ai is split. Thus, if m^ > 1, then Ai is split so, if L is of type I, rii > 1 and thus also each nii > I, Therefore each Ai is split. If L = {Z{L)^ X, y, u) is of type II, we only have to consider those simple components of A A ' ( ^ , L^) which have centre K (the others being split). By Lemma VII.2.2, the nonsplit simple components of AxiL^L^) are of the form A =

{{K[ZiL)])e,ixe)Mye)\iue)%

e a primitive central idempotent in AK{L^L'). The field {K[Z{L)\)e = K does not contain a fourth root of unity. Since a — (xe)'^^ /? = (ye)^, 7 = {ueY have orders which are powers of 2, it follows that a,/?,7 G { i l l It is easily seen that if one of these values is + 1 , then the equation x'^ —

284

XI. ISOMORPHISMS OF LOOP ALGEBRAS

OL'ip' — (iz^ + a/3w^ = 7/^ has a nontrivial solution. Hence, by Theorem 1.3.4, the algebra A is split, a contradiction.

Thus a = P = 'y = —l^soA

=

(A^-1,-1,-1).

D

We now establish the analogue of Theorem X.4.9 for finite RA 2-loops. 3.5 T h e o r e m . Let F and K be fields with characteristics

different from 2.

Let 7^2 be the class of finite RA 2-loops of type IL Then F is equivalent

to

K on 7^2 if ^ ^ ^ only if

(a) 0{F) =

0{K),

(b) ind2(F) = ind2(A') and (c) c{F) = PROOF.

c{K).

Assume t h a t F and K are equivalent on 72^2• First, we prove

that F and K are equivalent on the class of all finite abelian 2-groups. Assume t h a t A and B are finite abelian 2-groups such t h a t FA = Set / =

M\Q{Q^),

FB.

the Cayley loop, and \ei L - I X A and M = I X B. Then

FL = FM^ thus KL = A'M, since F and K are equivalent on 712- Hence, by Theorem 1.2, K[LIV] M/M'

^ K[M/M'],

Since L/L'

^ C2 x C2 x C2 x yl and

^ C 2 X C 2 X C 2 X ^ , wehaveA'[C2xC2xC2X/l] ^

Ii[C2xC2xC2xB].

Using the fact t h a t KC2 = K ® A^ we conclude t h a t KA

= KB.

Thus

F and A' are equivalent on the class of all finite abelian 2-groups so, by Theorem X.4.8, 0 ( F ) = 0{K)

and ind2(F) = ind2(A').

If c ( F ) 7^ c(A'), then, without loss of generality, c{F) = 1 and c{K) — 0, so, by CoroUary VII.2.3, F[M,e{Q)]

= F [ M i 6 ( Q , 2 ) ] , but K[Mie{Q)]

^

A^[Mi6(Q,2)]. Thus c ( F ) =r c(A'). Conversely, suppose t h a t conditions (a), (b) and (c) hold. For L and M in 7^2, assume FL ^ FM. AF(L,L')

^

AF(M,M').

By Theorem 1.2, F [ L / L ' ] ^ F[MIM']

Since both L/L'

and M/M'

2-groups, conditions (a) and (b) imply t h a t Ii[L/L^]

and

are finite abelian

= A ' [ M / M ' ] , by The-

orem X.4.8. Because of Theorem 1.2, it remains to show t h a t A x ( / / , L^) = A K ( M , M ' ) . We have Z{L)

^ V x 02^2 x • • • x C2nr and Z{M)

^ M' x

C2'n2 X • • • X C2m., where L' ^ C2 = M ' , n, > 0 for i = 2 , . . . , r and m^ > 0 for J = 2 , . . . , s. By Proposition 1.3, 2 ( A F ( i / , L')) = F[C2n2 X • . • X C2nr]

3. THE EQUIVALENCE PROBLEM

285

and Z{AF{M,

M'))

^ F[C2-2 X • • • X C2ms].

Hence F[C2r^2 X • • • X C2nr] ^ F[C2-2 X • • • x C2ms] ^ 0 ^ ^ F,-, where Fi = F or Fi = F{^2^^) wi^h 6i > 2 for i = 1 , . . . , /:. Similarly,

and 2:(AK(M, M'))

= K[C2m2 X . . . X C2m.].

Conditions (a) and (b) imply t h a t F and K are equivalent on the class of finite abelian 2-groups, by Theorem X.4.8. Therefore, K[C2n2 X • . . X C2nr] ^ A^[C2-2 X ' ' . X 0 2 ^ . ] ^ 0 ^ 1 A^', where K^ = K or A^ = A X 6 - . ) with a, > 2, z = 1 , . . . , ^. Write AF(L, L') = 0 f = i A{F^) and

AF(M,M')

^ 0 ^ ^ B^{F^), where A ( F t ) and /5,(/s) are

Cayley-Dickson algebras over Fi. Since Ap'{L^L')

=. A/7(M, M ' ) , we have

Ai(F^) = Bt(Fi) for i = 1 , . . . , A;, reordering, if necessary. Write A/^'(L, L') = 0 t i C^(^'^) ^nd

AA'(M, M O

^ 0 t i A ( A ; ) , where C,( A\) and P ( A\) are

Cayley-Dickson algebras over AV For A\ = A'(^2^t) with a^ > 2, we have Ci{Ki)

= Vi{Ki)

since these algebras are split over the same field Ki. It

remains to analyze the case Ki — K. If c[K) = c ( F ) = 1, then all the algebras Ci[K) and Vi{K)

are split

and thus isomorphic. If c{K) = c{F) = 0, then some of the algebras Ci{K) and Vi[K) may be division algebras while, by Theorem 1.4.16, the others are all isomorphic to (A^ 1,1,1). Thus it remains only to show t h a t the numbers of nonsplit algebras in

AK{L^L')

and in A/v'(M, M ' ) are equal.

By Propositions 3.3 and 3.4, we may assume t h a t char F = char A' = 0, so we may assume t h a t Q C F and Q C K. Also, c{K) = c{F) = 0 implies 0{F)

= 0{K)

— 0. Let mK{L) denote the number of direct summands of

KL which are Cayley-Dickson division algebras. From KL = K ® Q Q L , KM mK{L)

^ A ®Q Q M , FL ^ F ®Q QL and FM = rriQ^L) — mp^L)

and mK{M)

= rriQ^M)

FL = FM, we conclude t h a t 771^(1) - mF{M), and hence A K ( L , L') ^ A K ( M , M ' ) .

^ F ®Q Q M , we obtain = mjr{M).

From

Thus, mK(L) =

mK{M) D

286

XL ISOMORPHISMS OF LOOP ALGEBRAS

Now it is easy to see which fields will yield a positive solution to the isomorphism problem for finite RA 2-loops of type II. 3.6 Corollary. A field K of characteristic different from 2 determines the class 72.2 of all finite RA 2-loops of type II if and only if ind2(A) = I, 0{K) = 0 and c{K) = 0. Because of Theorem 3.5, K is equivalent to Q on TZ2 if and only if ind2(A^) = 1, 0{K) = 0 and c(K) = 0. Since we have shown in Theorem 2.11 that Q determines the elements of 7^2, the result follows. D PROOF.

To obtain a more general result, we now study the RA loops of type I. 3.7 Theorem. Let F and K be fields of characteristic different from 2. Let 7l\ he the class of all finite RA 2-loops of type I. Then F is equivalent to K on TZ\ if and only if (a) 0 ( F ) = 0[K) and (b) ind2(F) - ind2(A0. Suppose that F and K are equivalent on TZ\. To prove that conditions (a) and (b) are necessary, it is sufficient, by Theorem X.4.8, to show that F and K are equivalent on the class of all finite abelian 2-groups. Assume that A and B are finite abelian 2-groups such that FA =. FB. Let / be a finite indecomposable RA loop with Z{I) = C4 and / / / ' = C2 X C2 X C2 X C2. Let L = / X ^ and M = / X B. Since FL^ FA®FI ^ FB ® FI = FM and F and K are equivalent on 7^l, KL = KM, Hence, Ii[L/L'] ^ K[MIM'], by Theorem 1.2. Since LjV ^ C2 x C2 X C2 x C2 x A and MIM^ ^ C2 x C2 x C2 x C2 x 5 , it foUows that KA ^ KB, Thus K is indeed equivalent to F on the class of all finite abelian 2-groups. Conversely, suppose that conditions (a) and (b) hold. Let L and M be RA loops in T^i. We have Z{L) = 02^1 x C2"2 X ••• X €2^1, where ni > 1 and n^ > 0 for i = 2 , 3 , . . . ,^, and Z{M) = C2"^i X C2"^2 X ••• x C2mr, where mi > 1 and m^ > 0 for i = 2, 3 , . . . , r. As usual, we assume that the first factors of each decomposition contain L' and M', respectively. If FL ^ FM, then F[L/L^] ^ F[M/M'] and ^F{L,V) ^ AF(M,MO, by Theorem 1.2. By Theorem X.4.8, conditions (a) and (b) imply that K[L/L^] = K[M/M^]. Again, because of Theorem 1.2, we need only prove that AK{L,V) ^ AA'(M,M'). PROOF.

3. THE EQUIVALENCE PROBLEM

287

By Proposition 1.3,

[A (f 2^^! j : A J OTni—1

Z{/^K{M,

M'))

^ _ - — - _ _ - A ^ ( ^ 2 - i )[C2-2 X . . . X C2mrl [A(f2'^i j . ' A J oni-l

and o m i —1

Z{AF{M,

M')) ^ J—

^ ^ ^ ( 6 - 1 )[C2-2 X • • • X C2m.].

Writing 2 ' ^ i - V [ F ( 6 " i ) : -F] = 2" and 2 ™ ' - V [ ^ ( 6 ' " i ) : i^] = 2 ^ it foUows that Z{AFiL,

L')) ^ 2 " F ( 6 " i )[C2n2 X • • • X Ci",] S F ( 6 " : )[C'2X • • • X C 2 X C 2 " 2 X • • • X Ci",] a factors

and also t h a t Z(AF(M, M'))

= 2 ' ' F ( 6 " ' . )[C2m2 X • • • X C^mr] = F ( 6 ' " i )[C2 X ••• X C2XC2"'2 X • • • X C2mr]. 6 factors

Since >Z(A/r(L, L')) = Z ( A i ? ( M , M ' ) ) , comparing these equations gives Certainly conditions (a) and (b) imply t h a t 0 ( F ( ^ 2 " i ) ) = 0(A^(^2^i)) and t h a t ind2(F(^2"i)) = ind2( A'(^2"i ))• Therefore, A'(^2"i) = A^(^2"»i) stnd [ F ( 6 " i ) : F] = [ A X 6 " i ) : ^ 1 - Applying Theorem X.4.8 to

Z(AF{L,L'))

we also obtain

Z{/S.F{M,M'))

K{^2^i )\C2 X . . . X C2 xC2"2 x . . . x C 2 n t ] V

a factors

= A(6"^i )[C2 X . . . X C2^XC2^2 X . . . X C2mr]. 6 factors

^

288

XI. ISOMORPHISMS OF LOOP ALGEBRAS

Thus ZiAKiL,

L')) ^ Z{AKiM,M'))

^ ®Ui

I^^

where A',- = A'(^2<'i) f°r some d, > 0. Write A K ( I , L') ^ e i ^ i Ci{Ki) and A K ( M , M ' ) ^ @U^ Vi{Ki), each Ci{Ki)

and Vi{Ki)

is a Cayley-Dickson algebra.

m i > 1, Lemma VII.2.2 implies t h a t aD Ci{Ki) and Vi{Ki) hence isomorphic. Hence

AK{L,V)

Then KL ^ KM PROOF.

are split and

= A x ( M , M').

3.8 C o r o l l a r y . Let K be a field of characteristic ind2(/l') = 1 and 0{K)

where

Since ni > 1 and D

different from 2 such that

= 0. Let L and M be finite RA 2-loops of type L

if and only if L/V

^ M/M'

and Z(QL)

^

Z(QM).

The assumptions and Theorem 3.7 imply t h a t K is equivalent

to Q on 72-1; thus the result follows from Theorem 2.2.

D

For finite RA 2-loops in general, we have the following theorem. 3.9 T h e o r e m . Let F and K be fields of characteristics Let TZ be the class of all finite RA 2-loops.

different from 2.

Then F is equivalent

to K on

TZ if and only if (a) 0{F)

=

0{K),

(b) ind2(F) = ind2(A') and (c) c{F) = c ( A ) . PROOF.

Suppose F and K are equivalent on TZ. From Theorem 3.5, it

follows t h a t conditions (a), (b) and (c) hold. Conversely, suppose t h a t conditions (a), (b) and (c) hold. Let L and M be loops in TZ and write Z{L)

= C2"i X C2"2 X • • • X €2^1, where rij > 1 and

Ui > 0 for i = 2 , 3 , . . . , / , and Z{M)

= C2^i X C2^2 x • • • x C2mr, where

mi > 1 and mi > 0 for i = 2, 3 , . . . , r. Assume t h a t L^ and M ' are contained in the respective first factors of these decompositions. If FL = FM^ then, by Theorem 1.2, F[L/U]

^ F[M/M']

and

AF{L,V)

^

Theorem X.4.8, conditions (a) and (b) imply t h a t K[L/L'] Therefore, again by Theorem 1.2, it is sufficient to prove t h a t A x ( M , M').

By

AF{M,M').

^

K[M/M'].

A A ' ( / / , L^)

=

Because of condition (c) and Proposition 3.4, this proof is

essentially the same as t h a t of Theorem 3.7.

D

Chapter XII

Loops of Units

In this chapter, we study the unit loop U{1L)

= {a e ZL \ a/3 = I = /3a for some f3 G IL}

of the integral loop ring of a finite RA loop L,

Recall from §11.5.3 t h a t

U{7.L) is a Moufang loop. In the first section, we deal with some technical results which enable us, in some instances, to reduce the study of unit loop of ZL to the unit loop of Z[T(L)], where T[L)

is the torsion subloop of L. In the second section, we

give necessary and sufficient conditions for the unit loop to satisfy a group identity. The third section deals with central units. A description of the centre of the unit loop is given and finitely many generators for a subloop of finite index in this loop are given. In Section 4, we exhibit subloops of finite index in the unit loop and, in Section 5, we apply our results in order to describe the full unit loop ZY(ZL) for two specific RA loops. 1. R e d u c t i o n t o t o r s i o n l o o p s Let L be an RA loop with torsion subloop T.

Recall t h a t T is indeed a

subloop which is locally finite, normal and finite if L is finitely generated (Lemma VIII.4.1). The lemma which follows enables us to show t h a t if QT is a direct sum of division rings, then, modulo the trivial units, the unit loop of ZL is determined by the units of 2.T. The main part of the proof is as in the associative situation [Seh78, Lemma VI.3.22]. 1.1 L e m m a . Let L he a finitely generated subloop T and let K be a field. Suppose 289

RA loop with normal

that KT

torsion

= L^i ® • • • © D^ is the

290

XII. LOOPS OF UNITS

direct sum of division rings and that every idempotent of KT is central in KL. Then every unit fi G KL can be written in the form fJ' = ^ diii^ with di e Di, ii e L. Write Di = (KT)ei^ where Ci is a primitive central idempotent of KT and Yl^i — 1- Since each Ci is central in KL and since T is normal in Z/, the division ring Di is normal in KL in the sense that PROOF.

Dia = aDi, (Aa)/? = A(«/?), {aD^)(3 = a{DiP) and a{f3Di) = {af3)Di for any a^P £ KL, To see why, these equations in the case that a tain (A^i)^2 = Di{iii2) for ^1,^2 d = {Y^dtt)ei = '^dt{tei)^ dt G K^t

note that it is sufficient to establish and j3 are elements of L. So, to obG L^ for example, let d G Di^ write G T and use

[(t€^)ii]i2 = [t{e^i^)]£2 = [tiiiei)]i2 = [iti,)e,]i2 = itii){ed2) = [^'(^i^2)]ei,

= {til){i2e^) = [{thy2]e^ for some t' G T, by normality of T,

= t'[{e,e2)ei] = t'[e,{i,e2)] = (<'e.)(^i^2) to conclude that {dt\)i2 — d\i\i2) for some d' G Di. Since T is normal in L, L is the disjoint union of cosets of T (see §11.1.3); that is, L — UQGQ ^^ ^^^ some transversal Q. If g G L\T^ then TqOTq'^ = 0 since g''^ has infinite order, thus we may assume that q~^ G Q whenever qeQ. Let /i be a unit in KL. Then /i can be written /^ = X^/Xj^j with fij G A T and the qj distinct elements of Q, and, for any i,

(1)

e,/i = J^€i(/i^^^) = J^rf^g^,

dj = e^^j G -Dt, since 6^ is central. Since L/T is a finitely generated torsionfree abelian group, it can be ordered (see Lemma VIIL1.3) by an order relation <, say. Thus, in (1), we may assume that the qj are such that Tqi < Tq2 < • • • < Tq^ Similarly, we can write V

1. REDUCTION TO TORSION LOOPS

with d'j e Di and Tq[
291

Tq'^,

By normality of A , if ct,6 G A , there exist ai,a2,61,62,63 G A

such

that aqj • bq'i^ = ai{qj • 6^^) = ai(g^6i • q[) = ai{b2qj

' q'k) = ^1(63 • qjq'k) = ^263 • qjQk-

Thus ei/j. ' eifi~^ can be written Y^djk • ^ j ^ ^ for djk G £^i. On the other hand, Artin's Theorem (Corollary 1.1.10) gives eni • e^/i"^ = e^; thus, e, r= ^djk

-qjq'k-

The left hand side of this expression has support in T , while the terms of the right hand side have supports in Tqiq'i, Tqiq^,...

.Tq^ql.

The first coset here is the unique smallest, the last the unique largest and, if either u > 1 or t; > 1, these are diflferent. Since Di is a division ring, it follows also t h a t d n 7^ 0 and d^v 7^ 0. Hence u = v = I so t h a t e^/x = diqi, di G Di, and n

n

/i = ( ^ e,)/i = ^ 1=1

d.^i, rf, G A , g't G L,

1=1

as required.

D

1.2 C o r o l l a r y . Let L be a finitely generated torsion

subloop.

Suppose

normal

that QT = Di ® • • • © Dn is a direct sum of

division rings and that every idempotent U{1L) PROOF.

RA loop and T its

of QT is central in QL,

= [U{ZT)]L =

Then

L[U{ZT)].

We use the same notation as in the statement and proof of

Lemma 1.1. So let // be a unit in ZL. Let < be an order relation on the finitely generated torsion-free abelian group L/T and let Q be a transversal for T in L with the property t h a t q~^ G Q whenever q £ Q. Because of Lemma 1.1, we can write // = ^ i=l

diqi,

di G A ' , Qi € Q

292

XII. LOOPS O F UNITS

and, similarly, n

(2)

/x-^=5]gX,

d\eD,,q[eQ.

1

It is important to realize t h a t there is no reason here to expect the qi or q[ to be distinct. We can and do assume, however, t h a t Tqi
Tqn

and

Tq[
Tq'^.

Since for q^q' ^ L^ a ^ Di^ b G Dj and i / j , we have qa ' bq = q\ai

• bq) = q{a2b • q),

for some a i , a 2 G Di^ it follows t h a t q'a - bq — 0. Therefore, n 1

and each q^d^ • rfj^^ /

0. The left hand side has support in T while the

terms of the right hand side have supports in Tq[q\^... ordered Tq[q\

< • • • < Tq'^q^.

^Tq'^qn which are

If the first and the last of these cosets are

different, we have a contradiction. Thus each Tq[qi — T, so each q[qi G T and, by choice of the transversal Q, q[ — q~^ for each i. In the expression fi — Yl^i^ii

collect equal ^^ and write qeQ

where each /i^ is a sum of certain d^. Since /i has coefficients in Z, each ^q G ZT. Note t h a t ^q^q' = 0 \f q ^ q' since the division rings represented by the di in /i^ are all different from those represented in /i^/. Since q'^ = ^ ~ \ the q' in expression (2) for /i~^ will collect exactly as the qi in the expression for //, so t h a t we have

qeQ

with z/gZ^o/ = 0 if g' / ^'. Also, I = fifX ^ =

^ q.reQ

fiqq • r

^Ur.

1. REDUCTION TO TORSION LOOPS

293

The left side has support in T while figq • r~^i/r does not, unless r — q. So (3)

1=

^^iqq-q-^Uq,

Choose some fig^ ^ 0. We claim t h a t {li^q ' q-^i/q)fig^qo = 0

if q yi qo.

Indeed, observe first t h a t the element in question can be expressed as a linear combination of terms of the form {aq • q~^b)cqo where b and c come from different components Di. Now (4)

aq • q~^b = (aq • q~^)bi — ab\^

where 6i and b come from the same Di] hence [aq • q~^b)cqo — ab\ • cq^ — [abi ' ci)qo — (a^ •biCi)qo^ with a and ai in the same division algebra, and c and ci in the same division algebra. Since b\Ci = 0, (fXqq • q~^i/q)iigQqo = 0 and the claim follows. From (3), we now obtain /i^o^o = if^qoQo'Qo^ i^qo)l^qoQo sind, since x = xyx implies that xy is an idempotent, we see that i^igQqo • %^i^qQ is a (nonzero) idempotent. A calculation similar to (4) (using normality of T ) shows t h a t f^qoQo'Qo^^qo ^ ZT. Since idempotents in ZL are trivial (Corollary VIII. 1.7), f^qoQo'Qo ^qo = 1- So q^ Ug^ and, therefore, Ug^ are invertible. Similarly, j^ig^ is invertible. Since ^gfig^ — Q iox q :^ q^/xt follows that /x^ = 0 for q ^ q^ and fi = fig^qo G [U{ZT)]L,

So we have shown that U{ZL)

C [ZY(ZT)]L.

The other inclusion is obvious. Finally, if i e L smd ji e U{ZT), U{ZT),

so [U{ZT)]L

= L[U{ZT)]

then both i'^lit

and £/i£-^ are in

follows from //£ = £ ( £ - V ^ ) and i^i =

1.3 P r o p o s i t i o n . Let L be an RA loop with torsion either an abelian group or a hamiltonian

subloop T.

If T is

2-loop and if every subloop ofT

is

normal in L, then U{ZL) = [U{ZT)\L PROOF.

Clearly [ZY(ZT)]L C U{ZL),

=

L[U{ZT)].

To prove the other inclusion, we

have to show t h a t , if /x G ZY(ZL), then /i G [ZY(ZT)]L.

Replacing L by

the loop generated by the support of /i and three elements which do not associate, if necessary, we may assume t h a t L is finitely generated. Hence,

294

XII. LOOPS OF UNITS

by Lemma VIII.4.1, T is finite. We claim that QT is a direct sum of division rings. If T is an abelian group, this is obvious. If T is a hamiltonian 2-loop, then T = Qs X E OT T = MieiQs) x E^ where £ is a (possibly trivial) elementary abelian 2-group. So, by Corollaries VII.2.3 and VII.2.4, either QT = {QE)Qs = {®Q)Qs = ®QQs = e(4Q © H(Q)) or QT - {QE)[Mre{Qs)] = (®Q)[M^e{Q8)] = ©Q[Mi6(g8)] = ©(8Q ® (Q, - 1 , - 1 , - 1 ) ) . By Theorem 1.3.4, both H(Q) = ( Q , - l , - l ) and ( Q , - 1 , - 1 , - 1 ) are division algebras. So QT is indeed the direct sum of division rings; in particular, every idempotent of QT is central in QT. Hence, by Theorem VII. 1.4 and Proposition VII.2.1, every idempotent of QT is a linear combination of idempotents of the type jff, where ^ is a subloop of T. Since every subloop of T is normal in L, it follows from Lemma VI. 1.2 that every idempotent of QT is central in QL, so the result follows from Corollary 1.2. D

2. Group identities Let X — {a:i,X2,...} be a countable set and let F be the free group on X. If G is a group and w = K;(XI,. .. ^Xn) is a nonempty reduced word of F in the variables x i , . . . ,Xn, we say that K; = 1 is a group identity for G if w{gi^... ^gn) = 1 for all ^ i , . . . ,^7^ G G, Without restriction, we may assume that 11; is a word in at least two variables, for if x^ = 1 is a group identity for the group G, then so is (0:1X2)^ = 1. Furthermore, since a free group of finite rank can be considered as a subgroup of a free group of rank 2 (see, for example, [Rob82, Theorem 6.1.1]), we may assume that it; is a word in precisely two variables. Since Moufang loops are diassociative, this notion can be extended. 2.1 Definition. A Moufang loop L satisfies a group identity if there exists a nonempty reduced word w = w{xi^X2) in the free group F on {xi^X2} such that w(£i,i2) = 1 for all ^1,^2 ^ L,

2. GROUP IDENTITIES

295

2.2 Examples. 1. An RA loop L satisfies the identity {xyx'^y'^y — 1 (see Theorem IV.1.8). 2. An n-Engel loop is, by definition, a loop satisfying the identity (... ((x, y), y ) , . . . , y) = 1 (t/ repeated n times). 3. A Moufang loop L is said to be an FC loop if, for every a £ L^ the set [a] = {x~^ax \ x £ L} is finite. Suppose C = {^1,... , gk) is a finitely generated group such that U{ZG) is an FC group. For each i, 1 < i < k, let Ui = {u e U{ZG) \ (u^gi) = 1}, the centralizer of gi in ZY(ZG). Since U{7.G) is FC, each Ui has finite index in U{ZG). Hence U{ZG) is a finite extension of its centre Z{U{ZG)) = r\i
L2^'"2

in

Ln = {l}

of subloops Li, all normal in L, such that Li-i/Li is contained in Z{L/Li) for z = 1 , . . . ,n. Just as in group theory, a Moufang loop has an upper central series {1} = Zo{L) C Zi{L) C Z2{L) C ... Zi^\[L)lZi[L) — Z{L/Zi{L))^ which terminates at L in a finite number of steps if and only if L is nilpotent. In this case, the smallest n with Zn{L) = L is called the nilpotency class. A Moufang loop L also has a lower central series L = ML)

2 7i{L) D ML)

2 •••

where 71 (L) = L' and, for i > 1, 7.+i(L) = {{L, j,iL)),

(L,L, f,iL)),{L,7.(1),

I),{7^iL), L, L))

is the subloop generated by all commutators and associators of the indicated types. The loop L is nilpotent if and only if the lower central series terminates in a finite number of steps. When this occurs, the nilpotency

296

XII. LOOPS OF UNITS

class is the smaDest n for which jn{L)

= {1} (and this definition agrees

with the previous). 2.3 L e m m a ( M a l ' c e v [Mal53]). A nilpotent tency class n satisfies defined inductively

the group identity

loop L of nilpo-

x^ = yn, where Xn and yn are

by

xo = x,yo = y and, for m> PROOF.

Moufang

0, XTn-\-i = Xmym, 2/m+i ~ yrnXm-

We prove the lemma by induction on n. If n < 1, then L is

commutative, so L satisfies xi = yi.

Now assume t h a t L has nilpotency

class n > 1 and t h a t the result holds for nilpotent loops of nilpotency class less than n. Since the loop L/Z(L) hypothesis yields t h a t Xn-\Z[L)

is of nilpotency class n— 1, the induction — yn-iZ{L).

It follows easily t h a t

x^-i

and y-n-'i commute, so Xn = yn-

D

In [ M a l 5 3 ] , Mal'cev shows t h a t nilpotent groups can actually be characterized as those groups satisfying a certain group identity. To be more precise, let XQ, t/o, ^ i ^ ^2? • • • he a set of independent generators for a free group. Define words Xn and Y^ inductively as follows: XQ

=

XQ,

YQ

= yo

and, for n > 0,

It turns out that a group G is nilpotent of class n if and only if n is the least positive integer such t h a t Xn = Vn is a group identity for G. For a positive integer m, recall that a subset A^ of a ring A is said to be nil of bounded exponent

less than or equal to m if a ^ = 0 for all a G A^.

The following result is well known. Our proof, which appears in [ R o w 8 8 , Proposition 2.6.26], is due to A. Klein. 2.4 L e m m a . Let R be an associative

ring with I, let m be a positive

and let I be a nonzero nil right ideal of bounded exponent to m. Ifxel PROOF.

with x ^ - ^ 7^ 0, then {Rx'^'^Rf

integer

less than or equal

= {0}.

Note t h a t m > 2 since / is nonzero. To prove the result, it is

suflScient to show t h a t x'^~^rx'^~^

— 0 for any r G i2. So assume r £ R and

2. GROUP IDENTITIES

let r' = x'^~^r.

297

Then xr^ = 0 and ( r ' ) ^ = 0, because r' G / . Hence 771 — 1

For any i, 1 < i < m — 2, we have (^r j

X = (x

rj

X = X

rx

ri,

for some r^ G -Ra;. Thus

It follows t h a t 0 = ( r ' x ^ - i ) ( l + r ' ' ) , where r'' = J2T=~i^ ^ G /2x. Since Rx is nil, the element r ' ' is nilpotent; therefore, 1 + r'' is invertible. So we obtain

0 = r'x^-i = x^-Vx^-^ 2.5 P r o p o s i t i o n . [ G J V 9 4 ] an infinite identity.

commutative

n Let A be an associative

algebra with 1 over

domain R and suppose that LI (A) satisfies

a group

Then there exists a positive integer m such that, for all a^b^c ^ A

with a^ = be — 0, the right ideal bacA is nil of bounded exponent

less than

or equal to m. PROOF.

By assumption, 11(A) satisfies an identity in the variables x\

and X2 of the form

where A; > 1 and r^, 5^ are integers, all of which are nonzero except possibly r^^i.

If we make the substitution xi = xy^ X2 = yx (where x and y are two

variables), then we get a group identity for U{A) of the form (6)

{xyY^iyxY' '' •{xyY^iyxY^ixyY^^^ = 1. We first establish the proposition in the special case where b = c. So, let

a, 6 G ^ be such t h a t a^ = b^ = 0. We claim t h a t abaA is a nil right ideal of bounded exponent less than or equal t o m, where m is a number given by a formula in the Vi and Si. We consider separately the cases c h a r i ? = 2 and char/Z 7^ 2. Suppose t h a t char i2 ^ 2. We show t h a t k

m = 2 + 2|r^+i| + 2 ^ ( | r , | + |5,|).

XII. LOOPS O F UNITS

298

For u G A, we have (aua)^ = (bauab)^ = 0, hence 1 + aua and 1 + bauab are in U{A). We evaluate the group identity in U{A) by replacing x and y with 1 + aua and 1 + bauab respectively. We get ((1 + aua){\ + bauab)y^{{\ + bauab){l + aua)y'' • • • • • •((! + bauab){\ + aua)y^{{l

+ aua){\ + bauab)y^'^'^ = 1,

for all u e A and, after computing all products, an identity of the form f{aua^ bauab) = 0, where /(xi,X2) is a nonzero polynomial in the noncommuting variables xi,a:2 with integral coefficients and zero constant term. Replacing u by A^^, with A G iZ, and expanding, this identity can be written in the form Api + X^P2 + '" +

= 0,

AV

where p i , . . . ,p^ are polynomial expressions in aua and bauab with integral coefficients. Since R is infinite, there exist i distinct elements A j , . . . , A^ in R. Then Pi

P2

0

Pi

0

0

where j Ai

A2

A?

-^2

LM

A2

A2

• .

• A, 1 • A?

• ^i.

Since /Z is a domain, det A = ( Yli=\ ^i] Yliyji^i ~ Aj) 7^ 0. It follows that p^ z=: • •' = p^ = 0. Since each variable in (6) (when the left side is written as a reduced word) may appear with exponent either 1, —1, 2 or —2, pi = 0 implies 2'^{abauy = 0 for some integers r and 5, with 0 < r and 1 < 5 < m. To clarify this statement, suppose, for instance, that the r^ and the Si are all positive integers. Let aua = c and bauab = d. Since c ^ A with c^ = 0, we have (1 + c)^ = 1 + kc for any /: G Z, so we can write ((1 + Ac)(l + A d ) r ( ( l + Ad)(l + A c ) r • • • = 1

2. GROUP IDENTITIES

299

as ((1 + Ac)(l + Xd)y'-\l

+ Ac)(l + 2Xd) . ((1 + Ac)(l + Xd)y'-\l

+ 2 A c ) . . . = 1.

By a direct calculation, we get (A^cdp-^2A^cd(A^cd)"i-^2Ac--- + ( t e r m s of lower degree in A) = 0. Thus 2''X\cdyc^

+ (terms of lower degree in A) = 0,

where r > 0 , ^ > 1 , l < n <

|rfc+i| + E f = i ( k t l + \^i\) ^nd t G { 0 , 1 } . It

follows t h a t Q = p^ = 2^(cc/)^c^ Thus 2'^{auabauahY^^ t h a t 2''{abauy

= 0 and this implies

= 0, with 5 = 2n + 3 < m .

Since char i2 ^ 2, 2'^{abauy

— 0 implies {abauy

= 0 for all u ^ A. T h u s

abaA is a nil right ideal of bounded exponent less than or equal to m. Suppose now t h a t char/2 = 2. In the group identity (5), replace x^ by a:iX2 and X2 by 0:1X3. In this way, we obtain a group identity for U{A) of the form (7)

ix^X2y'{x^x^y^

• • • ( x i X 2 P ( a ^ i ^ 3 r ' ' ( ^ i ^ 2 r + ' = 1,

where each variable in the left hand side (written as a reduced word) may appear with exponent 1 or —1. Let u £ A.

Since chari? = 2, ((a + ba)u{a + o.b)y

1 + (a + ba)u{a + ab) is in U{A).

= 0, and thus

Evaluate the group identity (7) in U{A) by

replacing Xi with 1 + atxa, X2 with 1+6at/a6 and 0:3 with

\-\-[a-\-ba)u{a-\-ab).

We get ((1 + aua){\ + bauab)y^[{\

+ aua){\ + (a + ba)u{a + ab))y'

• • • == 1.

Now take X £ R and replace u by Xu. As above, we get an equality of the form Xpi + X^p2 + • • • + X^p£ = 0 and another Vandermonde determinant argument gives p£ = 0. By a direct calculation, it is easy to verify t h a t P£ = 0 implies {abauy

= 0 for some integer s with I < s < m and m again

given by a formula in the exponents r^ and Si. Thus abaA is a nil right ideal of bounded exponent less than or equal to m. This concludes the proof of the special case.

300

XII. LOOPS O F UNITS

T h e proposition in general now follows quickly. Let 6, c G A he such t h a t be — 0. Then (cn6)^ = 0 for any u E A and so, for a £ A with a^ = 0, cubacubA is a nil right ideal of bounded exponent less t h a n or equal to m. Hence bacA is a nil right ideal of bounded exponent less t h a n or equal to 3m+1.

D

Let A be an associative ring and S a multiplicatively closed subset of the centre of A such t h a t no element of 5 is a zero divisor in A (in particular 0 ^ 5).

The ring of fractions

localization

of A with respect to 5 , also called the

of A with respect to S (see, for example, [ C o h 7 7 ] ) , is the ring

s-^A = {s-\ \se s,ae A}, with addition and multiplication defined by s^^ai

+ S2^a2 - {s\S2)~^{s2ai

{s~^ ai){s~^ a2) =

+ Sxa2),

{siS2)~^{aia2)

for a i , a2 G /I and 5i, ^2 G 5 . If 5 = / ? \ { 0 } for some subring R of the centre of 4 , then we write R~^A rather than the more cumbersome {R\ 2.6 L e m m a . Let A be an associative

algebra with unity and let R be a

subring of the centre of A whose nonzero elements A.

{0})~M.

are not zero divisors

Suppose that bac = 0 for any a^b^c ^ A with a^ — be — 0. Then

idempotent PROOF.

Let e be an idempotent in / 2 ~ M . Then, for some 0 ^ r £ R^ Hence f{r - f) = 0 and {fu{r

any u ^ A, From the hypothesis, we obtain f{fu{r

— f)){r

- f)^

any u ^ A. Similarly, from (r — / ) ( ( r — f)uf)f

= 0 for

— f) — 0. Since

(r — / ) ( r — / ) = r{r — / ) , we have r'^fu{r — / ) = 0, and so fur

— fuf

= 0, we obtain ruf

and consequently eu = ue for any u £ A. 2.7 C o r o l l a r y . [ G J V 9 4 ]

subring

PROOF.

of R~^A

associative

fuf

algebra with

of the centre of A whose

are not zero divisors in A, IfU(A)

every idempotent

=

for D

Let A be a semiprime

and let R be an infinite

elements

every

of R~^ A is central.

f = re e A dind f^ = rf.

unity

in

nonzero

satisfies a group identity,

then

is central.

Assume t h a t there exist a^b^c ^ A such t h a t a^ = be = 0. By

Proposition 2.5, bacA is a nil right ideal of bounded exponent.

Because

2. GROUP IDENTITIES

301

A is semiprime, Lemma 2.4 implies bac = 0. Now Lemma 2.6 gives the result.

n

2.8 C o r o l l a r y . Let L be an RA loop with torsion 2-loop, then the following 1. U(TL)

conditions

satisfies a group

2. for any finitely group

are

identity;

generated group G contained

in L, K{ZG)

satisfies

a

identity;

subloop ofT

Moufang

loop and

every

= [U{ZT)]L

sat-

is normal in L.

if any of these conditions

isfies the identity PROOF.

If T is a

equivalent:

3. T is an abelian group or a hamiltonian

Furthermore^

subloop T.

(x^^y^)

holds, thenU{ZL)

= 1.

Since Z does not have zero divisors, the rings ZG and ZL are

semiprime for any subgroup G of L, by Corollary VI.3.6 and Theorem VI.2.7. Clearly, if ZY(ZL) satisfies a group identity, then so does U(ZG) for any finitely generated subgroup G in L; hence (1) implies (2). Now assume (2). By Corollary 2.7, for any finitely generated group G contained in L, any idempotent in QG is central in QG.

Hence, if £ G L

and g ^ T has order n, then the idempotent n~^{\ + g + g^ + - - • + g'^~^) is central in QG, where G is the group generated by g and i. It follows t h a t i{^g) = {g)i for any £ G ^ so, by Corollary IV. 1.11, the cyclic group (g) is normal in L. Since g is arbitrary in T , every subloop of T is normal in L, so T is an abelian group or a hamiltonian Moufang loop. Thus (3) holds. Finally assume (3). U{ZL) = [U{ZT)]L,

By Proposition 1.3 and because T is a 2-loop,

In c a s e T is a hamiltonian Moufang 2-loop, by Higman's

Theorem (Theorem VIII.3.2), U{ZT) Z{L)

= ± T , so U{ZL)

= ±L,

Since P e

for any £ G L, it follows in this case t h a t U{ZL) satisfies the identity

(x^,X2) = 1. So, for the rest of the proof, we assume T is an abelian group and show t h a t ji^ G U{ZT)Z{L) with V G U{ZT)

for any

JJL G U(ZL),

For this, write ji = ut

and ^ G //. By diassociativity, the loop generated by u and

^ is a group, thus

Because supp(i/) C T and because T is normal in L, iyt~^

G U{ZT).

It

follows t h a t /x-^ G [ZY(ZT)]Z(iy), as asserted. Since \U(ZT)]Z{L)

is an abelian

302

XII. LOOPS OF UNITS

group, we again obtain t h a t U{ZL) satisfies the identity (3^1,2:2) = 1, hence proving (1). T h e result follows.

D

Although we now show t h a t Corollary 2.8 can be proved more directly, even without the restriction on the torsion subloop, we chose to offer the above proof primarily because of Proposition 2.5 and its possible applications. One such application is given in [ G J V 9 4 ] and [ G S V 9 1 ] , where it is shown t h a t , if the unit group of a group algebra KG of a torsion group G over an infinite field K satisfies a group identity, then KG satisfies a polynomial identity. Furthermore, in [ P a s 9 5 ] , necessary and sufficient conditions are given for U{KG)

to satisfy a group identity.

2.9 C o r o l l a r y . Let L be an RA loop with torsion following

conditions

are

1. ZY(ZL) satisfies a group

in L, U{7.G) satisfies

a

identity;

subloop ofT Furthermore,

Moufang

2-loop and

every

is normal in L.

if any of these conditions

satisfies

PROOF.

the

identity;

3. T is an abelian group or a hamiltonian

U{ZL)

Then

equivalent:

2. for any finitely generated group G contained group

subloop T.

the identity

Assume U{ZL)

hold, then LI{ZL) — \JJ{2.T)]L and

[x'^^y'^) — 1. satisfies a group identity.

As in the proof of

Corollary 2.8, it follows t h a t (1) implies (2) and t h a t (2) implies t h a t T is either an abelian group or a hamiltonian Moufang loop with every subloop normal in L.

Since free groups do not satisfy group identities, it follows

from Corollary VIII.5.8 and Theorem VIII.3.2 t h a t T is a 2-loop in the hamiltonian Moufang case. The remainder of the statement follows from Corollary 2.8.

•

We conclude this section by describing the RA loops L for which U{7.L) has various properties, nilpotence for example.

For this, we need three

technical lemmas. Recall t h a t a group G is said to be solvable if G has a series C ^ G o 3 G i D • • O G , - {1} in which Gij^\ is a normal subgroup of Gi and Gi/Gi^\ 0,...,r-l.

is abelian for i =

2. GROUP IDENTITIES

303

2.10 Lemma. Let G be a group such that one of the following is satisfied:

conditions

1. for every finitely generated solvable subgroup H ofU{ZG), is nilpotent or H' is torsion; 2. the torsion elements ofU{ZG) form a subgroup.

either H

Suppose the set T of torsion elements of G is an abelian group and every subgroup of T is normal in G. If G' — { l , ^ } , then, for any g ^ G and t e T, gtg-^ = t ort~^. Let g^G^l^t^T and suppose t has order n. Since the cyclic group (t) is normal in G, we have g~^tg — f for some i, 1 < z < n and gcd(i,n) = 1. To prove the result, we assume that i / 1 (thus g and t do not commute) and show that i = n — I. Since g'Hgt'^ = f"\ it follows that f"^ = s. Hence g'^t^gt''^ = f2i-2 _ 2 So n = 2i — 2. These observations imply, in particular, that n is even and i is odd. Consider the Bass cyclic unit PROOF.

6 = (1 + ^ + ... + f-^ )^(^) +

f.

n Recalling the form of the inverse of such a unit (see §VIII.2) and noting that n I [1 - i(l + f)], we have b-^ ^(l+f

+ '" + f2)^(^)

+

1 n

= (1 + f + •. • + /^(^-i))^W +

1, n Hence b~^ = g~^bg^ since t is central. Thus g~^bgb~^ = b~^. If the torsion elements oiU{TG) form a subgroup, then b~^ is of finite order (as a product of two torsion units). Otherwise, let H be the group generated by g and b. Then the preceding yields b~^ ^ H' — 71 (77). Since g-'b^'^gb-^'^

= 6-2^6-2^ = ft-^"^^',

it follows by an induction argument that b'^"^ G 7m(^)? the mth group in the lower central series of H, for any m > 1. Since H is solvable, either H is nilpotent or H' is torsion and we see again that 6 has finite order. Since

304

XII. LOOPS OF UNITS

T is an abelian group, Theorem VIII. 1.4 yields t h a t 6 is a trivial unit. By Lemma VIII.2.7, i = n — 1 as required.

D

2 . 1 1 L e m m a . Let L be an RA loop with torsion subloop T. If the

torsion

units in the integral loop ring ILL form a suhloop, then every subloop of T is normal in L and T is either an abelian group or a hamiltonian

Moufang

2-loop. P R O O F . Let t e T, i e L and consider a = ?i{l - t). Clearly a'^ = 0, so 1 + Q: is a unit in ZL. Since the loop generated by i and ^ is a group, the element /3 = {I + a)~^^(l + a ) is a torsion unit in the integral group ring Z{t,i).

Note t h a t

/? = (1 - a ) ^ ( l + a) = t - at + a = t + a{l - t) = t + ?£{! - 2t + t^). Since L/V

is abelian, Proposition VIII. 1.4 implies t h a t the torsion units

of Z[L/L'] are triviaL Hence the natural image of/3 in Z[L/L'] is a trivial torsion unit.

It is then not hard to see t h a t supp(/3) contains a torsion

element, say g ^T.

Since, by assumption, the torsion units ofU{ZL)

form

a subloop, the element l3g~^ is also a torsion unit and with 1 in its support. So, by Proposition VIII. 1.5, l3g~^ and hence also /3 are trivial units. Since it has coefficient 2 in 2tit^ it follows t h a t It = t or it = tH for some i > 0, or it = fit'^ for some i > 0. Hence, either ^ = 1, iti'^

= f or i-'^Vi =

t'^;

thus C~^ti G (t). In particular, every subloop of T is normal in L, so T is either an abelian group or a hamiltonian Moufang loop. Let II = ^teT

l^tt be a torsion unit and suppose /x^ ^ 0 for some t G

T . Since the torsion units form a subloop, fit~^ is a torsion unit as well. Since T is locally finite (Lemma VIII.4.1), the support of /j.t~^ generates a finite subloop and therefore, by Proposition VIII. 1.1, fit~^ = ± 1 . Thus Consequently, for each

finitely

generated subgroup G of T , the (finite) group ± G is normal in

the torsion units of U{Z{L))

are trivial.

U{ZG)

so, for any /i G ZY(ZG), fi'^g/j,''^ = g for all g ^ G^ where n is the order of the automorphism group of ±G, Consequently U{ZG)

Thus /x^ is in the centre of

satisfies the group identity {x'^^x'2) — 1, so

U{ZG). U(ZG)

does not contain free subgroups of rank 2. By Theorem VIIL5.7, if G is not abelian, it is a 2-group. Since G is an arbitrary finitely generated subgroup of T , we obtain t h a t T is either abelian or a hamiltonian Moufang 2-loop.

D

2. GROUP IDENTITIES

305

We require a basic fact about divisible abelian groups. Recall that an abelian group D is said to be divisible if for each d E D and any positive integer m, there exists d\ £ D with d = d'^. Obviously, the additive groups Q and Q/Z are divisible. 2.12 Lemma. Let A be an abelian group with subgroup B and suppose f:B —> D is a group homomorphism into a divisible abelian group D, Then f extends to a homomorphism (fi: A -^ D. P R O O F . Let S be the set of all ordered pairs (H, Z) by setting (p\a) = 1. On the other hand, if a has finite order n modulo H ^ then a^ e H ^ and we can choose d £ D with d'^ = (p{d^). One then easily verifies that the map (p': (H^a) —> D given by (p'(ha^) = (p(h)d\ for h ^ H^ also extends (p. In either case, this contradicts the maximality of {H^ (p). D

The next lemma is standard in the study of dimension subgroups. Our proof is along the lines of the one given in [Seh78, Section III.l]. 2.13 Lemma. Let G be a group. 1. For any xi^... such that

^Xn E G^ and >2:i,... ,Zn ^ Z, there exists (3 £ Az(G)^

2. G n ( l + Az(G')2) P R O O F . TO

=G'.

prove 1, note that if m € Z, m > 0, and x £ G, then

m{x - 1) = {x"" - 1) + (x - l ) ( - x " ' - ^ - x'"-2

x+

m-l)

and -m{x - 1) = (x"'" - 1) + (x - l ) ( - m + x-^ + • • • + x-'"+^ + x""^).

306

XII. LOOPS OF UNITS

So m{x - 1) € (x^ - 1) + A2(G)2 for any m € Z. Since (a - 1) + (6 - 1) = {ab -I)-{a-

1)(6 - 1)

for any a^b £ G^ the first statement now follows by an induction argument. To prove 2, note that one inclusion follows immediately from the identity x~^y~^xy — 1 = x~^y~^{xy — yx) = x-^y-\{x

- \){y -l)-{y-

\){x - 1)],

which holds for any x^y £ G, To prove the reverse inclusion, it is sufficient to show that \i g e G\G', then g -I ^ Az{G)'^. For x e G,\et x = xG', Since 'g is not the identity element in the group GjG'^ there exists a nonzero homomorphism f^) -^ E — (^jZ. Because ^ is a divisible group, this map extends to a homomorphism / : G/G' —^ /T, by Lemma 2.12. Since f{'g) ^ 0, the map / : ZT —» H defined by a = ^ a^x 1-^ ^ ckxf{x) xeG xeG is a homomorphism of additive groups. Since f({x-l){y-l))

=

f{xy-x-y-l)

= 7(^)-7(x)-7(y)-7(T)-0, it follows that /(Az(G')2) = {0}. Since f{g - I) = J{g) - J{\) = Jig) ^ 0, clearly g-\ ^ Az(G)^ D R. Baer has proven that a solvable n-Engel finitely generated group is nilpotent (see, for example, [Rob82, 12.3.7, page 360]). The following result is based upon and unifies the main theorems of [GM95]. 2.14 Corollary. Let L be an RA loop with torsion subloop T. following are equivalent: 1. 2. 3. 4. 5. 6. 7.

U{ZL) is an RA loop. U{ZL) is nilpotent. U(ZL) is nilpotent of class 2. \U{ZL)\' is a group of order 2. U{ZL) is n-Engel for some n > 2. U{ZL) is 2-Engel. U{ZL) is FC.

Then the

2. GROUP IDENTITIES

307

8. [ll{ZL)Y is a torsion loop. 9. The torsion units in ZL form a subloop. 10. T is an abelian group or a hamiltonian Moufang 2-loop and x~^tx = t"^^ for any t ^ T and any x £ L, Moreover^ ifT is an abelian group and i £ L is an element that does not centralize T, then £~^ti = t~^ for all t E T. PROOF.

Consider also the following condition.

10'. T is an abelian group or a hamiltonian Moufang 2-loop and every subloop of T is normal in L. Any RA loop L whose unit loop satisfies any of (1) through (7) has the property that, for any finitely generated group G contained in L, U{ZG) satisfies a group identity. Hence, because of Corollary 2.9, any of these conditions implies (10'). The following implications are also clear: • • • • •

(1) (1) (1) (1) (1)

=> (3) (2); (4); ==> =^ (7); = ^ (6) = ^ (5); = > (8) => (9). = > •

Because of Lemma 2.11, (9) and, hence, also (8) imply (10'). Next, we show that any of (2), (4), (5), (7), (9) (hence any of the conditions (1) through (9)) implies (10). Note that in each case, we also have (10'). If (2) or (5) holds, then for any finitely generated solvable group H contained in ZY(ZL), the group H is nilpotent. In the case of (4) or (7), we know (see, for example, [Pas77, Lemma 4.1.6]) that H^ is torsion for any finitely generated solvable group H contained in U{ZL). So, in all cases under consideration, if T is an abelian group, applying Lemma 2.10 to 2-generated subgroups, we have i~^tt = t^^ for any t e T and i ^ L. If T is a hamiltonian Moufang 2loop, this condition is obviously satisfied as well. So, to prove (10), we now assume that T is an abelian group and that ^ G T is not central. For some i e L,we have i'^ti = t'^ with t'^ ^ 1. We have to prove that i'^tii = t~^ for any ti G T. Suppose to the contrary that ^i € T, i'^tii = ti and tl ^ I. Since ^i and t commute, the three elements ^, /, ^i associate, by

308

XII. LOOPS OF UNITS

Corollary IV.1.3, so they generate a group. Thus

and so tit'^

= tit

or

tit'^

=

tl'^t'^,

contradicting t'^ ^ I ^ t\. It suffices finally to show that (10) implies (1). For this, we first claim that, if (10) holds, then U{ZL) = HL^ where 7i is a subgroup oi Z{U{ZL)). By Corollary 2.9, U{ZL) = [ZY(ZT)]Z/, so the claim is obvious if T is central. If T is a hamiltonian Moufang 2-loop, then, by Higman's Theorem (Theorem VIII.3.2), U{ZT) = ±T and again the claim follows. In the remainder of the proof of the claim, we assume that T is an abelian group and not central. For any a = YjteT^ti € ZT, let a' = J2teT^tt~^' Note that (a/?)' = a'/3'. The assumptions in (10) imply that, if / is not central, then t~^ = f^ = st, where s is the unique nonidentity commutator-associator of L. Therefore, if a' = a, then a is a linear combination of elements t G Z{L) and elements of the form t + t* = (1 + s)t^ which are also central, by Corollary III. 1.5. Hence Tl = {a E U{ZT) \ a = a'} is a central subloop of ZY(ZL). We prove that ZY(ZL) = HL. For this, let a = Xllli ^iU G U{ZT), with each z, e Z and ti G T. It is sufficient to show that a G HL in the case that a has augmentation 1. By Lemma 2.13, m

m

a= 1 + 5^^,(^,-1)= (n^'O +/^ = ^o + /3 for some /3 G A(T)2, where ^o = UZi^TThus a = (I + 6)to, with 6 = f3t-^ G A(T)2and l + SeU{ZT). Let j = (1 +Sy(I+ S)-^ = Eterltt, each jt € Z. Then 7^7 = 1, so "^t^x^t ~ 1- Since 7 also has augmentation 1, 7 = ^ for some g £ T. Consequently, again by Lemma 2.13, ^ = (1 + sy{l + S)-' G T n (1 + A{Tf)

= r ^ {1}.

Hence I + 6 = {I + Sy e H, Therefore a G HL, So U{ZT) C HL and U{ZL) = HL^ as required. This proves the claim. Since H. is central, HL has the same unique nonidentity commutatorassociator as L and, since L has the LC property, so does HL. Corollary IV.2.2 says that U{ZL) is an RA loop. D

3. THE CENTRE OF THE UNIT LOOP

309

3. T h e centre of t h e unit l o o p In this section, as a consequence of the famous Dirichlet Unit Theorem, we show that the unit group of the integral group ring of a finite abelian group is finitely generated. Then we exhibit a subloop of finite index in Z{U(T{L)))^ where L is an arbitrary finite RA loop. We begin with some terminology. Let K = Q(0) be an algebraic number field and let f(X) G Q[-^] be the minimal polynomial of 9. Suppose that f{X) has ri real and r2 pairs of complex conjugate roots. The pair [ri,r2] is called the signature of K. If r2 = 0, then K is called totally real while, if ri = 0, K is said to be totally imaginary. 3.1 Theorem (Dirichlet Unit Theorem). Let R be the ring of integers of an algebraic number field K and let [ri,r2] be the signature of K. The unit group U{R) of R is the direct product of the cyclic group of roots of unity in K and a free abelian group of rank ri + r2 — 1. PROOF.

We refer the reader to [Kar88, Corollary IV.3.13].

D

Let ^ be a complex root of unity. Since Z[^] is a Z-order contained in the ring R of integers of Q(0? it follows from Lemma VIII.2.6 that the commutative groups U{R) and ZY(Z[^]) have the same torsion-free rank. Hence the Dirichlet Unit Theorem has at once the following application. 3.2 Corollary. Let n > 1 and let ^^ be a primitive nth root of unity. Then U{2[i^ri\) = (i^n) X F, a direct product, where F is a finitely generated free abelian group. Furthermore, if n > 2, the rank p{F) of F is ^(n) — 1; otherwise p{F) — 0. 3.3 Theorem (Higman). If A is a finite abelian group, then U{ZA) =

±AxF,

where F is a free abelian group of rank p{F)=\{\A\

+

n^{A)-2c{A)^\),

where n2{A) is the number of elements of order 2 in A and c[A) is the number of cyclic subgroups of A. In particular, the only torsion units in ZA are the trivial units.

310

XII. LOOPS OF UNITS

Let n be the order of ^ . From Theorem VII.1.4 and the ensuing remark, we know that QA = 0 1 a^Qi^d) stnd PROOF.

ZAC^^^^adZ[U]

= R.

where each ^d is a primitive dth root of unity and ad is the number of cyclic subgroups of order d. Recall that (d) is the Q-dimension of Q{^d)' By Lemma VIII.2.6, [1{(R): 1{(ZA)] < oo. Hence, because of Corollary 3.2, both these abelian groups are finitely generated with the same torsionfree rank. By Theorem VIII. 1.4, aU torsion units in ZA are trivial, hence U{ZA) = ±A X F , where F is a finitely generated free abelian group of finite rank p{F). Furthermore, Corollary 3.2 yields

p{F)= Yl a , ( ^ - l ) d\n4>2

d\n4>2

c/|n,d>2

= \ ( Z ) °<''^(^) - 2 ^ ad + n2{A) + l) d\n

d\n

= \{n~\-n2{A)-2c{a)

+ l),

D

Higman's Theorem can be used to determine when the free abelian part is trivial (that is, when all units are trivial), but we have already established this result by more elementary means in Chapter VIII. We now give the structure of the loop of central units of the integral loop ring ZL of a finite RA loop L. 3.4 Proposition. Let L be a finite RA loop. Then Z{U(ZL)) = where F is a free abelian group of rank 1 (\L\ 2 \ 2

\Z{L)\ + n^iL/L') 2

- n2{Z{L)IL') + {-c{LlL')

±Z(L)F,

+ n2{Z{L)) + l ) +

c{Z{L)IL')-c{Z{L)))

Let G be a nonabelian group such that L — M(G,*,^o) ^ii^d, as usual, let s denote the nontrivial commutator-associator of L. Write R — Z{ZL). Note that Z{U{ZL)) = U{Z{ZL))^ because both loops are equal to ZY(ZL) n Z{ZL). Clearly R C R^^ ® R ^ , and both R and R^^ 0 R ^ PROOF.

3. THE CENTRE OF THE UNIT LOOP

311

are Z-orders in Z{QL). By Lemma VIII.2.6, U(R) is a subgroup of finite index in the direct product U{R^^) X U{R^^)^ so its torsion-free rank is the same as that of the direct product. We concentrate on each direct factor separately. First, since ZL^^ = Z[LIV] is commutative, we have R^^ = Z[L/V], Theorem 3.3 therefore yields that U{R^^) = ±LFi^^, where Fi is a free abelian group of rank

By CoroUary VI.1.3, R ^ = Z[Z{L)]^.

Because

Z[Z{L)]CZ[Z{L)y-^®Z[Z{Lr-^ and since the rings Z[Z(L)] and Z[Z{L)]^^®Z[Z(L)]^ are each Z-orders in the ring Q[Z(L)], it follows from Lemma Vin.2.6 and Theorem 3.3 that U{R^^) = ±Z{L)F2^ where F2 is a finitely generated free abelian group. Furthermore, since Z[Z{L)]^^ =. Z[Z{L)IL']^ we also obtain the rank of

piF,) = i {\Z{L)\ + n2(2(L)) - 2c(Z(L)) + 1) - i {\Z{L)/L'\

+ n2iZ{L)/L')

- 2c(Z(L)/L') + 1).

So U{R) is a subgroup of finite index in a finitely generated abelian group of torsion-free rank p{Fi) + p{F2). Hence U{R) has torsion-free rank p{Fi) -\p{F2) and the result follows. D Theorem 3.3 gives a complete description of the structure of the unit group of the integral group ring of a finite abelian group. To obtain a presentation of this group, one would also like to have a description of the generators of the free abelian part. This, however, is a very difficult problem, as even the analogous problem forZY(Z[^]) is unsolved. Therefore, as a compromise, one hopes at least to find a description for generators of a subgroup of finite index in the unit group. In this direction, we have a fundamental theorem due to H. Bass and J. Milnor [Bas66]. For a detailed proof, we refer to [Seh93, Chapter 2].

312

XII. LOOPS OF UNITS

Recall that the Bass cyclic units in the integral loop ring of a finite RA loop are the units of the type 1 _ 7<^(^)

n where g £ L has finite order n, 1 < i < n, gcd(i,n) = 1 and ^ = 1 + ^ +

3.5 Theorem (Bass—Milnor). Let A be a finite abelian group. Then the group generated by the Bass cyclic units is of finite index in U{2.A), Next we give generators for a subloop of finite index in Z{U{2.L)). Recall (Corollary III.4.3 and preceding remarks) that r G ZL is central if and only if r = r* and, therefore, that elements of the form rr* are always central. In particular, if 6 is a Bass cyclic unit, then 66* is a central unit. 3.6 Theorem. Let L be a finite RA loop. Then the loop B = (66* I 6 a Bass cyclic unit in ZL) is of finite index in

Z(U{ZL)).

Since Z{L) is a finite abelian group, the (finitely many) Bass cyclic units, hence also the squares of the Bass cyclic units of Z[Z(Z/)], generate a subgroup of finite index in U{Z[Z{L)])^ by the Bass-Milnor Theorem. Call this subgroup J9i and note that B\ C B since 6-^ = 66* for any central 6. Since PROOF.

Z[Z{L)]CZ[Z{L)Y-±^®Z[Z{L)]

1-5

and because both rings are Z-orders in Q[>Z(Z/)], Lemma VIII.2.6 yields that 5 i , and thus also 5 , contain subgroups of finite index in the group U{Z[Z{L)])^^, Note that we have identified the latter group with ^ ^ + U{Z[Z{L)]y-^. As before, we have also 2(ZI)CZLl±^®Z[2(L)]lf^, each ring a Z-order in 2(QL), so, to prove the theorem, it is sufficient to show that B ^ = {v'^ \ v £ B] is oi finite index in U{ZL)^^ ^

3. THE CENTRE OF THE UNIT LOOP

313

U(Z[L/V]). We claim that if 6 is a Bass cyclic unit in Z[L/V]^ then there exists a Bass cyclic unit 6 G ZL with b^^ = 6 or 6 . Indeed, write _

1 — 7^("') 71

where ^ G L/V has finite order n, 1 < i < n, and gcd(i,n) = 1. Clearly g E L has order m, and m = n or 2n, Because of Proposition V. 1.1, if n is odd, we may assume that n — m. Consider then the Bass cyclic unit 1 — j4>{'^) m

If n = m (for example, if n is odd), then clearly b^^ hand, if m = 2n and n is even, then

= b. On the other

+ 2(i + 5 + --- + r-')*<"^^^^—9+^-^—^—-g. n

n

Since 'g^'g = 'g for any i > 0, it foUows that

V

n

n

= (1 + ^ + . . . + ^-l)20(n) ^ (^2i'^(n) ^ J _ .^(n)^ ^ ~ ^

/ f

= (T+5 + • • •+r'-')'*^"^ + H i + ^^^"^) (1 - ^*^"^) t

= (T+5 + • • •+r-')*^'"> + ^(1 - i*^'"')?. Because ^ 5 * = ii^gr''^'" for any i > 0, it follows that 'g = ^ 5 . Hence 6' = (T + p + • • • + 5^-1 )'^(-) + X ( l _ i't>(m)^^\±l = (1 + P + • • • + 5-^)^(-)l±^ + 2^(1 - i*('"))5i^ = 6if^, which establishes our claim. Since LjV is abelian, the claim and Theorem 3.5 once again imply that B^^ is indeed of finite index in U{ZL)^^ =

U(Z[LIV]),

D

314

XII. LOOPS OF UNITS

4. D e s c r i b i n g large s u b g r o u p s Let i be a finite RA loop. In this section, we exhibit a snbloop V of finite index in the unit loop of ZL. In many cases, it turns out that V is a normal complement of ±L in the sense that V < U(ZL), V H L = {1} and U{ZL) = ±LV. The results are based mainly on work by E. Jespers and G. Leal [JL91, JL93b]. We introduce two subloops of the unit loop of ZL. Let L be an RA loop with V = {1,5} and let Ui and U2 be the following subloops oiU{ZL): Ui = U{ZL) n {QL^^

+ ^ ) - U{ZL) n {ZL^^

+

^ )

U2 = U{ZL) n ( ^ + Q L i ^ ) = U{ZL) n (i±^ + ZL^f^). We list a few properties of these loops. As usual, the augmentation map QL ^^ Q is denoted €. 4.1 Proposition. Let L be a finite RA loop with L' = {^f^}-

Then the

following statements are true. 1. 2. 3. 4.

Hi = {fieU{ZL) \^= l + a(l + s),a € ZL}. ZY2 = {/i G ZY(ZL) I /i = 1 + a ( l - 6),a G ZL}. Hi is a central subloop ofU{ZL) and Hi 01^2 = {I}V = V(ZL) = {/i G U{ZL) I /i = 1 + a ( l - 5),a G ZL, 6(a) even} is a torsion-free normal subloop ofU{ZL) and V = H2/{l',s}. 5. UiV is a normal subloop of finite index in U{ZL). 6. / / LIV has exponent at most 4, then U{ZL) = ±LV; thus V is a torsion-free normal complement of ±L inU{ZL). P R O O F . For (1), let A = {^i e U{ZL)

| /x = 1 + a ( l + 5),a

G ZL}.

If

^ - \ -\- a ( l + 5) G A, then

/ ^ = ^ + ^ ( l + 2a), hence A CUi. Conversely, let ji — Ylg^LJ^gQ G ZYi, where each /x^ G Z. Then // = OL^^ + ^ ^ for some a = YlgeL^gd ^ ^ ^ ' ^^^^ ^9 ^ ^* Since 5(1 + 5) = 1 + 5, we may assume that supp(a) is in a transversal (containing 1) of L\ Consequently, for g G supp(a), it follows that asg = 0 and s ^ supp(a). Comparing coefficients, we see that if 1 ^ ^ G supp(a), then IXg = ^ag ^ Z, Therefore ag is even. Similarly, comparing coefficients.

4. DESCRIBING LARGE SUBGROUPS

315

/xi = i a i + i G Z. Hence a i is odd. Thus a = 1 + 2/3 with f3 G ZL. Therefore /i = (1 + 2 / ? ) ^ + ^ = 1 + /?(1 + 5); that is, fi e A, The proof of (2) is similar. For (3), since ZL^^ = Z[L/V] is commutative, Hi is a central subloop of U{ZL). Assume /i eHi nZY2. Since /x € ZYi, ( / / - 1)(1 - 5 ) = 0. Furthermore, (/i - 1)(1 + 5) = 0 since // G K2' It follows that /x - 1 = 0; that is, // = 1. To prove (4), let /x = 1 -f- a ( l - s) e U2 with a = Y^g^L^gd ^ ^^• Then /X5 = 1 — (1 + a ) ( l — 5), so multiplication by s changes the parity of 6(a). Thus U2 — V{1,5}. Next we show that s ^V. To the contrary, if s — 1 + a ( l — 5) where a = Y^g^L^gd ^ ^^ ^^^ ^(^) ^^en, then a^ = ags for ^ ^ {1,5}, so a = a'(l + 5) + a i + a^^ for some a' G ZL. Consequently, 5 = 1 + (ai + a55)(l — 5) and thus 0 = 1 + a i — a5, contradicting the fact that e(a) is even. In order to prove that V is a loop, it is enough to note that V is closed under multiplication and inverses. The former is clear and the latter follows because, if //"^ ^ V, then, by the above, /x~^ = i/s for some i/ G V. However this implies 5 = //^' G V, a contradiction. To prove that V is torsion-free, note that each element /x G V has 1 or s in its support, and also that €(/i) = 1. So, by Corollary VIII.1.2, if // is a torsion unit in V, then fj, = I or s. But the latter is impossible, as we already have seen. Hence V is torsion-free. It is now also clear that U2 is the direct product V X { l , ^ } . To finish the proof of (4), we show that V is a normal subloop of ZY(ZL). Let ly = I + a(l — s) E V with e(a) even. If //i,//2 € U(ZL) and if 9 is any of the inner maps T(/xi), i?(/xi,//2), L(/ii,/X2) (see §11.1.9), then 19 = I and s9 = s^ so 1/9 = I + (Q:^)(1 — s) which is in V because €(a) even implies €{a9) even. Hence V is indeed a normal subloop. To prove (5), we first note that ZYiV is a normal subloop of U{ZL) because U\ is central and V is normal. Since the index of V in U2 is 2, to show that U\V has finite index in ZY(ZL), it is enough to prove that U1U2 is of finite index in U{ZL), For this, we claim first that U{Z + 2ZL) C ZY1ZY2. For if ^ =z m+2j G ZY(Z + 2ZL), then fx~^ = n+2<5, where m, n G Z. Consequently, 1 = rnn + 2(7 + 6 + 2j6)^ so mn (and hence m) are odd. Write m = 1 + 2m',

316

XII. LOOPS OF UNITS

m' G Z. Then // = 1 + 2(m' + 7). Since ji = M 2^ "^ /^ 2^' ^^ obtain /x = 1 + 2(m' + 7 ) i ^ + 2{m' + 7 ) ^ ^ = 1 + (m' + 7)(1 + 5) + (m' + 7)(1 - s) = [1 + (m' + 7)(1 + s)][{l + (m' + 7)(1 - s)l which is an element of ZYiZY2- This establishes the claim. Since Z+2ZL C ZL and since the rings Z + 2ZL and ZL are each Z-orders in QL, it follows from Lemma VIIL2.6 that U{Z + ZL) is of finite index in ZY(ZL). By our earlier claim, U1U2 is of finite index in ZY(ZL), giving (5). By Theorem VIII.3.1, Z[LlV] has only trivial units. From this observation, it follows easily that U{ZL) = zbLV. Since V fl L = {1}, the result in (6) also follows. D In Theorem 3.6, we gave generators for a subgroup of finite index in the centre oiU{ZL), Since U\ is central, we therefore know this subgroup implicitly, up to finite index. We now show how to compute V when L has cyclic center. In so doing, we consider two separate cases. First we introduce some notation. Recall that GLL(2, R) denotes the general linear loop; that is, the loop of invertible 2 x 2 matrices in Zorn's matrix algebra 3 ( ^ ) over a commutative ring R. (See §11.5.5.) Let ^n be a primitive nth root of unity. Then Q(^n) is a Galois extension of Q and its Galois group is denoted Gal(Q(^n)/Q)- The norm n{a) of an element a G Q(^n) is defined by

^(^) = n ^(^)^EGal(Q(e„)/Q)

It is well known (see, for example, [Kar88, Proposition 1.2.29]) that a G Z[^n] is invertible in Z[^ri] if and only if 71(a) = ± 1 . For a subloop S of GLL(2, Z[^n]), let 5n(det)=±i = { ^ G 5 | n ( d e t A ) = ± l } and '^n(det)=±l = Sn{det)=±\l{I'>

-I]

where / is the identity matrix of 3 ( ^ ) - We also write 5'det=±i = { ^ G 5 | d e t A . r ±1}

4. DESCRIBING LARGE SUBGROUPS

317

and *5'det=±l = Sdet=±l/{Ij

-I}'

4.2 Theorem. Let L be a loop of type C\ with cyclic centre generated by an element ti of order 2'^^. Let ^ be a primitive 2'^^ th root of unity and let

V':QLi-^-3(Q(0) be the mapping defined in Proposition VIL2.7, Then V = V ^ ^ and the restriction of if) to V ^ ^ induces an isomorphism from V onto ' ^ ( V ^ Y ^ ) , where

1. V(vif^) = 1 + 2a 2v 2w 1 + 26

1 + 2Z[^] 2(Z[^])3

2(Z[^])3 1+2Z[^]

n(det) = l

a + 6 € 2Z[e], V + w G 2(Z[^])= if mi > 1, and

2. i^iV'-^) = l + 2a 2w

2v 1 + 26

1 + 2Z 2Z^ 2Z^ 1 + 2Z

J det=l

a + 6 G 2Z, v + w G 2Z^ if mi = 1. PROOF.

Assume first that mi > 1. Given an element

a = l + 2[(7o + Jix + 722/ + 73xy) + (70 + 71^^ + 72^ + 73^2/)^] l - s

318

XII. LOOPS OF UNITS

in ZY2, it follows that V>(aif^) =

2(-i7{ + l{, -iJ

+ iiJ, iix^ + 7 ^ 0

1 + 2(7o^ - ili)

and clearly 1 + 2Z[^] 2{Z\i\f 2(Z[^])3 1 + 2Z[^]

V^(aif^) e

n(det)=±l

Now we have to determine which matrices A =

1 + 2Z[^] 2(Z[^])3 2(Z\i\f 1 + 2Z[^]

1 + 2a 2v 2w 1 + 26

n(det)=±l

actually belong to '^{JA^^')Let v = (vi,i'2?^3) and w = \-i Then A G V ' ( ^ 2 ^ T2 ^ ) if ^-i^d only if the linear systems 1

i

1

-I

0 0

0 0

\A li

0 0 0 0

{wx^w^^w^.

a

b

ij

V2

LT^^.

W2 j

t -i 0 0

and

1 0 1 0 0 -i 0 i

0 0 1 1

[7/1

Vi

^

= V3

W/\

- '^3 J

each have a solution in Z[^]. These systems are equivalent to 1 i 0 0' [70^ 0 2i 0 0 li 0 0 1 i %' 0 0 0 2i

y\

a 1 a—b V2

y2 + W2\

and

i 0 0 0

1 0 0 2 0 0 0 —i 1 0 0 2

\A

V\

1

'^3

wA

yz + ^^3]

4. DESCRIBING LARGE SUBGROUPS

319

Hence, A e il^ ( ^ 2 ^ ) if and only if a+6 e 2Z[^] and v+w G 2(Z[^])^. Notice that a + 6 G 2Z[^] implies that det A € 1 + 4Z[^]; therefore, n(det A) = 1. Since V = U2IL'^ and '^(^y^) = —/, the result follows. The case m\ = 1 follows in a similar way. D We remark in passing that, when mi = 1, the condition a + 6 G 2Z is actually equivalent to det A = 1. 4.3 T h e o r e m . Let L he a loop of type £2 ™^^ cyclic centre generated by an element ti of order 2'^^ ^ mi > 1. Let ^ be a primitive 2^^ th root of unity and let

be the mapping defined in Proposition VIL2.8. Then V = V ^ ^ and the restriction of ^ to V ^ ^ induces an isomorphism from V onto '^{V^^), where

V'(vif^) = 1 +2a 2{wx,W2,W3)

2{v\,V2.,V3) 1 + 26

l + 2Z[e] 2{Z[^]f

2{Z[i]f 1 + 2Z[^]

I n(det)=l

a + be 2Z[^], V2 + W2, vii + t/^i, t;3^ + ^3 G 2Z[.<] I . PROOF.

The proof is similar to that of Theorem 4.2.

D

To compute V{ZL) for an arbitrary indecomposable RA loop L, one proceeds as follows. Given a unit ji = l + a ( l - 5 ) G V(ZZ/), write fi = ^ ^ + ^ ^ ^e, where e runs over all primitive central idempotents in A(JL, L ' ) . Since /ie G V((ZZ/)e) and Le is an indecomposable RA loop with cyclic center, it follows from Theorem VII.2.5, Proposition VII.2.7 or Proposition VII.2.8, and Theorem 4.2 or Theorem 4.3, that /x can be written as a sum of matrices (with entries in the ring of integers of a cyclotomic field extension of the rationals) and perhaps one element in a nonsplit Cayley-Dickson algebra. The problem is then to determine which of these sums are actually representations of units

320

XII. LOOPS OF UNITS

in V(ZL). This leads to systems of linear equations and then to other equations relating the components in the different matrices. Since the integral nonsplit Cayley-Dickson algebra has only trivial units—the only units are the elements in ±{l^i^j^k^£^i£^j£^ki} because the norm of a unit has to be 1—the appearance of such a component will imply some other equations relating the coefficients of /x. Note also that, to write fi G (ZZ/)e as a matrix using Proposition VIL2.7 or Proposition VII.2.8, one has possibly to replace /i by another element LJ such that (ct;e)^ = —e and to choose appropriately the group G of index 2 in L to ensure that {QG)e is a matrix algebra. The same method can also be applied to RA loops which are not indecomposable; that is, to loops of the form LxA where L is an indecomposable RA loop and A is an abelian group.

5. E x a m p l e s In this section we apply the method outlined at the end of the previous section to obtain the unit loops of the integral loop rings of two indecomposable 2-loops. In [Goo92], a description of the unit loop for all RA loops of order less than 64 is given. T h e unit loop of Z[M32(^t, 16)] Let L = M32{Ei,l6) = M(16r26,*, 1) = D\J Du.v^here D = lGT2b = {x,y,ti \ x^ = y^ = l^t'l = l,(a:,?/) = / i , / i central ). The group D is of type Vi and L is an indecomposable RA loop of type £ i . (See Chapter V, Sections 2 and 3.) Because of Proposition VII.2.7, L = D U Dw with w = tiu; thus w'^ = tj = s and ( t i ; ^ ) ^ = - ^ . Furthermore, any element in V{ZL) can be written uniquely in the form fj. = I + {(ao + /Soh -f- a i x + /3itix + 0:2 + /?22/ + (^s^V + l^shxy) + («o + f^oh + a[x + fi[tix + a'2 + /322/ + «3^2/ + l^3hxy)w] (1 - s) where all ai,l3i,a'-,fi[ G Z.

5. EXAMPLES

321

5.1 T h e o r e m . Let L = M32(Ei,l6), ThenU{ZL) = ±L[V{ZL)] V(Ziy) is isomorphic to the loop of matrices of the form l + 2Z[i] 2{Z[i\f 2{Z[i\f 1 + 21[i\

1 + 2a 2v 2w 1 + 26

where

det=l

with a + 6 e 2Z[j] andv + w 6 2(Z[t])^. Furthermore, the isomorphism maps /i = 1 + {(ao + /3o*i + oi\x + Afix + a2 + /322/ + otsxy + fi^tixy) + (ao + /3o^i + aia; + f3[t^x + a^ + /3^t/ + a'^xy + /9^iixt/)w} (1 - 5) fo

2 ( ( a 2 - / ? i ) + (ai+/32)i, 1 + 2((ao - ;33) + (as + /?o)0 (/?;+a^) + ( - a ' , + ; 3 ^ ) 0 ((A + 02) + ( - a i + /32)«, ( - a o - / 3 3 ) + (-/5o + a3)«,

PROOF.

orem 4.2.

1 + 2(a[, + 13':,) + (;9^ - a^)i

This follows from Proposition VII.2.7, Proposition 4.1 and TheD

The unit loop of ZfMieCQs, 2)] Let L = Mi6((58,2) = D4 U Z)4'« = M(£'4,*,a^), where we use the classical representation for the dihedral group £^4 of order 8: D^ = ( a , 6 | a^ = 6^ = 1,6a = a^6}. As before, D^^ is a group of type V\. To use the results of the previous section, we should use for D^ the presentation {x,y \x'^ = y^ = l , ( x , y ) = ti,t\

= 1).

As a consequence, we cannot directly apply the isomorphism given in Section 4. Instead we go through the whole procedure again, as in [JL93b]. 5.2 Lemma. The mapping VP: V ( Z L ) -

2Z + 1 (2Z,2Z,4Z)

(4Z,2Z,2Z) 2Z+ 1

det=±l

XII. LOOPS OF UNITS

322

defined by
a^))

2s+1 w

V 2t+l

where a = ai -\- a2Ci + a^b + a4ab f3 = (3i+fi2a

+ fisb + (i4ab

«s = a i + a2 - a s + a4 V = (4(a2 + a 4 ) , 2(/3i + /32 - /?3 + ^4). 2(-/32 + /^a)) w = ( 2 ( - a 2 + a a ) , -2(/3i - /32 + /J3 - /?4),4(/?2 + /?4)) t = ai - a2 + Ois - a4^ each ai^Pi G Z, is a PROOF.

monomorphism.

Let e n , 622, ^12 and 621 be the following elements of the group

ring QD4: €11 — ^(1 — a — b + ab) ^21 = ( - « + 0.b)

2

2

'

e\2 = Uo' + b)

2

'

^22 = U l + « + 6 - a6) l-a^

'

One can easily verify t h a t e n +^22 =

2^'> ^^^ identity element of A ( L , L'),

and t h a t eiie22 = 0, e n = ei2e2i, e22 = ^21^12 ^ind ej2 = ^21 — 0. Thus {€ii?^i27C2i,e22} play the role of the elementary matrices in M2(Q). Note also t h a t .i-o^

.1-0^

= €11 - ^22 + 2ei2 - e2i, €21 - ^11 + e22

and a6 l - g ^ = ^11 - ^22 + 2ei2. For L = M ( G , * , ^ o ) , remember t h a t RL = RG + RGu and A ( L , L ' ) = A ( G , G ' ) + A{G,G^)u

(see Lemma V L l . l ) . Here, A{G,G')

^ M2(Q). Now

we represent an element it; G ZY2(ZZ/) as an element of A ( L , L') = M2(Q) + M2(Q)/7. For this, note t h a t a'^{l-a'^)

= -{l-a'^),sow

= l + {a +

f3u){l-

5. EXAMPLES

323

a^), where a = a i + 0:20 + 0:36 + 0406, f3 — Pi + /?2a + l^sb + ji^ab and each a,,/3j G Z. Hence t(; = l + (a + / 3 u ) ( l - a ^ ) = i ^ + [(2a+l) + 2/?«]i^ = i ^ + [1 + 2(ai + a2a + 03^ + a^ah) + 2(/Ji + /?2a + IS^h + / ? 4 a 6 ) u ] i ^ = ^

+ [(1 + 2(^1 + a2 - "3 + a4))eii + 4(a2 + 04)612 + 2 ( - a 2 + Q;3)e2i + (1 + 2(ai - 02 + "3 - Q;4))e22] + [2(/3i + /32 - /33 + /?4)eii + 4(/32 + /34)ei2 + 2(-/32 + /?3)e2i + 2(/?i - ^^2 + /33 - /34)e22]«.

It follows that the mapping tp-.HiiZL) -^ M2(Q) + M2(Q)f/ defined by 1^(1 + ((ai + 020 + 036 + a4ab) + (A + /J2a + /Sgi + /34a6)ti)(l - a^)) 1 + 2 ( 0 1 + 0 : 2 - 0 3 + 0:4)

4(02 + 04)

2 ( - 0 2 + 03)

1+2(01-02 + 03-04)

+

2(/3i+/32-/93 4-/34) 2(-/32+/33)

4(/32 + /34) t/ 2(/?i-/32+/33-/?4)

is injective and preserves multiplication. The isomorphism from M2(Q) + M2{Q)U to 3(Q) given in §1.3.5 now allows us to obtain an injective, multiplication preserving mapping

ip': Ui{ZL)-^ S =

2Z + 1 (21,21,41)

(41,21,21) 2Z+1

defined by ip'('w) = [^*J"^ 2t+i ] ' where s,t,\/,\N are as in the statement of the lemma. Since the determinant function is multiplicative on 3(Q), it follows that the image of rp' is contained in 5det=±i • Finally, i)'(a^) = V'(l - (1 - a^)) = -I- Hence, by Proposition 4.1, rp' induces the required monomorphism 9 : V(1L)

2Z + 1 (21,21,41)

(41,21,21) 2Z+1

D det=±l

XII. LOOPS OF UNITS

324

5.3 T h e o r e m . The monomorphism (p defined in Lemma 5.2 yields the isomorphism V(ZL) ^ <^ A

A^ PROOF.

2s+1

(4x1,2X2,22:3)

(2i/i,22/2,4t/3)

2f+l

2Z+1 [(2Z,2Z,4Z)

(4Z,2Z,2Z) 2Z+1

I 2 I (X2 + 2/2) det=l

Because of Lemma 5.2, it is sufficient to show that A=

r

25+1 (2j/i, 2^2,42/3)

(4x1,2x2,2x3) e v(v(ZL)) 2<+l

if and only if 2 | (x2 + 2/2) and det ^ = 1. Clearly A € i^(V(Zi/)) if and only if the linear systems 1 0 0 1

1 1 -1 -1

-1 0

1 " ai 1 «2 0 «3 1 _ .^4.

s Xi

y\ _ t \

1 0 0 •1

and

1 1 -1

-1 0 1 -1

1 1 0 1

>r

XT]

/?2

2/3

/33

3:3

A.

.^2]

have an integral solution. These systems are equivalent to 2 0 0 0 1 0 0 1 1 0 0 1 1 -1 I -1

"ai"

' s + t "]

Ot2

Xi +t \

Ot3

xi + y\

_a4

t

and

\ 2 1 0 -1

0 0 1 0 1 0 1 1

0 0 1 1

"/Ji"

X2 - 2/2I

P2

J/3 - J/2

f33

ys + 2^3

A.

.

2/2 J

and hence have an integer solution if and only if 2 | {^s-\-i) and 2 | (x2 —1/2). Since det ^ — {2s-\-1)(2/+ 1) -\{2x\y\ +3:2^/2+ 2x3^/3) = ± 1 , the condition 2 I (5 + ^) is equivalent to det A—\ and the result follows. D

5. EXAMPLES

325

We now show that V{ZL) is closely related to V(Zi?4). For this, we introduce a certain subgroup of PSL(2,Z), the integral projective special linear group; namely, 2Z + 1 2Z

4Z 2Z+1

= {Ae

25+1 U

M2(Z) I A =

det=l

5, t,k,ieZ,

4k 2« + 1

det yl = 1 i / { / , - / }

and we also define three natural subgroups of V{ZL). 5.4 Definition. 25+1 (2t/i,0,0)

V^ = iA

(4x1,0,0) 2i+l s,t,xi,yi

25+1 (0,22/2,0)

Vo = lA

e Z, detA = 1 W { / , - / } ;

(0,2x2,0) 2t+l

s, t, X2,2/2 € Z, 2 I (X2 + j/2), det A = n / { / , - / } ;

V.

25 + 1

(0,0,2x3)

(0,0,4t/3)

2f+l

I

s,t,X3,y3e

Z, detyl = 1, W { / , - / } .

5.5 Proposition. 1. Each V, is a group isomorphic to 2Z + 1 2Z

4Z 2Z + 1

^ V(Z£>4)det=l

2Z + 1 2Z

4Z 2Z+1

det=l

XII. LOOPS OF UNITS

326

3. V{ZD4) is a free group of rank 3. Moreover, vi = I + {1 - a'^b)a^{l + a'^b) =

1 4 and 0 1

t;2 = 1 + (1 - aH)a{l + aH)

V3 = l + il-b)a\l

1 0 2 1

-3 -8 2 5

+ b)^

are free generators, where the matrix representations are those given by the isomorphism of part (2). Note that v\, V2 and v^ are bicyclic units o/ Z£>4. 1. This statement is obvious for Vi. That V2 is a group is easily verified, and conjugation by [J }] gives an isomorphism between PROOF.

{and

25+1

2A; 1

2e

2Z + 1

4Z

2Z

2Z+ 1

2t + 11

I

2|(A; + ^), det^ = l W { / , - / }

; thus V2 = Vi. 7

det=l

Furthermore, conjugation by [^J] gives an isomorphism between 2Z + 1 2Z 4Z 2Z+1

and det=l

2Z+1 2Z

4Z 2Z + 1

det=l

Hence V3 = V2. 2. This follows at once from Proposition 4.1 and Theorem 5.3 by restricting (f to the set {it; = 1 + a ( l - a^) \ a = ai -\- a2a + asb + a4ab^ e{a) G 2Z, w G U{ZD4)}, which is V(ZD4), and then applying the natural isomorphism between Vi

and [ i J S ^ i J s ^ . 3. The group G=

^ 2Z + 1 4Z 2Z 2Z + 1

det=l

5. EXAMPLES

327

is a subgroup of index 2 in the free group 2Z + 1 2Z 2Z 2Z + 1

det=l

By Corollary VIII.5.2, the latter is free of rank 2 and has free generators [II] and [21]- The Nielsen-Schreier Theorem (see, for example, [Rob82, Chapter VI]) shows that G is a free group of rank 3 and, using the Schreier transversal { [ Q ? ] ? [ O ~ I ] } ' ^^^ obtains the generators

[1 0' 2 1

1 2 0 1

2

"1 4' 0 Ij

and

r1 -2' 0

1

1 0' "1 2' 2 1 0 1

" -3

-8

2

5

Let

(0,0,0) 1

VP(1 + (1 - a^b)a{l + a^b)) = >f{l + (a + 6)(1 - a^)) 1 (0,0,0)

(4,0,0) 1

and ip{l + (1 - 6)a^(l + b)) = ip{l + {-a - ab){l - a^)) -3 (2,0,0)

(-8,0,0) 5 D

The isomorphism in part 2 then yields the result.

5.6 Remark. Examining the proof of part 3 of Proposition 5.5 and using the generators of V(Z7?4) given there, it follows that the generators of Vi are 1 (2,0,0)

(0,0,0) 1

1 (0,0,0)

(4,0,0) 1

-3 (2,0,0)

(-8,0,0) 5

XII. LOOPS OF UNITS

328

that those of V2 are r

-1

(0,-2,0)

[(0,2,0)

3

1 (0,0,0)

(0,4,0) 1

-5 (0,2,0)

(0,-18,0) 7

1

(0,0,0)

5 (0,0,-8)

(0,0,2) -3

and that those of V3 are 1

(0,0,2)'

(0,0,0)

1 J ' [(0,0,4)

1 J

5.7 Corollary. The Moufang loop V(ZL) is generated by the free groups V\j V2 and V3. Thus V{'LL) has generators vi

=

1 + ( - a + a6)(l - a^),

V2 =

l + (a + 6 ) ( l - a 2 ) ,

V3 =

l +

V4 =

l + [{a +

v^

1 + [(1 _ a - 6 +a6)i/](l - a^),

=

{-a-ab){l-a^), b-ab)-u]{l-a'^),

^6 zr

1 + [(3a + 36 - Sab) + ( - 5 + 4a + 46 - 4a6)iz](l - a^),

vj

=

1 + [ ( - a +a6)tx](l - a^),

vs

=

1 + [(a + 6 ) i i ] ( l - a ^ ) ,

vg =

1 + [(-2a - 26 + 2a6) + ( - 3 a - 26 + ab)u]{l - a^).

Let S be the subloop of V(ZL) generated by the groups Vi, V2 and V3 and let PROOF.

25 + 1 [(2yi,2i/2,4t/3)

(4x1,2x2,2x3)1 2/+1 J

^

^'

in particular, 2 | (X2 + 2/2) ^i^d det A = 1. Consider the following statements. 1. We may assume 2^ + 1 > 0. 2. If Xi = 0 for some i, 1 < i < 3, then we may assume yi = 0 too. Similarly, if some yi = 0, then we may assume Xi = 0. In particular, upper and lower triangular matrices of V(ZL) belong to S. 3. If gcd(25+ l,xi) = 1, then Ae

S,

4. If (25 + 1) I xi, then we may assume xi = y^ = 0.

5. EXAMPLES

329

5. If 0 < gcd(25 + 1, a:i) = 2^1 + 1 < 25 + 1, then we may assume A — \

2si-f-l

(0,2x^,24)1

6. In assertions 4 or 5, if {2si + 1) | 2:3, then A ^ S. 7. In assertions 4 or 5, if 0 < gcd(25i + 1,0:3) = 2s2 + 1 < 2^1 + 1, then we may assume 252 + 1

A =

(4a:;',2a7^',0)

These assertions, if true and used a number of times, show t h a t A £

S.

The rest would then follow from Theorem 5.3 and Remark 5.6. Thus it is sufficient to verify the seven statements. 1. Since / equals —/ in V(ZL), this is clear. 2. We show t h a t if Xi = 0, then we may assume yi = 0. So suppose 2s + 1

(0,2x2,2x3)

(2t/i, 2^/2,4^/3)

2^ + 1

A = and let

2s-\- 1

(0,2x2,2x3)

(0,22/2,4^/3)

2^ + 1

B = Clearly B G V(ZL).

While the following calculation is obvious, it is included

as motivation for further computation in the proof and to indicate how this proof can be adapted to show the last p a r t . Consider, therefore.

B+

0

(0,0,0)

(22/1,0,0)

0

B\I+B

0

(0,0,0)

(22/1,0,0)

0

-1

1 (25+l)(22/i,0,0)

(0,82/12/3,-4J/12/2) 1

1

(0,0,0)

(2s + l)(22/i,0,0)

1

(0,82/12/3,-42/12/2) 1

B It follows t h a t A = B

1 (0,0,0)

XII. LOOPS OF UNITS

330

Let

C =

1

(0,0,0)

[(25+l)(2yi,0,0)

and

1 J

Then C , P G 5 and A = B{C[D{EF)]},

E =

1 (32j/2j/2y3,0,0)

(0,0,0) 1

1 (0,0,0)

D

(0,82/1^3,0) 1

where

and

F =

1 (0,0,0)

(0,0,-4yij/2) 1

Note that both E and F are in S. Thus ^ G 5 if and only if fi 6 5 , so, if xi = 0, we may also assume j/i = 0. Similar arguments also work for X2 and Xz as well as for the rest. 3. Since gcd(25 + 1,^i) = 1, there exist x,y £ Z such that x{2s + 1) + ySx\ = I. Let B =

r

X (2j/,0,0)

(-4x1,0,0)1 25+1

Then 5 e 5 and AB =

1 .1

x' 0*l

[y' 2t' + l

is in V(ZL). Hence, we may assume that A =

1

(4x1,2x2,2x3)

(22/1,22/2,41/3)

2t + \

in which case A

1 (0,0,0)

(-4x1,0,-2x3) 1

1 (2yJ,22/^,42/^)

(0,2x2,0) 2^'+ 1

By statement 2, we may assume 2/J = 2/3 = 0, so ^ G S. 4. In this case, Xi = x(2s + 1) for some x G Z. Consequently A =

1 25+1 (0,2X2,2X3)1 [(22/1,22/^, 4y^) 2t'+ I (0,0,0)

(4x,0,0) 1

is in V(ZL), where 2/2 = 3/2— 4xx3, 2/3 = 2/3 + 2xx2 and t' = t — 4x1/1. Thus, by statement 2, we may assume Xi = 2/1 = 0 .

5. EXAMPLES

331

5. Let 0 < 2^1 + 1 = gcd(25 + l,a;i). Then there exist x,y G Z such that x{2s + 1) + ySxi = 2^1 + 1. Multiplying A on the right by the matrix

(-42^,0,0)] 2s+l 251+1

L(2y,0,0)

(which belongs to S) shows that we may assume A=

2^1 + 1 (2t/i,22/^,4t/^)

(0,2a:'2,2x9| 2/' + l

and (2si + 1) | Xj. The result now follows from statement 2. 6. In this case X3 = a;(25i + 1). Let B =

1 (0,0,0)

(0,0,-2a;)| 1

be an element of S. Then 251 + 1 {2yl2y'^,4y'^)

AB =

(0,2x^,0)1 2t" + \

is in V(ZL). The result then follows from statement 2. 7. Let 0 < 2^2+1 = gcd(25i + l,x^) < 25i + l. Then there exist x,y G Z such that x{2si + 1) + 2/8x3 = 2s2 + 1. Let B

(0,0,-2(^))l (0'0'4j/) If^

(which belongs to S). Then AB

252 + 1 i2y'{,2y'^,4y'^)

{4x'{,2x'^,0) 2t" + \

Again, the result follows from statement 2.

€ ViZL). D

Chapter XIII

Idempotents and Finite Conjugacy

In Chapter XII, we found exphcit conditions under which the unit loop of the integral loop ring of an RA loop is an FC loop. The main thrust of this final chapter is to investigate this problem for alternative loop algebras over fields of any characteristic. Crucial in tackling this problem is knowledge of when all idempotents are central. This is what will be explored in the first section. As an application, one can also determine when all nilpotent elements are trivial. In the final section, the FC problem is answered fully.

1. Central i d e m p o t e n t s In this section, we give necessary and suflScient conditions under which all idempotents of loop algebras of RA loops are central. Most of the main results and proofs are based on [GM]. In the case of a group algebra KG^ S. Coelho solved the analogous problem when K is a field of positive characteristic (provided the set of torsion elements of G is a locally finite group) [Coe87], and S. Coelho and C. Polcino Milies settled the issue for fields of characteristic zero [CM88]. (See also Example 1.11 in this section.) For loop algebras over fields of positive characteristic our proofs are almost verbatim those of the group ring case [Coe87]. A field K is said to be a splitting field for a finite abelian group G if the group algebra KG is the direct sum of fields, each isomorphic to K. Clearly, a field K is a splitting field for a finite abelian group G of order n if and only if KG is semisimple artinian with n orthogonal nonzero primitive idempotents. 333

334

XIII. IDEMPOTENTS AND FINITE CONJUGACY

1.1 L e m m a . Let K he a fields G = (x) a cyclic group of order n and ^n CL primitive

nth root of unity over K. Suppose char/if does not divide n. If j

and k are integers with 0 < j ,fc< n — 1, then 1- ^j — n ^7=o(^^xy 2. CjCk = 0 ifj ^ k. In particular, PROOF.

K{(n)

^^ ^^ idempotent

in K{^)G

and

i^ ct splitting field for G.

Let 0 < j ,fc< n - L Then

\i j ^ k^ then ^ = ^n~

is a nontrivial root of unity of order m with

m dividing n, hence also an n t h root of unity.

Thus Y^^=Q ^^ = 0 and

YllZo i^ — ^ ^^d thus CjCk — 0. On the other hand, if j = A;, then we obtain

1=0

t=0

The next result is a special case of a well known theorem of R. Brauer. (See, for example, [ C R 8 1 , §15.16 and §17.1].) 1.2 C o r o l l a r y . Let K be a field of characteristic abelian group of order m . If in i^ o, primitive

zero and let G be a finite

nth root of unity over K and

m I 71, then A'(^n) is a splitting field for G and the primitive K{in)G

are the elements

idempotents

of

of the form ,

{n/\H\)-\

in E «!.""-)•. i=0

where H runs through the subgroups of G such that G/H G/H

= {xH) PROOF.

is cyclic, x £ G,

andO<j<j^-l, Write F = A^^n)- From Theorem X.2.1, we know t h a t

id a primitive dth root of unity and ad = nd/[F{id)'

^ ] , where nd is the

number of elements of order d in G. Since F contains a primitive m t h root

1. CENTRAL IDEMPOTENTS

335

of unity, each field F{^cl) is F and thus

So F is indeed a splitting field for G. Now let e be a primitive idempotent of FG. Then Ge is a finite subgroup in the component Fe = F and each of its elements is a root of unity whose order divides m. So Ge = {x) is a cyclic group of order i dividing m and Ge = G/H for some subgroup H of G. It follows that {FG)e ^ F is an epimorphic image of the group algebra F{x) = (FG)H, From the first part of the proof, we know that F is also a splitting field for (x). Furthermore, by Lemma 1.1, eo, e i , . . . , e^_i are the primitive idempotents of F{'x), The above epimorphism then shows that, for some J, 0 < j < ^ — 1,

where x is an element of G with xe = x. Hence n

^-^ 1=0

as desired. Conversely, let ^ be a subgroup of G with GjE = (Hx) = (^), a cyclic group of order £, and let

t=0

Since {FG)H ^ (F{x))/, where

F[G/H],

by Lemma VL1.2, it foUows that {FG)e

^

1=0

Because of Lemma 1.1, the element / is a primitive idempotent of F{x). Hence e is a primitive idempotent of FG, D 1.3 Corollary. Let K he a field of characteristic p > 0 with prime subfield V. Assume A is a finite abelian group whose order is not divisible by p. If e is an idempotent in KA^ then e G VA^ where V denotes the algebraic closure ofV in K.

336

XIII. IDEMPOTENTS AND FINITE CONJUGACY

Recall that for an RA loop L with (locally finite) torsion subloop T, and for a prime p > 0, we denote by Tp/ the subloop of L consisting of elements whose order is not divisible by p and by Tp, the subloop of elements whose order is a power oi p. It was shown in Proposition V.1.1 that T = T2 X T21 and that T21 is a central subloop of L. 1.4 Lemma. Let L be an RA loop with torsion subloop T, and let K be a field of characteristic p > 0. If all the idempotents of KT are central in KL and t £ T has order not divisible by p, then the cyclic group (t) is normal in L. Let ^ € // be an element of order n with p )( n. By assumption, the idempotent e = ^ Y17=o ^^ i^ central in KL^ so, for every x G //, we have PROOF.

_i

e = xex

-l ^ nn—\ 1

^ =n Lyxfx-\ . t=0

Hence xtx ^ G supp(e); that is, xtx ^ = f for some 0 < i < n — 1. By Corollary IV.1.11, {t) is a normal subloop of L. D 1.5 Lemma. Let L be an RA loop with torsion subloop T. If K is a field of characteristic 2, then every idempotent of KT is central in KL. Since T21 is a central subloop of L, it is sufficient to show that each idempotent of KT belongs to KT21. So let e G KT be an idempotent. Since a subloop generated by finitely many elements has finite torsion subloop (Lemma VIII.4.1), the subloop generated by the support of e and three nonassociating elements of L is a finite RA loop. Hence, replacing Z/ by a finitely generated RA subloop containing the support of e, we may assume that T is finite. Because of Corollary VL4.3, KT2> = ®T=\ ^^t? ^^e direct sum of fields Ki. Hence PROOF.

KT = K[T2. X T2\ = (/1T2OT2 ^ ®Zi I<^T2^ Write e = Y^^=\ ^i^ where each e^ is an idempotent in A\T2. Then the augmentation of Ci is 0 or 1, so, either Ci or 1 + ^i belongs to A/v'.(T2), the augmentation ideal of KiT2. It follows from Theorem VL3.4 that AKX'^^) is a nil ideal. Consequently Ci = 0 or Ci = I. Thus e G 0 ^ i Ki = KT2'^ •

1. CENTRAL IDEMPOTENTS

337

1.6 Theorem. Let L = M{G^ *?5o) ^^ o,'^ R^ loop with torsion subloop T. Let K be a field of characteristic p > 0. Then every idempotent of KT is central in KL if and only if p = 2 or 1. Tp/ is an abelian group and 2. ifTpf is not central, then the algebraic closure Fp of Fp in K is finite and, for all t G Tpi and i £ L, there exists a positive multiple r of [F^: Fp] such that iti'^ = t^^ prove the necessity of the conditions, assume that every idempotent of KT is central in KL and that p > 2, By Lemma 1.4, every subloop of Tp/ is normal in L, so, Tp/ is either an abelian group or a hamiltonian Moufang loop. In the latter case, Tpt contains the quaternion group Qg^ hence, FpQg C KT. By Corollary VII.2.4, KT contains H{Fp), Since t(Fp) = 1, by Proposition XI.3.3, H(Fp) is split (and hence isomorphic to M2{Fp)). Thus KT contain noncentral idempotents, a contradiction. Therefore Tp> is an abelian group, proving (1). To prove (2), let t G Tp, and let K^ be a finite subfield of K. Then K^{t) is semisimple and thus, by Theorem X.2.1, A^'(^) = ®Ki, where each Ki is a cyclotomic field extension of K^. Furthermore, at least one of these fields, say A^i, is of the form A^i = K\^) with ^ a primitive root of unity of order o{t). Note that the natural projection K^{t) -^ Ki sends t to ^. Let i £ L, Since subloops of Tpf are normal in L, iti~^ = f for some i > 0. Since idempotents of KT are central, the inner automorphism a ^-^ iai~^ of (the group algebra) K'{t^i) induces a field automorphism V^: A^i —> A'l. Since Ki is a finite field of characteristic p, 'tp is necessarily a power of the Frobenius homomorphism x \-^ x'^, (See, for example, [Kar88, Proposition II.5.3]). Hence, ^^(0 = r for some r > 0. Since K' C A'l and the elements of K' are fixed under V^, we obtain also that fc^"^ = k for every k G K'. Hence, [A'': Fp] | r. Note that r depends on t and K'. P R O O F . TO

Now assume that Tpi is not central and let t G Tp/, i ^ L he such that it / tL We have shown that Ht~^ = t^"^ for some positive r. If [Fp: Fp] is infinite, there exists a finite subfield K' of Fp such that [A^^: Fp] — 2r. Hence, using again what we showed earlier, there exists a positive integer v such that

338

XIII. IDEMPOTENTS AND FINITE CONJUGACY

Since lU ^ = f^"^ ^ we have PU '^ = f^ "^ and so, because i'^ is central, t'P^'' = t. Hence

where v exponents are involved, and therefore M ^ i=: ^, a contradiction. Thus [Rp-. F] is finite and (2) follows. We now show that the conditions are sufficient, lip = 2, then Lemma 1.5 implies that all idempotents of KT are central in KL, So assume p > 2. Thus both Tp and Tpt are abelian groups. (In fact, Tp is central.) Let e be an idempotent in KT, Write this element in the form

where kij E K^ gi ^ Tp and hj € Tpf. Let / be the image of e under the natural mapping exp'- KT -^ KTpi. Then

/—

2L^

^^ - -^ *

Clearly e- f = 6 e AK(T,TP) = ker6Tp. It follows from Theorem VI.2.7 that 6'P'' = 0 for some n > 0. Since e and / commute, e = e^"" = / P " + 6^"" = /p" = / . So e G A'Tp/ and thus, by Corollary 1.3, e = Ylj ^j^j ^ FpA^ where A is a finite subgroup of the abelian group Tpi. If A is central, clearly e is central. Suppose A is not central and let i e L. Write ,4 = (^i) x • • • x {th) as the direct product of cyclic groups. Because of (2), there exist numbers Ui > 0 and ix > 0 such that Uii'^ = tf' ior I < i < h and e{Ut2 • "th)i~^ = (^1 • • 'thY • Since A is commutative, the subloop (A^i) of L generated by A and ^ is a group. (See Corollary IV.2.4.) Hence

Therefore, t^ * = t^ for any i = 1 , . . . , /i and so, for any a £ A^ iai~^ = a^"*. Since tx is a multiple of [Fpi F], for any k G Fp, k^"" — k. So we obtain,

M-' = Y.^y; = J2kfhf = eP" = e, J

as desired.

3

D

1.7 Corollary. Let L = M^G^^^go) be an RA loop with torsion subloop T of finite order n. Let K be a field of characteristic p > 0. Assume K

1. CENTRAL IDEMPOTENTS

339

contains a primitive nth root of unity. Then all idempotents of KT are central in KL if and only if p = 2 or Tpf is central Because of Theorem 1.6, it is sufficient to show that, if all idempotents of KT are central in KL^ if p ^ 2 and if ^ G i/, ^ G Tp/ and iti-'^ = tP\ then tP'' = t. For this, let (^rn be a primitive mth root of unity, where m = o{t). By Lemma 1.1, PROOF.

^ m—l i=0

is an idempotent of KT, Since this idempotent is central, 771 — 1

^

m

.

(Note that i and t generate a group, so conjugation by t defines an automorphism on K{i^t).) Comparing coefficients of ^P"^, we obtain Sm

sm

SO m I (p^ — 1) and t^"^ — t, as desired. Let K be a finite Galois extension of the field F with Galois group Q = Gal(AYF). If X G A, then the trace of x, denoted tr(a:), is defined by tr(x)- X)a(x). Clearly tr(x) is fixed under any element of the Galois group G; therefore, tv{x) G F. For a group G and automorphism cr G ^, there is a natural extension of a to an F-automorphism of KG defined by (^{Ylg^G^gd) — T.geG^MdClearly

creG

geG

geG

We shall require characterizations of split quaternion and split CayleyDickson algebras in addition to those given in Chapter I. 1.8 Proposition. For a field F, the following conditions are equivalent. (i) H(F) = (F, — 1, — 1) 25 a split quaternion algebra. (ii) H(F) has nonzero nilpotent

elements.

D

340

XIII. IDEMPOTENTS AND FINITE CONJUGACY

(iii) H(F) has nontrivial idempotents. (iv) The equation x^ -\- y^ = —\ has a solution in F. Furthermore J if F — Q(^n)^ where ^n is o, primitive nth root of unity, then any of these conditions is also equivalent to (v). (v) The multiplicative order of 2 (mod n) is even. P R O O F . That (i), (ii) and (iii) are equivalent follows from Proposition 1.4.15. That (i) is equivalent to (iv) follows from Theorem 1.3.4. Now assume that F = Q(^n) with ^n ^ primitive nth root of unity. To prove that (iv) and (v) are equivalent for this field is much harder. We refer the reader to either [Mos73] or [FGS71]. D

In [FGS71], it is also shown that, for F = Q(^n)? (iv) is equivalent to the existence of a solution to a:-^ + y^ = — 1 in Q(^p) for some prime divisor p of n, where ^p is a primitive pth root of unity. Furthermore, it is shown that the latter holds for all primes p = ±3 (mod 8) and for some primes p = 1 (mod 8). The condition fails, however, for any prime p = 7 (mod 8). 1.9 Proposition. For a field F, the following conditions are equivalent. (i) (ii) (iii) (iv)

(F, — 1, — 1, — 1) Z5 a split Cayley-Dickson algebra. (F, —1,—1,—1) has nonzero nilpotent elements. (F, —1,—1,—1) has nontrivial idempotents. The equation x^ + y'^ + z^ + w^ = —I has a solution in F.

Furthermore, if F = Q{^n), where ^n is ^ primitive nth root of unity, then any of these conditions is also equivalent to (v). (v) n > 2. in Proposition 1.8, the equivalence of (i), (ii) and (iii) follows from Proposition 1.4.15 and that of (i) and (iv) from Theorem 1.3.4. Suppose that F = Q(^n) foi* some ^n-> ^ primitive nth root of unity. Again, to prove that (iv) and (v) are equivalent for K — Q(^n) is much harder. The result follows from the following fact, which is also shown by C. Moser in [MosTS] (see also [Pfi95, p. 41]): When n > 2, — 1 is a sum of four squares in P R O O F . AS

Q(^n).

•

1.10 Theorem. Let L — M{G^^.,go) be an RA loop with torsion subloop T. Let K be a field of characteristic zero. Then every idempotent of KT is central in KL if and only if the following conditions are satisfied.

1. CENTRAL I D E M P O T E N T S

1. Every subloop ofT 2. IfT

341

is normal in L.

contains a noncentral

element

of order n and if S,n i^ ^

primitive

nth root of unity over K, then the map a defined by (7(^n) = ^n is in G3l(K{^n)/K)' 3. T is either (a) an abelian group, or (b) a hamiltonian

group with the property

of odd order k, then the field K(^k)

solutions

of the equation x^ + y^ = —1, or

(c) a hamiltonian

Moufang

contain solutions PROOF.

that if it contains

element

RA 2'loop

does not

an

contain

and the field K does

not

of the equation X'^ + y'^ + z'^ + w^ = — 1 .

Assume t h a t every idempotent of KT

is central in KL,

By

Lemma 1.4, every subloop of T is normal in L. Hence T is either an abelian group or a hamiltonian Moufang loop. Suppose T is a hamiltonian Moufang loop. Let H he di finite (hamiltonian) subloop of T . either Qs or

M\Q{QS)',

By Theorem n . 4 . 8 , H E

S X E x A, where S is

is an elementary abelian 2-group and ^ is a finite

abelian group of odd order. If a G A is an element of order A;, then, using Lemma VILO.l, we see t h a t KT contains the subring K[{a) X 5] ^ K{a)

®K

KS.

Because of Theorem X.2.1, K{a) contains the field K{^k) ^s a direct summand.

By Corollary V n . 2 . 3 and Corollary V n . 2 . 4 , KS

contains either

(A', —1, — 1, —1) or (A', —1, —1) as a simple component. Hence KT

contains

the simple subring /i ( 6 ) ® A - ( A ' , - 1 , - 1 , - 1 ) = ( A ' ( 6 ) , - 1 , - 1 , - 1 ) , in the case t h a t T is a hamiltonian Moufang RA loop, or the simple subring A'(a)®A-(A',-l,-l) = (A'(6),-l,-l), in the case t h a t T is a hamiltonian group. Since all idempotents of

KH

are central, these simple rings contain only trivial idempotents, so they are division rings. By Propositions 1.8 and 1.9, the field K((k)

does not

contain solutions of the equations x'^ + y'^ + z"^ -\-w"^ = —1 and x^ + t/^ = —1 respectively. Furthermore, as

Q(^A:)

^ ^^ (^A:)^ we also get k = 1 when H is

a hamiltonian RA loop. Hence condition (3) follows.

342

XIII. IDEMPOTENTS AND FINITE CONJUGACY

We now show t h a t condition (2) must hold as weD. Suppose / G T is a noncentral element of (necessarily even) order n and let x be an element of L such t h a t xtx~^

= st^ where s is the unique nonidentity commutator-

associator of L. Since (t) is normal in L, 5 G (^), so s is the unique element of order 2 in {t). Therefore s = W^^. Consider the group algebra A^(^) as a subring of the group algebra K{^ri){t)'

Since both rings are direct sums of

fields, the nonzero idempotents of K{t) are precisely those elements of A^(^) which are sums of primitive idempotents of A'(^n)(^)- Let ei = and G = Gal(A'(^n)/A')-

^J27=oi^rity

By Lemma 1.1, the set {cr(ei) | <J G ^ } is a

collection of orthogonal primitive idempotents in K{^n){t)'

Therefore,

creQ

is an idempotent in K{t)

and hence central in KL^ by assumption.

In

particular, 2^xa{ei)x~^ creQ Since each xa{€i)x~^

= xex~^

= e.

is also a primitive idempotent of

{xa{ei)x-^

\aeG}

= {a{ei)

K{^ri){t)',

|a G ^}.

Hence xe\X~^ = cr(ei) for some a £ Q. Now st is in the support of e i , with coefficient ^^n n^n

; thus t is in the support of xeix~^,

also with coefficient

• Since the coefficient oft in cT{e\) is <7(^^n)? we obtain t h a t

This finishes the proof of (2). Conversely, assume t h a t conditions (1), (2) and (3) hold. To prove t h a t every idempotent of KT is central in K L we may assume t h a t L is finitely generated; thus T is finite. We claim t h a t KT

is a direct sum of division rings and hence every

idempotent of KT is central in KT,

This is obvious if T is abelian, so assume

t h a t T is not abelian. Because of (3), T is hamiltonian, more specifically, T = QsxExA

or T = Mie{Qs)

x £ , where E is an elementary abelian

2-group (possibly trivial) and A is an abelian group (possibly trivial) all of

1. CENTRAL IDEMPOTENTS

343

whose elements have odd order. Now, K[Mi6{Q8) >^E] = ^

{KE)[Mie{Qs)] (®K)MieiQs)

= (®AO®(e(A',-l,- -1,-1))

by Theorem VII. 1.4 by CoroUary VII.2.3.

Hence, by assumption (3.c) and Proposition 1.9, K[Mie(Q8) X £^] is a direct sum of division rings, as desired. On the other hand, using again several times the Wedderburn decomposition of the rational group algebras of finite abelian groups, and because of Corollary VII.2.4, we obtain K[Qs xExA]

= iKE)[Qs x A] ^ i®K)[Qs X A)] ^ ®(KA)Qs

= ®(e^||^|adii'(^d)g8)

S T ®(®4^i«'^(^''(^<')'-i'-i))' for certain integers a^, primitive dth roots of unity ^d and some sum J^ of fields. By assumption (3.b) and Proposition 1.8, KT is a direct sum of division rings, again as desired. To prove the result, it is sufficient to show that every primitive idempotent e of KT is central in KL. Since s is central and of order 2, either e = e i ^ or e = e ^ . Since e G Z{KT) = KT^^ + Z{KT^) and since KT^^ is in the centre of KL^ (by Corollary VI.1.3), we may assume that e G Z{KT^-^) = {K[Z{T)])^ (again by CoroUary VI.1.3); in particular, e G K[Z{T)\. Since {K[Z{T)])e is a field, we obtain that, for some subgroup H of Z ( T ) ,

Z(T)e^Z{T)IH={x), X G Z(T)^ X = xH and Z(T)/H a cyclic subgroup of this field, say of order m. Hence (K[Z{T)])e = {K[Z{T)])He\ where e' is the natural image of e in {K[Z{T)])H ^ K[Z{T)/H]. By CoroUary 1.2, we know the primitive idempotents of K(^rn)[Z{T)/H]. It foUows that e' is a sum of idempotents of the type ^ m—l t=0

344

XIII. IDEMPOTENTS AND FINITE CONJUGACY

where ^m is a primitive mth root of unity over K and 0 < j < m — 1. Hence e is a sum of idempotents of the type m —l

i=o

with 0 < i < m - 1 and n = \2{T)\, Note that if j = 0, then Cj = Z(T) is a central idempotent in KL. Since ej is a direct summand of the primitive idempotent 6, it follows that e = e^; hence e is central. So, for the remainder of the proof, we may assume also that j 7^ 0. It is easily verified that {K[Z(T)])ej = K{^^)' Because there is a natural field epimorphism from {K[Z{T)])e onto {K[Z{T)])ej^ it follows that the fields {K[Z{T)])ej ^ K(CL) and {K[Z{T)])e ^ K{U) are isomorphic. Hence gcd(j, m) — 1. Replacing ^^ by ^^, if necessary, we may then assume that ei is a summand of e. Since any A'-automorphism a of A'(^^) can be extended in a natural way to a A'-automorphism of A^(^77i)[2^(T)], and since (7(e) = €, it follows that C7(ei) is also a summand of e. Clearly cr(ei) = e^ with gcd(j, m) = 1 since (jiXm) is also a primitive mth root of unity. Then

is a summand of e, where Q = Gal(A'(^r^)/A'). Since e is primitive,

e = 5^a(ei). To prove that e is central in A'^L, we have to show that yey~^ = e for any y £ L. li y and x commute, then, since H is central in K{^rn)L (by condition (1), H is normal in L), it is clear that ycr(ei)y~^ == <^(^i); hence yey~^ = €. So suppose that x and y do not commute. Then, since {x) is a normal subgroup of L and because s (which has order 2) is the only nonidentity commutator, it follows that s = x^/^, where k is the order of X, Because of assumption (2), there exists r £ Q with r(^fc) = ^^ ' ^ Without loss of generality, we may assume that ^m — ik • ^^ '^ restricts to a A'-automorphism of A''(^^). It follows that

1. CENTRAL IDEMPOTENTS It remains to show t h a t r((j(6i)) = y(T{ei)y~^.

For this, write

i=0 where each ej(t) G Ki^m) / = y€jy~^

345

teZ(T)

^nd j is some integer with I < j < m — 1. Write

= ^t£Z(T)f(^)^'

^^

compute the coefficients

f{t).

Let i be an integer and let h ^ H, Write i = vm +1 for integers v and t with 0 < ^ < m — 1. From the definition of e^, it follows t h a t CjytlX ) = CjytlX

X j = n^'^ ~ n^'^'

because x ^ ^ G ^ . On the other hand, since x^ is central precisely when i is even and because yx^y~^ = sx^ = x^^'^x^ when i is odd.

/(/ix')= ; if i is even. It then easily foUows t h a t

/(/la;') = -^

i^((V2)+l)..-

jf . i3 ^^^^

Hence

thus yej{t)y-^ as required.

= ^i^jit)) for all / G 2:(T) = {H, x), so y(7(ei)?/-i = r((7(ei)) D

1.11 E x a m p l e . Contrary to what is stated in [ C M 8 8 ] , the following example shows t h a t condition (2) in Theorem 1.10 cannot be weakened to "A' does not contain roots of unity of order n whenever T contains a noncentral element of order n " . Let G = {x,y

\ yxy'^

and it follows t h a t Z{G)

= x^,x^ = {x^,y^)

= 1).

Then x^ and y^ are central

and G/Z{G)

^ C2 x C2. Hence L =

M{G^ —l^y'^) is an RA loop with T{L) = {x) a cyclic group of order 8. Let K = Q(^8 + ^8^^) = Q ( \ / 2 ) . As a real field, K has only ± 1 as roots of unity.

346

XIII. IDEMPOTENTS AND FINITE CONJUGACY

Write 7

ei^lY^iM' i=0

7

and

e, =

lJ2i^;'xr. i=0

These two elements are orthogonal primitive idempotents in K(^s){^) and ^1 + ^2 ^ J^{^) is an idempotent, with coefficient of x equal to y/2 and coefficient of x^ equal to —\/2. It follows that the coefficient of x in y{ei + e2)y~^ is — \/2. Hence this idempotent is not central in KG and thus not central in KL^ although all the other conditions of Theorem 1.10 are satisfied. If n is even and ^n is a primitive nth root of unity over Q, then the map in ^ ^(''/2)+^ = -^n defines an element of Gal(Q(^n)/Q). Thus the next theorem is an immediate consequence of Proposition 1.8 and Theorem 1.10. 1.12 T h e o r e m . Let L = M{G^^^go) be an RA loop with torsion suhloop T. The following conditions are equivalent. 1. Every idempotent of QT is central in QL. 2. Every suhloop of T is normal in L and T is either an abelian group, a hamiltonian Moufang RA 2'loop or a hamiltonian group with the property that ifT contains an element of odd order n, then the multiplicative order of 2 (mod n) is odd,

2. Nilpotent elements We apply the results of the previous section in order to decide whether or not an alternative loop algebra contains nontrivial nilpotent elements. 2.1 Theoremi. Let K he a field of characteristic p > 0 and let L = ^{G^ *,5'o) be an RA loop with torsion suhloop T. Then KL has no nonzero nilpotent elements if and only if the following conditions are satisfied. 1. L does not contain elements of order p (in particular, p / 2). 2. Every suhloop ofT is normal in L. 3. If p z=z 0 and T contains a noncentral element of order n, then, for ^^y in) 0. primitive nth root of unity over K, the map a defined by cT(^n) = ^n'^^""^ is in Gal{K{(n)/K)Furthermore, (a) T is an abelian group, or

2. NILPOTENT ELEMENTS

(b) T is a hamiltonian

347

group and, if it contains

an element

of odd

order k, then the field K{Ck) does not contain solutions

of the

equation x^ + y^ =: —I, or (c) T is a hamiltonian solutions

Moufang RA 2-loop and K does not

contain

of the equation x'^ + y'^ + z^ + w'^ = —1.

4. Ifp > 2, then (a) Tpi is an abelian group and (b) if Tpt is not central, then the algebraic closure Fp of Fp in K is finite

and, for all t G Tpi and i £ L, there exists a positive

multiple r of[Fp: Fp] such that M~^ PROOF.

= t^"^.

Assume t h a t KL has no nonzero nilpotent elements.

Then,

from the description of the prime radical in Theorem VI.3.4, we see t h a t L does not contain elements of order p. We claim, furthermore, t h a t all idempotents of KT are central in KL. KT and let x G KL.

Indeed, let e be an idempotent in

Since the ring generated by x and e is associative,

(ex(l - €))2 = ((1 - e)xey

= 0. Hence ex{l - e) = {\ - e)xe = 0. So

ex = exe = xe and it follows from Corollary III.4.2 t h a t e is central in

KL.

T h a t the mentioned conditions are necessary now follows from Theorems 1.6 and 1.10. We now show t h a t the conditions are also sufficient. By Theorems 1.6 and 1.10, all idempotents of KT are central in KL.

To prove t h a t any nilpo-

tent element of KL is trivial, we may assume that L is finitely generated and hence t h a t T is finite. Since T does not contain elements of order p , the algebra KT is Jacobson semisimple (Corollary VI.4.3). It follows t h a t KT = 0 g ( A'^T)e is a direct sum of division rings, where e runs through the finitely many primitive idempotents of KT.

Since each such e is central in

KL^ we obtain KL = 0 g ( A ' L ) e , a direct sum of two-sided ideals, and so it is suflScient to show t h a t each algebra {KL)e

has only trivial nilpotent

elements. As in the proof of Lemma XII. 1.1 leading to equation (XII. 1), any nonzero element x in {KL)e

can be written uniquely as a finite sum

X = diqi + ••• + dkqk, where each qi belongs to a transversal of T in L, and each di is a nonzero element of {KT)e.

Moreover, (diqi){djqj)

= d(qiqj) for some nonzero d G

348

XIII. IDEMPOTENTS AND FINITE CONJUGACY

{KT)e. Now a "leading term" argument as in the proof of Lemma XII.1.1 shows that {KL)e does not contain nonzero nilpotent elements. D 2.2 Corollary. Let L — M(G, *,^o) be an RA loop with torsion subloop T. Then QL has no nonzero nilpotent elements if and only if 1. every subloop ofT is normal in L and 2. T is either (a) an abelian group, or (b) a hamiltonian group and, ifT contains an element of odd order n, then the multiplicative order of 2 (mod n) is odd, or (c) a hamiltonian Moufang RA 2-loop.

3. Finite conjugacy In this section, we find KL of an RA loop to depend on whether or torsion, we distinguish is based upon a paper

necessary and sufficient conditions for a loop algebra have a finite conjugate (FC) unit loop. Our results not L is a torsion loop and, for loops which are not the semisimple and nonsemisimple cases. This work by E. G. Goodaire and C. Polcino Milies [GM96a].

3.1 Lemma. Let L be an RA loop with unique nonidentity commutatorassociator s. Assume x and y are noncommuting elements of L such that y has order a power of 2 and the group (x,y) is finite. If K is a field of characteristic 2 and z is a central element of L with z ^ {x^y), then y has a conjugate y(z) inU{KL) with 1. zv G supp(y(2:)), for some v G (x^y), and 2. supp(y(z)) C |Jt=o ^^{^-iV)^ where k is the smallest positive integer such that z^ G {x^y) if such an integer exists, and otherwise k = oo. If Z{L) is infinite, y has infinitely many conjugates of the type y{z). Since char K = 2 and y has order a power of 2, it follows from Theorem VI.3.4 that I + y £ V{KL)^ the prime radical of KL, Hence, zx{l + y) is nilpotent, say of index n, and therefore /i r= 1 + zx(l + y) is an PROOF.

3. FINITE CONJUGACY

349

invertible element. Then fiyfJ^~^ is a conjugate of y and we note that My/^~^ = (1 + ^^(1 + y))y . (1 + zx{l + y) + {zx{l + y)f + • • • + {zx{\ +

yW'^)

= y + zx{l + y)y + yzx{\ + y) + • • • + y{zx{l + y))""^ + zx{l + y)yza:(l + y) + • • • + 2:x(l + y)y{zx{l + y ) r ~ ^ If no positive power of z belongs to (a:, y), then the terms involving different powers of z have no elements of their supports in common. The sum of the terms containing only the first power of z is zx{\ + y)y + yzx{\ + y) = z{xy + yx){l + y) - z{xy + sxy + xy'^ + sxy'^), which has four distinct elements in its support; in particular, fJ^yf^'^ has zxy in its support and, furthermore, supp(/x?//x~^) C [j^i^Q z^{x^ y). On the other hand, suppose there exists a positive integer k such that z^ G (a:,y). Let k be the smallest such number. Since z ^ (a:,y), by assumption, fc > 1. It follows that

= y + z{xy + yx){l + y)[l + z\x{y + 1))^ + z''\x{y + 1))^^ + • • •] +z^xy + yx){l + y)[x{l + y) + z^{x{y + 1))^+^ + . . . ] + . . . +z^-'{xy + yx){l + y)[{x{l + y))'-^ + z^{x{l + y))^^'^ + • • • ] . By the choice of A:, the terms involving z^^ have no elements in their support in common with terms involving z^'^^^ for i = 2 , . . . ,fc— 1. Since

is an invertible element and because z{xy + yx){l + y) ^ 0 (as above), it follows again that z{xy + yx){l + y)[l + z\x{y + I))' + z^'{x{y + l)f'

+ • • •] ^ 0

and so zv € supp(//7/^~^) for some v G {x^y). Furthermore, it is clear that supp(/xy/x-^) C UfrJ z'{x,y). Finally, if Z{L) is infinite and {zi | i = 1,2,...} is an infinite set of central elements such that ZizJ^ ^ (x, y) for i ^ j , then {y{zi) | i = 1,2,...} is an infinite set of conjugates of y. D

350

XIII. IDEMPOTENTS AND FINITE CONJUGACY

3.2 L e m m a . The unit group of the classical quaternion

algebra H(Q) is

not FC. PROOF.

Write H(Q) = Q H - Qi + Qj + Qij with (i, j ) = Qs- Then, for

any nonzero rational g, the element 1 + gi is invertible in H(Q) with inverse YjrrO- — qi)- Furthermore 1

(1 + qi)j{l + qi)-^ = -^-hiiJ + W ) ( l - ^0 1 + ^2' 1

r(i + qij + qij + g^t^j)

2'

1+ 9 ^ Since ^ i ^ 7^

((1-^2)^.^2^.^').

? ;^2 for g'' 7^ ^ 9 ^ ~ \ it follows t h a t j has infinitely many

conjugates.

D

Lemma 3.2 is a consequence of a more general result of A. I. Lichtman concerning the existence of noncyclic free subgroups in noncommutative division rings [Lic82]. In particular, it is known t h a t the unit group of H(Q) contains a noncyclic free group. 3.3 L e m m a . Let K be an infinite field of characteristic

2. Then KQs

is

not FC. PROOF.

Write Qs = ( a , 6 | a^ = l , a ^ = 6^, 6a = a^b).

Then, for any

k G A^ the element 1 + A;(l + a) is invertible with inverse (1 -\- k{\ + a))^, since (1 + aY = 0. Now (1 + k{\ + a))b{\ + k[\ + a))-^

= (1 + k{\ + a))b{\ + k{\ +

a)f

= 6(1 + k{l + a^))(l + A;(l + a) + k^{l + a)^ + k^(l +

af)

= 6(1 + k{a + a^) + k'^{l + a + a'^ + a^)), so, if A: 7^ 0 and k y^ 1, it is clear t h a t ba appears in the support of this element with coefficient k. Hence 6 has infinitely many conjugates.

D

3 . 4 L e m m a . Let K be a field and L a loop such that KL is alternative.

If

U{KL) are

is FC and the set K U Z{L)

central.

is infinite,

then all idempotents

of KL

3. FINITE CONJUGACY PROOF.

351

Assume e is a noncentral idempotent of KL,

Let t ^ Lh^

such

t h a t either et{\ — e) ^ Q or {I — e)te ^ 0. Suppose n — ei{\ — e) / 0. (The other case is proved similarly.) Then n^ = 0, so ^ = 1 + n is a unit in

KL.

For 0 7^ z G A^ U Z{L)^ the element 1 + zn \s s. conjugate of jj, because [ze + (1 - eMze

+ (1 - e)]"^ = [ze + (1 - e)]{\ + n)[z-^e = [ze + zn+{\-

+ (1 - e)]

e)][z-^e + (1 - e)]

= 1 + zn. Since the support of n is finite and because, by assumption, there exist infinitely many units z G A U 2 ( L ) , it follows t h a t /i has infinitely many conjugates, a contradiction.

D

For torsion RA loops L and for any field K^ it is particularly simple to determine when U{KL)

is F C .

3.5 T h e o r e m . Let L he a torsion U{KL)

is FC if and only if K and L are

PROOF.

U{KL)

RA loop and let K he a

field.

Then

finite.

If A' and L are finite, clearly U{KL)

is F C . Conversely, assume

is F C .

We first show t h a t the characteristic of K is not zero. contrary. Then ZL C KL and thus U{ZL)

Assume the

is FC as well. Since L is not

an abelian group, it follows from Corollary XII.2.14 t h a t L is a hamiltonian Moufang loop. In particular, QQs = 4Q © H(Q) is a subring of KL.

This is

impossible, however, because of Lemma 3.2. Thus char A' = p > 0 and the prime subfield of K is Fp, the field oi p elements. Assume t h a t p ^ 2. Because L2, the 2-torsion part of L, is an RA loop as well and because L is locally finite (Lemma VIII.4.1), L contains a finite RA 2-subloop LQ. Because p j( |Lo|, the loop algebra KLQ is a direct sum of fields and Cay ley-Dickson algebras, by Corollary VI.4.8. Furthermore, by Proposition XI.3.4, these Cayley-Dickson algebras are either split or of the type (A^, —1, —1, —1). The latter are also split because (Fp, —1, —1, —1) is split (Proposition 1.9 and Proposition XI.3.3). Consequently, KL

con-

tains noncentral idempotents. By Lemma 3.4, K U Z{L) is finite and, since \L/Z{L)\

= 8, KL is also finite.

Finally we consider the case p = 2.

Since L contains an RA 2-loop

and the torsion subloop of L is locally finite, certainly L contains a finite

352

XIII. I D E M P O T E N T S AND FINITE CONJUGACY

noncommutative 2-group (x^y).

Lemma 3.1 yields t h a t 2{L)

is finite; t h u s ,

L is finite. So it remains to show t h a t K is finite. Assume the contrary. Then, by Lemma 3.3, L does not contain Qg- Since L is not an abelian group, not every subloop of L is normal and so, since elements of odd order are central, there exist y^t £ L^ o(t) a power of 2, such t h a t y~^ty — st ^

(t).

(Here, as always, s denotes the unique nonidentity commutator-associator of L.) Let k G A' and let /j. = I + k X^^ig o(t)-\

^^- Then fi~^ = ji and

o(t)-\

i=0

2=0

o(t)-l

o(t)-l

o(t)-l

= y + ky Y,it' + y~'t'y) + f''y{i2 y~'^'y){Il ^02= 0

2=0

2= 0

We compute the coefficient of yt in this sum. Suppose y{y~^fy)

— yt. If i

is even, then f is central, so f~^ = 1, contradicting the fact t h a t t has even order. On the other hand, if i is odd, then t = y'^t^y the fact that s ^ (t). Thus y{y~^t^y) with i,j

contradicting

/ yt for any i. Suppose y~^VyP

—t

G { 0 , . . . ,o(^) — 1}. If i is even, then f"*"-^ = ^, so j is uniquely

determined by i.

If i is odd, then y~^Vyt^

again contradicts s ^ (t). k + ^^k^

— st\

= t implies sV^^

— t^ which

It follows that the coefficient of yt in ^y^~^

is

which is fc or A: + ^^ since p = 2 and t has order a power of t.

In any case, yt G supp(/it/^~^) for infinitely many k. Hence y has infinitely many conjugates, a contradiction. So, indeed. A' is

finite.

D

We now investigate the situation with loops L which are not torsion and, again, we begin with a series of lemmas. 3.6 L e m m a . LeA K he a field of characteristic loop with unique nonidentity

commutator-associator

E = {e ^ KT \e IfU{KL)

p > 0 and let L be an RA

= e,e central in

s. Let KL}.

is FC, then there exist only finitely many nonzero elements

form e(l — s) with e ^ E. In particular, with s ^ H and if H contains PROOF.

of the

if H is a torsion subgroup of

no elements

of order p, then H is

Z{L)

finite.

Let x and y be noncommuting elements of L. Suppose e is an

idempotent in KT which is central in KL.

Then (1 — e) + xe is a unit in

3. FINITE CONJUGACY

353

KL with inverse (1 — 6) + x~^e and [(1 - e) + xe]y[{l - e) + x~^e\ = (1 - e)y + esy = y - ey{\ - s). li f ^ E and e(l — 5) 7^ / ( I — 5), then these idempotents clearly yield different conjugates of y. Hence the first part follows. For the second part, assume H is an infinite torsion subgroup of Z{L) without elements of order p and with s ^ H. Then H contains an infinite strictly ascending chain of finite subgroups

Each idempotent Hi — Hj is central in KL. Furthermore, since s ^ Hi^ it follows that

for i 7^ J, in contradiction with the first part.

D

3.7 Lemma. Let K he a field of characteristic p > 0. Let L be an RA loop with torsion subloop T ^ L. IfU{KL) is FC, then T is an abelian group. Since squares of elements in L are always central, the centre Z{L) of L contains a nontorsion element, so Z(L) is infinite. Because of Lemma 3.4, all idempotents of KL are central. Suppose p > 2. Then it follows from Theorem 1.6 that the loop Tp/ is an abelian group and so elements of even order commute. Since elements of odd order are always central, this yields that T is commutative and hence an abelian group. (See Corollary IV. 1.3.) If p = 2 and the loop T2 is not central in T, then, since T is locally finite, T contains a finite noncommutative subgroup. By Lemma 3.1, ll{KT) is not FC, a contradiction. Thus T ^ T2 X T2' (Proposition V.1.1) is an abelian group. Finally, if p — 0, since all idempotents of KL are central, either T is an abelian group or T contains Qs as a subgroup, by Theorem 1.10. However, the latter is not possible since H(Q) C QQg Q KT and ZY(H(Q)) is not an FC group, by Lemma 3.2. D PROOF.

3.8 Lemma. Let K be a field of positive characteristic p and let L be an RA loop with torsion subloop T ^ L. IfU(KL) is FC and L contains an element of order p, then p — 2 and T = L' x A for some finite abelian group A of odd order.

354

XIII. IDEMPOTENTS AND FINITE CONJUGACY PROOF.

Let x and y be noncommuting elements of L, let z be a non-

torsion element in Z{L)

and let ^ G i^ have order p. By L e m m a 3.7, the

loop T is an abelian group. Suppose p /

2. Then, since elements of odd order are central, ^ =

Yl^Zo 9^ ^^ ^ central element of KL with ^

= 0 = (z^x^^

for any i > 0, so

1 + z^x^ is invertible and (1 + z'xg)-'y{l

+ z'xg)

= {1 - z'xg)y{\ = y + z'giy^

+

z'xg)

- xy) = y + z'g{\

-

s)yx,

where V — { l , ^ } . Since s ^ (^), it is clear t h a t yx G supp(^(l — s)yx)

and

thus z'yx e s u p p ( l + z'x^y^y{l

+

z'xg).

Because z has infinite order, y has infinitely many conjugates, a contradiction. Thus p — 2 and so, by Lemma 3.6, T21 is finite. Next we show that if t is an element of order 2, then t is central in L. Assuming the contrary, \et y ^ L be such t h a t yt 7^ ty. Thus yt — sty.

Let

i > 1 be an integer and consider the following conjugate of y:

[1 + z\l + t)]-'y[\ + z\\ + t)] = [1 + ^^(1 + t)]y[\ + z\l + t)] = [y+z'{y

+ syt)][l +

z'{l+t)]

= y[l + z\l

+ st)][l + z'{l + t)]

= y[l + z\t

+ St) + z'^\\

+st + t + s)].

Since z has infinite order, z'^yt is in the support of this element. It follows t h a t y has infinitely many conjugates, a contradiction. So elements of order 2 are central. Next we show that if t is an element of order 2, then t — s. For this, let i > 1 be an integer and consider the following conjugate of y: [1 + z'x{\

+ t)]-^y[l

-f z'x{\

+ t)] = [1 + z'x{\ ^y

+ z'xy{\

+ t)]y[\ + z'x{l

+ t)]

+ t){\ + s)

= y + z'xy{\-\-t

+ s + ts).

If 5 / ^, then z'^xy belongs to the support of this element, again yielding a contradiction. So indeed t = s. Thus every subgroup of the abelian group T2 contains s.

3. FINITE CONJUGACY

355

Remember that T = T2 X T21 (Proposition V.1.1). We prove that T2 is cyclic of order 2 and hence that T2 = {1? 'S}. If this is not true, there exists g E T2 with g'^ = s. Assume g is central and let n = 1 + ^. Then n"* = 0 and, for any integer i > 0, (1 + z'xn)y{l

+ z'xny^

= (1 + z'xn)y{l

+ z'xn + {z'xnf

+

{z'xnf)

= y + z^n{l + s)yx (l + z'^nx + (z*n)^a;^) = y + z'n{\ + s)yx = y + z'{l + g + g^ + g^)yx. Since z^yx is in the support of this conjugate of y, it follows that y has infinitely many conjugates, a contradiction. So g is not central. Let h £ L be such that g and h do not commute. Then [h(l + g)f = hih + gh){l + g) = h{h + shg){l +g) = h\l

+ sg)il +g) = h''g{\ + s).

Since (1 + s)^ = 0, we obtain [h{\ + g)Y = 0. Consequently, for any integer i > 0, [\ + z'h{\+g)]g[l

+ z'h{\ + g)]-'

= [1 + z'h{l + g)]g[l + z'/i(l + 5) + (z'/i(l + g)f + (z'/i(l + p ) f ] = 5 + [z'/i(l + 5)ff + gz%\

+ 5) + 3(z'/i(l + 5)]2

+z'/i(l + 5)P-^'/^(l + ff) +ff[-^'Ml+ 9)f +z'/i(l + 5)5[-^'Ml + 9)? + ^'Ml + 9)9Wh{'^ + ^)]^ = 5 + z'(/i5 + gh){\ + ff) + ^2^(^/1 + hg){\ + p)/i(l + 5) +z^\gh + /ip)(l + 5)/i(l + g)h{l + 5)

= ff + z^h{\ + 5)(5 + 52)[i + z^h{\ + 5) + ^ " ' ( ^ 1 + 9)?]Since (1 + 5)(p + g'^) is central in KL (Corollary III.1.5) and because (1 + ^){g + g^){^ + 5^) = 0, it follows that [1 + z^h{\ + ^)]^[1 + z^h{\ + g)]-^ =g + z'h{\ + s){g + y^). So z'^h is in the support of this conjugate, again a contradiction.

356

XIII. IDEMPOTENTS AND FINITE CONJUGACY

We have shown t h a t p = 2 and T2 = L\

Thus T = V X A, where

A = T21 is a finite abelian group of odd order.

D

3.9 L e m m a . Let K be a field of characteristic loop with torsion subloop T = L' = { l , ' ^ } .

2 and let L be an

Then, for any /J^^i^ E:

RA

U{KL),

fx~^i/fx = 1/ or si/. It is well known t h a t the units of the Laurent polynomial ring

PROOF.

K[X,X~^]

are trivial; t h a t is, of the form kX'^ with 0 ^ k e K and i G Z.

(This can be shown easily, as in the proof of Lemma VL2.1.) It then follows by an induction argument t h a t all the units in Laurent polynomial rings in several commuting variables are also trivial. So, if yl is a finitely generated torsion-free abelian group, then all units in KA are trivial as well, because KA = K[Xi^X^^^,..

, X r i , X ~ ^ ] . Obviously this implies t h a t all units in

KA are trivial for an arbitrary torsion-free abelian group A. By assumption, L/L^ is a torsion-free abelian group, so every unit of K[L/L^] is of the form ka for some k G K\ A; 7^ 0, and a = aV G LIL'. KL{\

-f s) is the kernel of the natural epimorphism eii: KL

every unit /i G U{KL)

-^

Since

K[L/L^]^

can be written in the form fi — ka + a ( l + 5), where

k £ Ii\ k ^ 0^ a ^ L and a G K L. As both a^ and a{l + s) are central in KL (Corollary III. 1.5), it is clear t h a t fx~^ — k~^a~^ + k~'^a~^a{l

+ s).

Consequently, for any g £ L^ figfi~^ = [ka + a ( l -|- s)]g[k~^a~^ = [kag + a ( l + s)g][k~^a~^ = {ka)g(k-^a-^)

+ A:"^a"^a(l + s)] + A:"^a~^a(l + s)]

+ kag[{k-'^a-^

a(l + s)] + a ( l +

s)g{k-^a-^)

= aga~^ + {k~^ag)a~'^ a{l + s) + [fc"^a(l + s)g]a~'^ = aga~^ -f k~^[{aga~^

+ g)a~^]a{\

+ 5),

using diassociativity to write aga~'^ as {aga~^)a~^,

Since (1 + 5)^ = 0 and

aga~^ -\- g \s 2g or (1 + 5)^, we obtain /J^gfJ^'^ = aga~^ G {g-tSg}. Thus the lemma holds in the special case i/ = g £ L. In general, if 1/ G U{KL)^

then

u = kiai + a i ( l + 5), 0 7^ fci G K^ ai G I^, a i G KL^ and fii/fj,"^ = li[kiai + a i ( l + s)]ii~^ — k\\ia\[i~^ Since \ia\\i~^

-f- a i ( l + s).

— a\ or sa\^ as already seen, and since s{\ -\- s) — \ -\- s., the

lemma follows.

D

3. FINITE CONJUGACY

3 . 1 0 T h e o r e m . Let K he a field of positive

357

characteristic

p and let L be

an RA loop with torsion suhloop T ^ L. Suppose T contains order p. Then U{KL)

an element

of

is an FC loop if and only ifp = 2 and T = V x A^

where A is a finite abelian group of odd order. PROOF.

T h a t the conditions are necessary follows from L e m m a 3.8. For

sufficiency, first note t h a t as a finite group of odd order, A is in t h e centre of L. Since \A\ is relatively prime to p., the commutative group algebra KA is semisimple artinian and thus a direct sum of m fields Ki — {KA)ei., Ci is a primitive idempotent of

where

KA.

Suppose // and u are units of KL,

Let 5* be a finitely generated RA

subloop of L containing supp(/^) U supp(i/) U A.

Since SjS'

is a finitely

generated abelian group, we can write SIS'

^ BJS'

X

B2/S\

where B\ and B2 are subloops of S containing S\

B2/S'

consists of those

elements of odd order in S/S'

and B\IS'

has no elements of odd order.

Hence the finite group B2/S'

is isomorphic to v4 = T2', so there is an

epimorphism (^: S -^ A with (p{a) = a for all a G A. Therefore, S — SiA^ S\ = ker(/?, and, since A is central and Anker Thus KS — {KA^S\

— 0 ^ | KiS\.

(f — {1}, we have S — S\X

A.

Since T = L^ x A^ the torsion subloop

of 5*1 is 5J = L' = {1,5}. Also, S is an RA loop, so, by Lemma 3.9, for i — 1,... ,m, {liei){uei){fiei)~^ Hence ^vji'^

— Y^^=\^i{^)^^i^

= uci

or

svci.

with ^i{s) G { l , ^ } . Consequently, v has at

most 2 ^ conjugates in KL and U{KL)

is indeed an FC loop.

D

3.11 L e m m a . Let K be a field of characteristic

different from 2 and let L

he an RA loop with torsion suhloop T. IfU{KL)

is an FC loop and T is an

infinite abelian group, then any subloop of T is normal in L. PROOF.

Let s be the nonidentity commutator-associator of L.

Since

noncentral subloops H of L are normal in L if and only ii s ^ H (Corollary IV.1.11), it is sufficient to show t h a t if t G T is not central in L, then s ^ {t).

Suppose the contrary and let ^ G T and y e L he such t h a t

yty-'^ = st ^ (t). Then [{t - l)yT]'^ = 0, so 1 + (^ - l ) ^ ? i s a unit in

KL.

358

XIII. IDEMPOTENTS AND FINITE CONJUGACY

Because tt = t^ we obtain [l + {t-

l)yt]-h[l

+ (t-

l)yt] = [1 - (^ - l)yt]t[l = t-{t-l)yt

+ (t-

+ t(t-

l)yt]

\)yt

= t ~ 2tyt + yt + t^yt = t - 2sy?+ 2yt because t'^ is central. Since yt G supp(2t/?) but s ^ {t)^ we have y in the support of the conjugate [I + (t - l)yt]-h[l

+ {t -

l)yt].

Since T is commutative, (tiy^t) = {y-,t) / 1 for any t\ ^ T (see Theorem IV.1.14 and Corollary IV.1.3), so, since T is infinite, t has infinitely many conjugates, a contradiction. D For a prime p, we denote by Z(p^) the ;?-Priifer group, that is, the additive group Z(p^) = Q ( P ) / Z

where

Q^^^ = { ^ | a,n G Z}.

Clearly Z(p^) is a divisible abelian group. Less clear, but a standard fact, is that every proper subgroup of Z(p°^) is finite, a property which characterizes the p-Priifer group within the class of infinite abelian p-groups. (See [Rot95, Chapter 10] for details.) For an abelian group A and a positive integer n, let >1^ = {a^ I aG A], Obviously, A^ is a subgroup of A. 3.12 Lemma. [Pas77, Lemma 14.3.2] Let A be an abelian p-group.

Then

1. any divisible subgroup of A is a direct factor of A; 2. A has a unique maximal divisible subgroup dA; 3. if dA ^ {I}, then dA is a direct product of Z(p^)-groups. Let D be a divisible subgroup of A. By Lemma XII.2.12, the identity map I: D -^ D extends to a homomorphism (p: A ^^ D and it is easily verified that A = D x ker(/?, giving (1). PROOF.

3. FINITE CONJUGACY

359

To prove (2), let dA be the subgroup generated by all divisible subgroups Di of A, i G X, some index set. Then {dAf

= {D\ \iei)

= {Di\ieJ)

= dA.

Thus dA is also divisible and therefore the unique maximal divisible subgroup. For (3), consider the nontrivial subgroup E = {d ^ dA \ d^ = 1}. Since every nontrivial element of E has order p^ E = Ylieli^'^)

^^ ^^^ direct product

of cyclic groups of order p. Since dA is divisible, for each z G X we can choose a sequence of elements of dA defined by di^i = di and d^^^-j^ = di^n- Then Di = {di^n I ^ > 1) =

Let F = {Di\i

AP"^)'

e I).

Clearly F is a divisible

subgroup of dA^ so dA = F X S for some subgroup S of dA^ by p a r t (1). Since E C F^ the group S must be trivial, so F = dA. It remains now to show that dA is in fact the direct product of the subgroups Di. To see this, suppose t h a t a\a2' - -at — 1 with aj G Dj. If p'^ is the largest order of the elements aj and \i p^ > 1, then 1 = (aia2 •• -atY^''

= a{^

oF^

"'O^r

^ (^i) X (^2) X ••• X (d^),

a contradiction. Thus p ^ = 1 and the product is indeed direct.

D

The following lemma is essentially proved in [SZ77]. 3 . 1 3 L e m m a . Let F C K be fields of characteristic positive

integer such that k'^ ^ F for each k ^ K.

p = 0 or gcd{m^p) PROOF.

p > 0 and let m be a If \F\ > m and

either

— I, then K = F.

Let a e K. Then

(1 + far

- 1 = m / a + ( ^ ) ( / a ) 2 + • . . + ( / a ) - = cj e F

for each f £ F. Since | F | > m, there exist distinct elements / i , . •. , / m in F. Then fj^i

+ fj^2

+ '•' + f'j'xm

= c/^,

\ <3

<m,

360

XIII. IDEMPOTENTS AND FINITE CONJUGACY

is a system of linear equations in the variables x i , . . . , Xm with determinant

n • • /r

/l

/2

/I

fi

Jm

J m

m fm

So its (unique) solution is ma^i^^a^^...

7^0.

,a^.

Therefore a ^ F.

F ^ K.

Thus D

The following lemma is taken from [Mil78]. 3 . 1 4 L e m m a . Let K be an infinite field of characteristic be a group which does not contain elements group, then each torsion element PROOF.

p > 0 and let G

of order p. IfU{KG)

is an FC

of G is central.

By Lemma 3.4, each idempotent of KG is central, a fact we

will use several times. Let ^ G G be a torsion element of order r and let g E G. The unit group U[K{t^g))

is clearly an FC group and hence a finite extension of its centre.

(See §XII.2.2.) Therefore, there exists a positive integer n such t h a t //^ is central in K{t^g)

for any fi G U{K{t^g)).

Since r is invertible in A', the

idempotent e = r ~ ^ ( l + t + - - - -\- T " ^ ) is central in KG.

Hence (t) is a

normal subgroup of G. First we deal with the case p = 0. Because the group algebra K{t) is semisimple, we can write K{t) = ®Ki as the direct sum of field extensions Ki of K. Because (t) is normal in G, conjugation by g defines an automorphism (f of A'(^). As each Ki = K{t)ei

for some idempotent e^ and because

Ci is central in KG^ ^i^^'i) = ^^f Let Kf

be the fixed subfield of A'^; t h a t

is, K^ consists precisely of those elements fixed by (f. Then, by the above, k'^ G A'^^, for any ki G Ki.

Since Q C Kf^ the field K^ is infinite, so, by

Lemma 3.13, Ki = K'f. Thus ip is the identity m a p , so tg = gt. Secondly, we consider the case p > 0 and write n = p^m with z > 0 and g c d ( p , m ) = 1. Assume first t h a t K is an algebraic extension of its prime subfield Fp. Since K is infinite, there exists a finite subfield E of K with |A^| > m. By Theorem X.2.1, E{t)

= © £ i , where each Ei = £^(6)? ^i ^ primitive root

of unity over E. Since (t) is a normal subgroup of (^,^), conjugation by g

3. FINITE CONJUGACY

defines an automorphism (/?: E{t) -^ E{t),

361

Since each Ei = E{t)ei for some

primitive idempotent €{ G E{t) and because €{ is central, we obtain t h a t ^{E^) = E„ Let Ef denote the fixed subfield of Ei. ^p m ^ ^^^. Since all the fields Ef

Given a^ € £^t, we know t h a t

are finite and because the Frobenius

homomorphism on a finite field is an isomorphism, there exists bi G Ef such t h a t oJ^^

— b^ ; thus a ^ = bi G E^ for all z. It now follows from

Lemma 3.13 t h a t Ei = Ef.

Hence ip is the identity and so tg — gt.

Now assume t h a t K is not an algebraic extension of Ep. Let 6 G K be an element which is transcendental over Ep. Since U{Fp{0)G)

C U{KG)

also an FC group, we may assume t h a t K = Fp{6). Set F — Fp{6^).

is

Then

E — {/jP^ I k G A'} and, furthermore, if e is an idempotent of A (^), then e = e^^ G F{t'P'').

So both group algebras K{t)

and E{t) have the same

primitive idempotents. Since both algebras are also semisimple, we have A'(^) = e A ' , and

F(^P')

= e F , where F, = F{t'P')e,,

primitive idempotent in F{t).

R\ = K{t)e,,

e, a

Also, F^ = A'f .

Again, we consider the automorphism (/?: A'(/) —> K{i) which is conjugation by ^ G C and denote by K'f and Ff the fixed subfields of A\ and F^, respectively. For an element fi G F^, we have //^ ^ = a^ G Ff. a^ = 6^ for so ne b^ G A ^ Since ^{ai)

— ai^ we obtain b\

Write

— ^{biY''

and

thus b, G A f . Hex.:e //^ = 6, G A f H F, = /;^. So Lemma 3.13 yields t h a t F^ = F'f \ thus the restriction of (f to F(^'^ ) is the identity. So gt''^ = f^ g. Since the order of t is relatively prime to p, we again see t h a t / and g commute.

D

3.15 L e m m a . [ P a s 7 7 , Lemma 14.4.3] Let p be a positive G — T{p^).

Suppose K is a field of characteristic

does not contain

a root of unity of order p^ for some positive

Then any idempotent idempotents extension

prime

e G AK{G)

e^ Furthermore, of K generated

is a finite sum of orthogonal

for each i, {KG)ei

integer

k.

primitive

= F where F is the field

by all p^th roots of unity for all n.

finite abelian group whose order is invertible

and let

different from p which

in K, then for any

If B is a idempotent

e G A/v'(G'), the algebra {K[G X B\)e is a finite direct sum of fields. PROOF.

Let e G A x ( G ) be an idempotent and let E — {KG)e.

Suppose

M is a maximal ideal of the algebra E. Denote by A the natural homomorphism E -^ E/M.

Because G is commutative, A(F) is a field extension of

362

XIII. IDEMPOTENTS AND FINITE CONJUGACY

K generated by the multiplicative group X{Ge). As a homomorphic image of Z(p°°), we have A(Ge) = {1} or X{Ge) ^ l{p°°).

If t h e former, however,

we would have A(Ai^(G)e) = 0, a contradiction, because e G A(e) ^ 0. Thus \{Ge) ^ Z(p°°), so \(E)

A.K{G)

and

^ F. Clearly, we may assume t h a t

XiE) = F. Since, by assumption, K does not contain a root of unity of order p^ for some positive integer k^ it follows from Theorem X.4.4 t h a t there exist roots of unity ^j and ^2 of orders p^^ j > 2, and p^^ respectively, such t h a t [K{^j):

/i^(^2)] > P- Since ^j € A(Ge), we can choose g E G with

X(ge) = ^j. Let H be the finite subgroup of G generated by supp(e) U {g}. Then H = (h) is cyclic with, say, X{he) = ^ ^ , a root of unity of order p ^ ; clearly, m > j . Consider the epimorphism a: KH by (T{a) = A(ae).

Because KH

-^ K(^m)

given

is semisimple, there exists a primitive

idempotent / G KH with cT{f) = 1. Clearly fe = f,so

E = Ef®

E{e - / )

and X(E(e — / ) ) = 0. We claim t h a t A: Ef -> F is an isomorphism. T h e mapping is obviously onto, so we need to compute its kernel. Let a G Ef = {KG)f

be contained in the kernel of A and let Hi — ( / f , s u p p ( a ) ) .

Then

a

H\ is cyclic, generated by /ii, say, and we may assume t h a t h\ [H\: H] = p^ and \{hie) Because a G {KH\)f,

— h. Hence

= ^m+a for some suitable p'^'^^th root of unity. we can write a — X^^_o (^i^\ with a^ G

{KH)f.

Thus

0 = A(a) = Y. M«.)C+a and A(a,) G AX^m) = ^ - But, by Theorem X.4.4, [F{Ui-a):

F] = [AX^m+a): A^^m)] - P ^

so ^m-\-a cannot satisfy a nontrivial polynomial over F of degree less than p^. Therefore X{ai) = 0 for all i. Furthermore, because A is an isomorphism on {KH)f^

we conclude t h a t ai — 0 and hence that a = 0.

It follows from the foregoing t h a t

E = Ef®E{e-

f) =

Ef®M.

So any maximal ideal Af of £ is complemented in E by a minimal ideal and EjM

= F. Hence E is artinian and semisimple. This proves the first part

of the lemma.

3. FINITE CONJUGACY

363

Assume now t h a t 5 is a finite abelian group whose order is invertible in K.

Let e G A K ( G ) be an idempotent.

By the first p a r t , {KG)e

=

A \ © • • • © Kr is the direct sum of fields, each isomorphic to F. Then {K[G X B])e = {KG)eB

= {K\ © • • • © Kr)B

= 0^^^

lUB.

Since each group algebra KiB is semisimple artinian, it follows t h a t (K[G X B])e is the direct sum of

fields.

3 . 1 6 T h e o r e m . Let K he a field of characteristic

D p > 0 and let L be an

RA loop with torsion subloop T ^ L. Suppose T does not contain of order p. Then U{KL)

is an FC loop if and only ifT

and one of the following

conditions

1. KT

is an abelian group

holds.

is finite and, for all t E T and x e L, we have xtx~^

some nonnegative 2. T is finite and

elements

multiple r of[K:

= t^"^ for

Fp].

central.

3. T is central and of the form

Z ( 2 ^ ) X A with A a finite group and

V C Z ( 2 ^ ) , and there exists an integer k > 0 such that K does not contain a root of unity of order 2^. PROOF.

Note t h a t p / 2 since s £ T has order 2, where L' = {l?'^}.

Suppose U{KL) Z{L)

is an FC loop. Since squares in L are central and L ^ T^

is infinite. Hence, by Lemma 3.4, all idempotents in KL are central.

Furthermore, from Lemma 3.7, we see t h a t T is an abelian group. Assume t h a t T is infinite.

From Lemma 3.11, each subloop of T is

normal in L, so, if 7/ is a subloop of T with s ^ H^ then H is central in L. Because T does not contain elements of order p . Lemma 3.6 yields t h a t T2' is finite. Also, letting j^^ be a maximal subgroup of T2 not containing 5, then H is also finite. Hence T2/H is an infinite abelian 2-group with the property t h a t 5 = sH belongs to all its nontrivial subgroups. Thus T2/H no direct products and so, by Lemma 3.12, T2/H divisible, it follows t h a t T2 = TfH

contains

^ Z ( 2 ^ ) . Since Z ( 2 ^ ) is

for all n > 1. Hence [T2: Tf]

< \H\.

Choose n so t h a t [T2: Tl""] is as large as possible. Then

so T2" is divisible. Thus, if D is the maximal divisible subgroup of T2, then [T2: D] is finite and, by Lemma 3.12, T2 = D x B with B

finite.

Since

r = T2 X T2' we have T = D x A for some finite subgroup A. Since D is

364

XIII. IDEMPOTENTS AND FINITE CONJUGACY

the direct product of Z(2^)-groups and because T2IH = Z ( 2 ^ ) , we obtain t h a t D ^ Z(2°°). Thus T = Z ( 2 ^ ) x A for some finite group A.

Because

squares of elements of L are central, the group Z ( 2 ^ ) is central. Because of Lemma 3.6, we also have U = {1,'S} C Z ( 2 ^ ) . Thus s ^ A^ so A and hence T as well are central. We now show t h a t the field condition in (3) is satisfied. Suppose t h a t K contains a primitive 2^th root of unity (^ k > 1. Then, by Lemma 1.1, for any x G Z(2'^) of order 2^,

is an idempotent in KL^ and central as earlier noted. Furthermore, e ( l — s) ^ 0 since ^^

^ = —^ ^ ^. Because of Lemma 3.6, we know t h a t there

are only finitely many idempotents with these properties. Therefore, there exists A: > 0 such t h a t K does not contain a root of unity of order 2^. Suppose now that T is finite but not central. Let t £ T and g ^ Lhe such that the group G = (t^g) is not commutative. Since U{KG) group U{KG)

C U[KL)^

the

is an FC group. Furthermore, G does not contain elements

of order p. Hence Lemma 3.14 yields t h a t A' is finite. Since all idempotents of KL are central, we also obtain t h a t for a l H G T and x G Z/, xtx~^

= fP""

for some integer r > 0, a multiple of [A^: Fp] (Theorem 1.6). This finishes the proof of necessity. To prove the converse, assume t h a t T is an abelian group and suppose first t h a t (2) is the case; that is, T is finite and central. Since \T\ is not divisible by p = char A' (T contains no elements of order p), the group algebra KT is semisimple artinian and, because it is commutative, KT — A^i ® • • • ® Kn is the direct sum of fields AV Certainly every idempotent of KT is central in KL^ so we can apply Lemma X I L l . l and write any ji G U{KL)

in the form // — Y^^=\ ^i^i'> ^i ^ ^^^ ^^^ Qi ^ ^- Since KT is central

in A'L, we have {kiqi){kjqj)

= (kikj){qiqj)

so, in particular, [kiqi)(kjqj)

—0

\ii ^ j . Since kik~^ = e^, the identity of A^^, and Yll=\ ^i — 1? i^ follows t h a t /i~^ = Y17=i K^^7^'

If ^ = Z)r=i ^i'^i is another unit, hi G Ki, r, G L, then,

remembering t h a t hi and ki are central, we have ^i~^u^ — Y^^=\ Since L' — { 1 , ^ } , each q~^riqi is either r^ or svi. t h a t the set {/I'^u/i

\ /j, G U{KL)}

is finite, so U{KL)

^mi^'^i^i-

It follows immediately is F C .

3. FINITE CONJUGACY

365

Now assume t h a t (1) is the case but t h a t T is not central (otherwise, we are in case (2) which has just been settled). As before, KT = Ki®is the direct sum of fields !{{.

- -Q Kn

Since p is odd and elements of odd order

are central, not all 2-elements can be central; thus, Tpi is not central. Since Tpf C T , Tpf is an abelian group. Also, K is finite. Thus both assertions (1) and (2) of Theorem 1.6 hold and again we conclude t h a t every idempotent in KT

is central in KL,

Once again, we have t h a t any unit /x G KL

can

be written in the form /x = X^^Li ^i^if where ki G Ki and qi G L. At this point, we require the fact t h a t each field Ki is normal t h a t for all a.P e

in KL in the sense

KL, A > = aK^,

{aKi)f3 = a{KiP)

{Kia)(5 =

K^{af3),

and a(PKi)

=

(af3)K^,

This follows from normality of T in L, centrality of idempotents of KT in KL (which implies t h a t the identity €{ of each Ki is central) and routine calculation (see the proof of Lemma XII. 1.1). In particular, q~^ Ki — Kiq~^, so we have (kiqi)~^

— q~^k~^ G KiL, thus /i~^ = Y17=\i^i^i)~^-

^^ before,

if zy = X^^_| h^Ti, hi G A\, n G // is another unit in A'L, then n

ji'^u^

= ^{kiq^)~^{h,ri){k,q,)

where, for x G U{FL), by T{x)

n

— R{x)L{x)~'^.

^ ( ^ ) - U{FL)

= -^ U{FL)

Recall t h a t T{x)

^{Kri)T{k^q^) is the inner m a p defined

is a pseudo-automorphism with

companion x~^ (Theorem II.3.3) and t h a t we apply this map to the right of its argument. Thus (*)

[{h,r,)Tik,q,)]c

with c = (kiqi)'"^.

= [h,T{k^q,)][r,T(k,q,)

Now hiT{kiqi)

• c]

G A\, by normality of A'^, and similarly,

for certain A:i,... , /^s G Ki, we have rtT{k,q^)

^ {q~^k-^)ri{kiqi)

=

[{kiq-^)r^]{qik2)

= [{f^3{Q^^'^i)]{Qif^2) = k4[{q~^ri){qik2)]

=

k4[{q~^r^qi)ks].

Since Vi and qi are in L and since L' = {l^'^}, the element q~^riqi is either ri or svi. Thus riT(kiqi)

G Ki{6riKi),

where 6 = 1 or S = s. From (*), it is

366

XIII. IDEMPOTENTS AND FINITE CONJUGACY

apparent that [ihiri)T(kiqi)]c € Ki{[Ki{6r,Ki)]c}

C {Ki[Ki{6TilQ]}c

C

[Ki{6rilQ]c

SO that {hiT,)T{kiq,) e lUiSnlQ

C {KT)[6r,{KT)]

=

{KT)[r,{KT)]

because 6 eT, Thus //"^r/// G T,7=iiI^T)[ri{KT)], showing that {//"^^M I /JL € U{KT)} is finite, because KT is finite. Once again, we have that U{KL) is an FC loop. Finally, assume that (3) holds. Let // G U{KL). Let ^ be a finitely generated RA subloop of L containing supp(/x) U A U L'. Denote by T{H) the torsion subloop of H, Since H is finitely generated, T{H) is finite. Since also A C T ( ^ ) C T =: Z(2^) x yl, there exists t e Z(2^) such that T{H) = (t) X ^ . Because there exists an integer A: > 0 such that K does not contain a root of unity of order 2^, we know from Lemma 3.15 that 1 - ? = ei +

\- en,

the sum of the orthogonal primitive idempotents in {I — t )A [(^) X A]. Let 1/ G U{KL) and, as above, let H\ be a finitely generated RA subloop of L containing H U supp(i/) U supp(ei) U • • • U supp(en). Then A'[r(^i)] = A^®- • -e AV, the direct sum of fields A\ = (A^[T(/i'i )])/,• and each fi a primitive idempotent in K[T[H\)]. As each {K[T{H\)\)ei is a domain which is algebraic over A', the ring {K\T{H\)])ei is a field as well. So each e^ is a primitive idempotent in K\T[H\)\ and we may assume that r > n and /i = e i , . . . ,/n = en- Write ii = X)Li M/^ ^^^ u = J2Ui ^fiBecause of Lemma XILl.l, ij.fi = aiQi and vfi = fiihi^ where a^,/?, G Ki and Qi^hi ^ L^ i = l^... ^r. Because the fields A\ are central in KL^ r

Again, since V = {1,'S}, higih~^ = Sigi, Si = 1 or Si = s. So aihiQih'^ ^ aiQi if and only liSi - s and ji{\-s) ^ 0. Since 1-5 G {l-t)K[T{H)xA],

3. FINITE CONJUGACY

367

the latter implies fi{ei + - —\- e^) ^ 0; hence, ji G { 6 i , . . . , e^}. So n

v^iu~^ = ^

r

aigi6i + ^

t=l

t=n-(-l

n

/x/,- = ^fieiSi

+ (1 - (ei + • • • + e^))//

t=l

So the conjugates of /x belong to the finite set n

{Y^fMeiSi-\-(l-{ei Hence ll{KL) is an FC loop.

+ "' + €n))ti\6i

G {l,^}, 1


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G. Pickert, Projekiive Ebenen, Springer-Verlag, Berlin-Heidelberg-New York, 1975, second edition. M. M. Parmenter and C. Polcino Milies, Isomorphic group rings of direct products, Archiv der Math. 31 (1978), 11-14. S. Pedis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950), 420-426. D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, New York, 1982. Klaus W. Roggenkamp, Observations on a conjecture of Hans Zassenhaus, Groups-St. Andrews 1989, Vol. 2, Lecture Note Ser., vol. 160, London Math. Soc, Cambridge Univ. Press, 1991, pp. 427-444. Joseph J. Rotman, An introduction to the theory of groups, fourth ed.. Graduate texts in mathematics, vol. 148, Springer-Verlag, New York, 1995. Louis Rowen, Ring theory, vol. 1, Academic Press, New York, 1988. J. Ritter and S. K. Sehgal, Integral group rings with trivial central units, Proc. Amer. Math. Soc. 108 (1990), no. 2, 327-329. J. Ritter and S. K. Sehgal, Construction of units in integral group rings of finite nilpotent groups. Trans. Amer. Math. Soc. 324 (1991), no. 2, 603-621. R. Sandling, Group rings of circle and unit groups, Math. Z. 124 (1974), 195-202. R. Sandling, The isomorphism problem for group rings: a survey. Lecture Notes in Math., vol. 1141, Springer, Berlin, 1985. R. Sandling, The modular group algebra of a central-elementary-byabelian p-group, Arch. Math. (Basel) 52 (1989), 22-27. R. D. Schafer, An introduction to nonassociative algebras. Academic Press, New York, 1966. S. K. Sehgal, On the isomorphism of integral group rings I, Canad. J. Math. 21 (1969), 410-413. S. K. Sehgal, Topics in group rings, Marcel Dekker, New York, 1978. S. K. Sehgal, Units in integral group rings, Longman Scientific & Technical Press, Harlow, 1993. E. Spiegel, On isomorphisms of abelian group algebras, Canad. J. Math. 27 (1975), 155-161. E. Spiegel and A. Trojan, On semi-simple group algebras II, Pacific J. Math. 66 (1976), no. 2, 553-559.

376

[SZ77] [Tit72] [TW80] [Wei91] [Whi68] [Wil78] [ZM75] [ZSSS82]

BIBLIOGRAPHY

S. K. Sehgal and H. J. Zassenhaus, Group rings whose units form an FC group, Math. Z. 153 (1977), 29-35. J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270. A. D. Thomas and G. V. Wood, Group tables, Shiva Publishing, Orpington, 1980. A. Weiss, Units in integral group rings, J. Reine Angew. Math. 415 (1991), 175-187. A. Whitcomb, The group ring problem, Ph.D. thesis, Chicago, 1968. A. Williamson, On the conjugacy classes in an integral group ring, Canad. Math. Bull. 21 (1978), no. 4, 491-496. A. E. Zalesskii and A. V. Mikhalev, Group rings, J. Soviet Math. 4 (1975), 1-78. K. A. Zhevlakov, A. M. Slin'ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative. Academic Press, New York, 1982, translated by Harry F. Smith.

Index

algebra

semi-, see s e m i - a u t o m o r p h i s m a u t o t o p i s m , 58

Cayley-Dickson, 15, 17 composition, 27, 169 generalized q u a t e r n i o n , 17

Baer radical, 148

nonassociative, 5

Baer, R., 306

q u a d r a t i c , 27, 103, 104

Bass cyclic unit, 201

q u a t e r n i o n , 17

Bass, H., 311

real quaternion, 13

Bass-Milnor T h e o r e m , 312 B a u t i s t a , R., 256 B e r m a n , S. D., 195, 209 bicyclic unit, 200 bilinear

simple Cayley-Dickson, 18 q u a t e r n i o n , 18 split, 19 composition, 36

form, 23 m a p , 38 Brauer, R., 3, 334 Bruck, R. H., 3, 49

Zorn's vector m a t r i x , 21 algebraic field extension, 248 integer, 202 n u m b e r field, 202, 309

Cayley loop, 68, 81

a l t e r n a t i n g function, 8 a l t e r n a t i v e ring, 7

n u m b e r s , 14 Cayley, A r t h u r , 3 Cayley-Dickson

Artin Generalized T h e o r e m of, 12 T h e o r e m of, 13

algebra, 15 process, 16 central series lower, 295 upper, 295 centrally nilpotent, 295 centre

associative bilinear form, 166 associator loop, 52 ring, 5 a u g m e n t a t i o n , 149 ideal, 149

of a loop, 53 of a loop ring, 87, 88, 101

m a p , 149 A u t ( a u t o m o r p h i s m conjecture), 234, 237

of a ring, 6 characteristic, 88 Chein, O., 69, 80, 144 Chinese Remainder T h e o r e m , 249, 253 class conjugacy, 86

automorphism F-, 230 inner, 228 of a loop, 58 pseudo-, see p s e u d o - a u t o m o r p h i s m 377

378 nilpotency, 295 s u m , 86 Classification T h e o r e m , 140 Coelho, S., 333 Cohen, D. E., 256 C o h n , J. A., 195 commutator loop, 52 ring, 5 unique nonidentity, 87 commutator-associator subloop, 53 unique nonidentity, 87 c o m p a n i o n , 58 component simple, 167 composition algebra, 27 split, 36 p e r m i t , 27 conjugacy class, 86 conjugate, 86 c o m m o n , 239 of a q u a t e r n i o n , 14 rationally, 236 conjugation, 65 C u r t i s , C. W . , 2, 36 cyclotomic extension, 249 field, 175 polynomial, 175, 249 Dade, E., 226 de Barros, L. G. X., 19, 271, 282 Deskins, W . E., 226, 258 d e t e r m i n a n t , 22 diassociative, 52, 68, 125 Dieudonne, J., 166 T h e o r e m of, 166 direct p r o d u c t , 109 s u m , 25 Dirichlet Unit T h e o r e m , 208, 213, 309 divisible g r o u p , 305 division ring, 7 normal, 290, 365

INDEX dual space, 25 Engel group, 306 loop, 295 equivalent fields, 261, 282 Euler (^ function, 175 exponent b o u n d e d , 296 of a Moufang loop, 132 extension, see field e x t r a loop, 109 F C , 295 Fernandes, N. A., 237 field algebraic extension, 45, 248 cyclotomic extension, 249 finite extension, 248 Galois extension, 248 normal extension, 248 separable extension, 248 fixed subfield, 360 flexible, 8, 63 form associative, 166 bilinear, 23 m a t r i x of, 23 nondegenerate, 23 q u a d r a t i c , 26 q u a d r a t i c , 26, 104 s y m m e t r i c , 23 free group, 215 s u b g r o u p , 214 Frobenius h o m o m o r p h i s m , 337, 361 Frobenius, G., 3 function alternating, 8 multiplicative, 14 C-loop, 116 Galois field extension, 248 group, 248, 316 general linear

INDEX group, 83 loop, 83, 316 Giambruno, A., 234, 297, 300 Glauberman, G., 125 group divisible, 305 Engel, 306 Galois, 248, 316 general linear, 82 hamiltonian, 68 identity, 294 inner mapping, 57, 86 multiplication, 50 ordered, 196 Priifer, 358 principle congruence, 215 projective special linear, 215, 325 solvable, 302 special linear, 82, 214 group algebra modular, 247 Hall, M., Jr., 69, 99, 138, 144 hamiltonian group, 68, 93 Moufang loop, 68 Hartley, B., 195, 220 Higman, G., 195, 204, 208, 225 Theorem of, 208, 309 Hurwitz, A., 2, 35 Theorem of, 30 ideal left modular, 148 nil, 147 nilpotent, 147 prime, 147 idempotent central, 300, 333 primitive, 174 trivial, 199 identity flexible, 8, 63 group, 294, 301, 302 left alternative, 7, 63 Moufang, 9, 63

379 right alternative, 7, 63 TeichmuUer, 6 index, 67 finite, 202 of a subloop, 62, 202 inner automorphism of a ring, 228 map, 57, 65, 86 integral element, 155 extension, 155 loop ring, 223 inverse left, 50 property, see right or left quasi-, 148 right, 50 invertible, 81 involution of a loop, 80 of an algebra, 14 isomorphism normalized, 226, 259 problem, 223, 225, 247 isotope, 116 isotopic, 116 isotropic, 24 totally, 24 Jacobson radical, 148 Jacobson, N., 223, 230 Juriaans, S. O., 237 Kothe radical, 148 Karpilovsky, G., 158 Klein, A., 296 Kleinfeld function, 8 radical, 148 Klinger, L., 237 lack of commutativity, see LC Lagrange's Theorem, 51, 67, 114, 115 Laurent polynomial ring, 153, 356 LC, 94-96, 98, 119 Leal, G., 125, 271, 314

380

left inverse property, 60 Lichtman, A., 350 linear function, 104 linearization, 7 Livingstone, D., 195 localization, 300 locally finite loop, 212 loop, 50 2-, 125 Cayley, 68, 81 e x t r a , 109 F C , 295 G-, 116 general linear, 83, 316 h a m i l t o n i a n , 301, 302, 307 Moufang, 68 indecomposable, 126 inverse property, 60, 63 locally finite, 212 Moufang, 64 n-Engel, 295 nilpotent, 295 opposite, 55 quotient, 57 RA, 107 ring, 85 ring alternative, 107 special linear, 83 torsion, 125 t y p e 1, 271 t y p e 11, 271 unit, 82 L u t h a r , I. S., 237 Mfc-law, 115 Mal'cev, A.I., 296 Marciniak, Z. S., 218 Maschke, H., 3, 4, 147, 164 T h e o r e m of, 167 May, W . , 158 m e t a b e l i a n , 225 metacyclic, 226 Milnor, J., 311 modular algebra, 247 case, 267

INDEX ideal, 148 Molien, T . , 3 Moser, C , 340 Moufang identities, 9, 63 Moufang, R u t h , 2, 63 T h e o r e m of, 68, 82 multiplicative function, 14 n-Engel, 295 n-squares problem, 35 Nielsen-Schreier T h e o r e m , 327 nil, 147 nilpotency class, 295 nilpotent, 147, 295 centrally, 125, 295 element, 147 Noether, E., 3 nonassociative, 5 ring, 5 nondegenerate norm, 104 norm, 27, 103 quaternion, 14 normal complement, 314 division ring, 290, 365 field extension, 248 subloop, 55 normalized form, 189 isomorphism, 226, 259 unit, 236 Norton, D. A., 68 T h e o r e m of, 75 nucleus left, 53 middle, 53 of a loop, 53 of a loop ring, 101 of a ring, 6 right, 53 order (in an algebra), 201 O s b o r n , J. M., 88 p-sequence, 262

INDEX Paige, Lowell, 4, 88 T h e o r e m of, 88 P a r m e n t e r , M. M., 208, 209, 212, 234 Passi, I. B. S., 237 P a s s m a n , D. S., 197, 198 Perils and Walker T h e o r e m of, 177, 252 Perils, S., 226, 252 Peterson, G., 234 Pflugfelder, H. O., 49 Pickel, P. F., 195, 220 Pickert, G., 49 polynomial ring, 153 power SLSSociative, 64, 88 Priifer group, 358 prime radical, 148 ring, 147 primitive i d e m p o t e n t , 174 root of unity, 175, 248 principal congruence group, 215 product direct, 109 projective special linear group, 215, 325 p s e u d o - a u t o m o r p h i s m , 57, 65 quadratic algebra, 27, 103, 104 form, 26, 104 quasi-inverse, 148 quasi-regular, 148 quasigroup, 49 quaternion, 1 algebra, 13 R A loop, 107 radical, 148 Baer, 148 hereditary, 149 Jacobson, 148 K o t h e , 148 Kleinfeld, 148 prime, 148 Smiley, 148 u p p e r nil, 148

381 Zhevlakov, 148 rank, 132 rationally conjugate, 236 reflection, 230 representation right regular, 167 right inverse property, 60 ring Laurent polynomial, 153, 356 nonassociative, 5 of algebraic integers, 202 of fractions, 300 polynomial, 153 prime, 147 simple, 7 ring alternative, 7 Ritter, J., 212, 213, 237 Robinson, D. A., 229 Roggenkamp, K. W., 226, 237 root of unity, 175 primitive, 175, 248 Sandling, R., 226 Schur, 1., 3 Scott, L., 226, 237 Sehgal, S. K., 212, 213, 218, 234, 237, 238 s e m i - a u t o m o r p h i s m , 65 semiprime ring, 147 semisimple algebra, 148 case, 267 Senior, J. K., 99, 138, 144 separable field extension, 248 polynomial, 248 signature, 309 simple c o m p o n e n t , 167 ring, 7 Smiley radical, 148 Smith, P. F., 208 special linear group, 214 projective, 325 special linear loop, 83

INDEX

382 Spiegel, E., 261, 266 split, 19 composition algebra, 36 splitting field, 333 subloop, 51 c o m m u t a t o r - a s s o c i a t o r , 53 index of, 62 n o r m a l , 55 subring unital, 201 s u p p o r t , 86 TeichmuUer identity, 6 tensor p r o d u c t , 39 T h r a l l , T . M., 252 T i t s , J., 218 torsion element, 125, 196 loop, 125 subloop, 212 unit, 196, 198 totally imaginary, 309 real, 309 t r a c e , 27, 103 associative, 104 c o m m u t a t i v e , 104 T r a m a , P., 237 t r a n s l a t i o n m a p , 50 transversal, 52, 290 trivial, see unit T r o j a n , A., 261, 266 unit, 81 Bass cyclic, 201, 312 bicyclic, 200 central, 198 torsion, 199 loop, 82 Engel, 306 F C , 306 nilpotent, 306 normalized, 236 torsion, 196, 198 trivial, 195, 196, 204 Bass cyclic, 204

bicyclic, 200 central, 212 unital, 5 unity, 5 Valenti, A., 234, 237, 297, 300 Walker, G. L., 226, 252 Weiss, A., 226, 237 W h i t c o m b , A., 224, 225 L e m m a of, 224 Williamson, A., 209 Wright, C. R. B., 125 Z-order, 201 Zassenhaus, H. J., 236 conjectures of, 236 ZCl (first Zassenhaus conjecture), 236, 239 ZC2 (second Zassenhaus conjecture), 237, 241 ZC3 (third Zassenhaus conjecture), 237, 241 Zhevlakov radical, 148 Zorn's vector m a t r i x algebra, 21 Zorn, M., 22

Notation

What follows is a list of symbols and other notation used in this book. Page references guide the reader to the place where the synnbol is defined or first mentioned.

Symbol

Meaning

P

the prime numbers

N

the natural numbers, 1,2,3,...

Z

the integers

Q

the rational numbers

R

the real numbers

C

the complex numbers

Fp

the field of p elements, p a prime

(j)

the Euler function, p. 175

^„

unless specifically stated to the contrary, this always denotes a primitive nth root of unity, p. 175, p. 248

(F, Qf,/^)

a generalized quaternion algebra over a field F, p. 17

»(F)

( F , - l , - l ) , p . 17

H(R)

(R,—1,—1), the classical division algebra of real quaternions, p. 17

(F, Of,/?, 7)

a Cayley-Dickson algebra over a field F , p. 17

C

the Cayley numbers, p. 15

Mn{R)

the ring of n x n matrices over a ring R

383

384

NOTATION

R[Xi,

X 2 , . . . , Xn]

the ring of polynomials in c o m m u t i n g indeterminates X\^. . . , Xn with coefficients in a ring R

R[Xi, X f \ . . . , Xn, X~^]

the ring of Laurent polynomials in c o m m u t i n g indeterminates X i , . . . , X n , p. 153

R^

the set of ordered n-tuples over a ring R

RL

the loop ring of a loop L over a ring R, p. 85

M 04) A^

the tensor product of modules M and A'' over a c o m m u t a t i v e ring with unity, p. 39

Cn

the cyclic group of order n

Dn

the dihedral group of order 2n

Qg

the quaternion group of order 8

Mi6{Qs)

the Cayley loop, p. 68

Z{p^)

the p-Priifer group, p. 358

M(G,*,^o)

the Moufang loop determined by a nonabelian group G with involution * and central element (70, p. 80

M{G, - 1 ,

.9(1

the special case of M(G,^,

go) where * is the in-

verse m a p on G, p. 81

M(G, 2)

M ( r ; , - l , l ) , p. 80

In

where / is an ideal of a ring R, and n G N, this is {r e R\ rire

Lp

/ } , p. 157

where L is a loop (or group) and p is a prime, this is the set of elements of L whose order is finite and a power of p, p. 157

^p'

where L is a loop (or group) and p is a {)rime, this is the set of elements of L of finite order relatively prime to p, p. 157

[E: F]

where E/F

is a field extension, this is the dimen-

sion of E" as a vector space over F , p. 175 G?i\{E/F)

the Galois group of a Galois field extension

E/F,

p. 248 F{a)

where F is a field and a is an element contained in some extension field of K of F , this is the smallest subfield of K containing F and cv, p. 45

NOTATION

Z[a]

385

the subring of Q(of) generated by an elennent a E C, p. 202

J\fil{R)

the upper nil radical of a ring R, p. 148

V{R)

the prime radical of a ring R, p. 148

J[R)

the Jacobson radical of a ring R, p. 148

GL(2, R)

the general linear group of degree 2 over a ring i^, p. 82

SL(2, R)

the special linear group of degree two over a ring R, p. 82

PSL(2, R)

the projective special linear group of degree two over a ring R^ p. 215

3(/^)

Zorn's vector m a t r i x algebra over a ring R, p. 21

GLL(2, R)

the 2 X 2 invertible elements of 3(7?.), p. 83

SLL(2, R)

the elements of GLL(2, R) of determinant 1, p. 83

Sdet=±\

the matrices of determinant 1 in a subloop S of GLL(2,Z[<en]),p. 316

'^det=±i

^^^ quotient loop 5'det=±i/{/, — ^}) where / is the identity m a t r i x of Zorn's vector matrix algebra, p. 317

5'^i(det)=±i

those matrices in a subloop S of GLL(2, Z[<^n]) with the property t h a t the norm of dei A is ibl, p. 316

^nidet^-±\

^^^ quotient loop 5'n(det)=±i/{^, — 0 , whcre / is the identity matrix of Zorn's vector m a t r i x algebra, p. 316

x^

the right inverse of an element x in a loop, p. 50

x^

the left inverse of an element x in a loop, p. 50

R[x)

right translation by x, p. 50

L[x)

left translation by x, p. 50

T{x)

R{x)L{x)-\p. 57

R{x,y)

R{x)R{y)R{xy)-\p.b7

L[x,y)

L{x)L{y)L{yx)-\p.b7

(a, 6)

the c o m m u t a t o r of elements x and y in a loop, p. 52

386

{X,Y)

NOTATION

for subsets X and F of a loop, this is the set of all c o m m u t a t o r s (x, y), x ^ X^ y G V, p . 52

{a,b,c)

the associator of elements a, 6 and c in a loop, p. 52

(X,Y,Z)

for subsets X , y , Z of a loop, this is the set of all associators (ic, y^z)^ x ^ X ^ y ^Y

^ z E. Z ^ p. 52

[a,b]

the c o m m u t a t o r of elements a and 6 in a ring, p. 6

[X,Y]

for subsets X and F of a ring, this is the set of all c o m m u t a t o r s [x, y], x E -^, y G F , p. 6

[a, b, c]

the associator of elements a, 6, c in a ring, p. 6

[X, Y, Z]

for subsets X , F , Z of a ring, this is the set of all associators [x, y, z], x G ^ , y G F , z £ Z^ p. 6

L'

the commutator-associator subloop of a loop L, p. 53

Z{X)

the centre of a loop or ring X , p. 6 and 53

M{X)

the nucleus of a loop or ring X , p. 6 and 53

M{L)

the multiplication group of a loop L, p. 50

Inn(L)

the inner mapping group of a loop L, p. 57

[L: / / ]

where // is a subloop of an inverse property loop L, this is the index of H in L, p. 62

Aut(L)

the group of automorphisms of a loop L

CM

where A^ is a normal subloop of a loop L, this is the kernel of the m a p RL —> R[L/N],

which is

the linear extension of the natural m a p L —^ L/N, p. 149 6

ei^ where L is a loop, this denotes the augmentation m a p RL -^ R, p. 149

A ( L , N)

where L is a loop with normal subloop TV, this is kere^v, p. 149

A(L)

for a loop L, this is A ( L , L ) , the augmentation ideal of a loop ring RL, p. 149

N

where N is a finite loop contained in a loop ring, ^his is E n G N ^ ' P- 1^1

NOTATION

387

N

where N is a. finite loop contained in a loop ring RL with \N\ invertible in i?, this is AnN^ p. 151

^

{9) — Yl7=o 5'*' where g is an element of finite order n in a loop ring, p. 200

p

(^) rr 1 Yl^=o 9^' where g is an element of finite order n in a loop ring RL^ with n invertible in R^ p. 218

U{R)

the set of units in a ring 7?, p. 81

U{RL)

the set of units in the loop ring RL

U\{RL)

the set of normalized units in the loop ring RL^ p. 236

TU{RL)

the set of torsion units in the loop ring RL, p. 236

TU\ [RL)

the set of normalized torsion units in the loop ring RL, p. 236

char R,

the characteristic of a ring R, p.88

p[A)

the rank of an abelian group A] that is, the minimal number of generators for A, p. 132

o(x)

the order of an element x in a loop, p. 128

ind2(A)

an invariant of a field K related to its 2-sequence, p. 265 this is 1 if x^ 4- 1 == 0 has a solution in the field K and 0 otherwise, p. 265

0[K) i{K)

this is 1 if x^ -fy^ = —1 has a solution in the field K and 0 otherwise, p. 265

c[K)

this is 1 if x^ -h y^ -\- P -f tf;^ = _ i has a solution in the field K and 0 otherwise, p. 282