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VOLUME 3
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM * LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
Locally Finite Groups OTTO H. KEGEL and BERTRAM A. F. WEHRFRITZ Department of Pure Mathematics Queen Mary College University 5f London
1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
0 North-Holland Publishing Company - 1973 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the
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0 7204 2450 X 0 7204 2454 2
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Introduction
A group G is said to be locally Jinite if every finite subset of G generates a finite subgroup. The class of locally finite groups is placed near the (not too well defined) cross-roads of finite group theory and the general theory of (infinite) groups. Many theorems about finite groups can be phrased in such a way that their statements still make sense for the class of all groups, or at least for some suitably restricted class of groups. Having observed this for some particular theorem, one naturally asks whether this statement still represents a true theorem in this wider class. Rather than investigating the “domain of validity” of each individual theorem, we shall restrict our attention to the class of locally finite groups. This class is sufficiently restricted for many interesting results on finite groups to carry over to it, and yet it is wide enough to cause most of the results of an arithmetic nature not to carry over and to allow new phenomena to arise, often of a set-theoretic nature. A central result on finite groups, the statement of which still makes sense for arbitrary groups is Sylow’s theorem: For every prime p and every finite group G any two maximalp-subgroups of G are conjugate. That this does not give rise to a true statement about arbitrary (infinite) groups has long been recognized. The discussion of the validity or failure of this statement in this or that class of groups and of weakened forms o f it, by which one might salvage some of its content, runs through many papers and has given birth to a number of classes of locally finite groups. An example would be the class of locallyfinite-normalgroups, that is the class of all those groups in which every finite subset is contained in some finite normal subgroup. For any two maximal p-subgroups P and P I of a locally finite-normal group G there exists an automorphism u of G mapping P onto PI.Further, such an autornorphism can be chosen to be a sort of limit of inner automorphisms. A proof of this result is given in Section K of Chapter 1. We shall not pursue the problem of extending and modifying Sylow’s Theorem any further (the expositions in Section D of Chapter 1 and Section VII
VIIl
INTRODUCTION
A of Chapter 3 reflect quite different interests). One reason for our reluctance to get ensnared in this problem is the following generic construction, due to H. Heineken [11: Theorem: Let n be any non-empty set of primes not containing the prime g, and suppose that P and P I are any two inJinite n-groups. Then the canonical images of P and P , in the group G = ( P * P 1 ) / K ' K qare maximal n-subgroups of G, where K denotes the kernel of the canonical projection of the free product P * PIo f P and P , onto the direct product of P and P I . Further i f P and P I are locallyfinite, then so is G. Note that K is a free group and that the factor group KIK'K' is an infinite elementary abelian g-subgroup of G. We leave the proof of this result as an exercise, but note that what is essentially a special case of this theorem is presented in a different guise as 3.3. On the other hand there are theorems about finite groups that lead to true statements about locally finite groups. The proof of such an extension usually relies heavily on the finite case. An example of such a result is the following (a proof of which may be found in Section J of Chapter 1). Frobenius' Theorem: If a locally finite group G contains a subgroup H such that H n H g = (1) f o r every element g E G\ H, then there is a characteristic subgroup K of G such that HK = G and H n K = (1). Once a field of study has separated itself out, it tends to assume an independence and impetus of its own. There are results which deal just with infinite, locally finite groups. The first general example of such a result, and probably one of the most important, is due (independently) to Hall and Kulatilaka [l] and Kargapolov [ 6 ] (see Chapter 2). Theorem: Any infinite, locally jinite group contains an infinite abelian subgroup. Although at first sight it may seem unlikely, even this result depends upon the piecing together of various theorems about finite groups. Here the celebrated theorem of Feit and Thompson [l], which states that any (finite) group of odd order is soluble, plays a crucial role. There is a further dichotomy of the theorems about locally finite groups, that cuts across the above types. This dichotomy divides these theorems between results such as Frobenius' Theorem, which depend only on the structure of the finite subgroups involved in the group in question, and results like the property of the maximal p-subgroups of a locally finite-normal group
INTRODUCTION
IX
quoted above, that depend rather on the way in which these finite subgroups are embedded in the group. It is the interplay between these two types of hypothesis that give the theory of locally finite groups much of its richness and flavour. With these general notions in mind, we set out to write a rather personal account of the theory of locally finite groups. Our selection of topics was influenced partly by our interests, but mainly by the desire to include the results of Chapter 5, not all of which have appeared in print before. Consequently, locally finite groups satisfying minimal conditions of various kinds occupy a major portion of our text. The apparatus that we need to attack the problem of determining the structure of locally finite groups satisfying the minimal condition on subgroups can be applied to a wider range of locally finite groups. However, in some sense, any group for which these methods give information is structurally rather small. About large groups we know very little. Chapter 6 serves to illustrate this point; the main topic here being a very special class of large groups, namely theuniversal groups of P. Hall [3]. In the course of presenting our material we have taken the opportunity to include a number of (at present) open questions, some quite wide, some very specific, and some we hope of general interest. We have also included a certain number of exercises throughout the text. A one-semester introductory course to general group theory can be based on the material of Chapter 1. This book is not an introductory work on groups. Although we do not expect a potential reader necessarily to be a specialist in group theory, we shall assume that he has a working familiarity with anything that might reasonably appear in undergraduate courses on group theory. For example we will expect the reader to be aquainted with the material covered in, say, the first two hundred pages of Rotman [I] or of KuroS [l], Vol. 1. We shall also assume that the reader has absorbed the basic definitions and the most rudimentary facts concerning soluble, nilpotent, locally soluble and locally nilpotent groups. These may be found in KuroS [I], Vol. 2, especially Sections 57, 59, 62 and 64. A more recent account may be found in the lecture notes Robinson [I]. These have had only a limited circulation, but the author is preparing a more formal edition.* Also a third volume of KuroS [I] will appear shortly, bringing this work up to date. We shall have to assume a number of results on finite groups. These are of two kinds. Firstly, we shall use a large number of very standard results for
* This has just appeared: D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, 1972.
X
INTRODUCTION
which we shall give references usually to Huppert [l] or Gorenstein [l]. As far as these results are concerned, the less experienced reader should find no difficulty in reading them up as he goes along. Secondly, we shall need a small number of very deep results, indeed. These we shall always state in the text in such a way that the meaning will be readily comprehensible (we hope), even when the reader has not the least notion how to proceed with a proof. A good example of the type of result we have in mind here is the main result of Feit and Thompson [11, which we have quoted above. We owe very special thanks to Susan McKay, who heroically read through an early version of the entire manuscript and thereby saved us from many errors, obscurities and embarrassments. We are also indebted to Kurt Hirsch, who, by making his card index available to us, saved us many tedious hours work while preparing the bibliography.
Notation Throughout this list, G denotes a group, S a subset of G, p a prime and .n a set of primes. centralizer of S in G normalizer of S in G subgroup of G generated by S (S) set of all conjugates in G of elements of S SG ZG centre of G cl-th term of upper central series of G z, G JG intersection of subgroups of G of finite index QiG Frattini subgroup of G cG see p. 16 bG see p. 15 ItG see p. 15 PG see p. 15 OG, O,G, 0,G see p. 15 OPG,0 " G see p. 15 p-size of G, see p. 92 IGI, Max, G set of maximal p-subgroups of G set of Sylow p-subgroups of G SYl, G X(G) see p. 89 lattice of centralizers of G ZC(G) Min class of all groups satisfying the minimal condition on subgroups Class of all groups whose abelian subgroups all lie in Min Min-fl class of all groups G for which g C ( G )satisfies the d.c.c. R
s NG s CG
If H, K and G, for I E I are all groups, then we denote by
lI G,
X G,, H x K
* G,, H Hx K 0 v, H t K
*K
Cartesian product of the G, direct product of the G, (resp. of H and K ) free product of the G, (resp. of H and K ) split extension of H by K (when defined) direct sum of the vector spaces V,. restricted (standard) wreath product of H by K XI
CHAPTER 1
Basic Methods, Concepts and Examples
This long chapter is mostly a compendium of results that we shall need at some point or other. It therefore lacks form and continuity. We have not been strictly utilitarian, but in places have included a little more than is necessary for our later applications, where the topic seemed to warrant it. We do not envisage, however, that the majority of our readers will wish to read straight through Chapter 1. For example, the knowledgeable reader, after a brief glance at Sections E, F and G,should start at once at Chapter 2. If at a later stage he finds that he is not familiar with certain results in Chapter 1, then he can read them up as he goes along. Section A contains the definition and some common examples of locally finite groups, and elementary properties of series and local systems. Section B studies locally soluble chief factors of locally finite groups and introduces a number of canonical subgroups (such as the Hirsch-Plotkin radical) for locally finite groups. Section C is a purely utilitarian section on stability groups. Much of Section D, a discussion of the conjugacy problem for the maximalp-subgroups of a locally finite group, is just background reading and will not be used subsequently. Sections E, F and G are of fundamental importance for the entire book. These sections introduce and give important characterizations of Cernikov groups. Section H discusses much weaker minimality conditions than the preceding three sections. They will not appear again, Section H simply completes the discussion begun in Sections E, F and G . We shall occasionally need to know quite a lot about the structure of certain very special types of groups. This information we collect together in Sections I, J and L. Section I discusses (locaily) dihedral, (locally) generalized quaternion and quasi-dihedral groups, Section J, locally finite Frobenius groups and Section L, periodic linear groups. There is a standard type of argument using inverse limits that we use a number of times throughout the book. Section K explains the basis of this argument and gives some simple examples of its use. 1
2
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1,
5A
Section A. n-Finite groups, series and local systems
The whole of this text is concerned with the class of periodic (= torsion) groups, that is, the class of those groups in which every element, or every 1-generator subgroup, has finite order. This notion suggests the definition, for each natural number n, of the narrower class of n-finite groups, that is, the class of all those groups in which every n-generator subgroup is finite. The intersection over all natural numbers n, of the classes of n-finite groups, is the class of locally finite groups. Every finitely generated subgroup of a locally finite group is finite. In Golod [ l ] it is shown that for every natural number n there exists an infinite (n+ 1)-generator group which is n-finite; in particular, there are periodic groups that are not locally finite. Moreover, even a periodic group of finite exponent need not be locally finite; this is the content of the main result of the important paper Novikov and Adjan [ l ] (this result, with different constants, had been announced in Novikov [ l ] and settles a famous problem posed by Burnside). We collect some formal properties of these classes of groups. 1.A.1 Lemma. For every natural number n subgroups and quotient groups of n-finite groups are n-finite. Subgroups and quotient groups of locally finite groups are locally Jinite.
These statements being completely obvious, one wonders about extensions; here the situation is less satisfactory. 1.A.2 Lemma. Extensions ofperiodic groups by periodic groups areperiodic. Extensions of locally finite groups by locally finite groups are locallyfinite. Proof. Of these two statements only the latter needs proof. Let N be a normal subgroup of the group G such that Na n d GIN both are locally finite groups. Let F be any finite subset of G. We have to prove that the subgroup (F) is finite. If X is any subset of the group G and n is any positive integer, put P I = {x, x2 . . . x,; x i E X for i = 1,2, . . ., n}. Since the index I(F)N : NI is finite, there exists a finite set Tof elements of (F) such that F and some transversal of (F)N over N is contained in T. Then clearly there exists a finite subset S of N such that Tlz1 G TS. A simple induction shows that T [ n + 1E3 TSrnlfor each natural number n. But G is periodic, so
u TI"' c T ( S ) . m
(F)
=
(T) =
n= 1
Since the normal subgroup N is locally finite, the subgroup ( S ) of N is
CH.
1,
5-41
n-FINITE GROUPS, SERIES AND LOCAL SYSTEMS
3
finite. Hence the set T ( S ) is finite and consequently also the group < F ) . (A proof can also be based on M. Hall [ 2 ] ,Corollary 7.2.1.) I Note that the above proof yields that extensions of locally finite groups by n-finite groups are n-finite. It follows from the fundamental theorem of abelian groups that periodic abelian groups are locally finite. Thus 1.A.2 implies that periodic soluble groups are locally finite and hence that periodic locally soluble groups are locally finite. For generalizations of this see Kurog [I1 6 59. Obviously, the subgroup generated by the elements of finite order in an abelian group is a locally finite group. This situation is slightly generalized in the following result due to Dietzmann [l]. 1.A.3 Proposition. If the finite set F of elements offinite order in the group G is invariant under all inner automorphisms of G, then the subgroup < F ) is a finite normal subgroup of G . Proof. Let n be the number of elements in F, and let m > 1 be the least common multiple of the orders of the elements of F. It suffices to show that each element of the subgroup < F ) can be expressed as a product of at most n(m- 1) elements of F. Suppose that the element g E < F ) can be expressed as g = fif i , . . .,L withf, E F and t > n(m - 1). Then, by the pigeon hole principle, one of the elements of F, sayf, appears at least m times in this expression. L e t d be the first term in the above product with f = fi. Put f j = f -'fj f for all j < i; then f j E F and 9 =ff;, * . , K I f i + l Y * * .,.A.
-
In this way one has movedfi from the i-th position in the product to the first position. Apply this trick to the first among the remaining factors fi+ ',. . .,f,equal to f, and continue in this manner. One finally obtains an expression g =f"f:, * .,A?" =f;", .yjt:m ~
f
-
with fewer than t factors, since f" = 1, and f; E F. This result has an immediate corollary.
I
1.A.4 Corollary. For the periodic group G the following two properties are equivalent: a. Every conjugacy class of elements of G contains only finitely many elements; b. Every finite set of elements of G is contuined in afinite normal subgroup of G.
4
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
5,
5A
This corollary characterizes a particular class of locally finite groups among the periodic groups. Periodic groups with the two equivalent properties of 1.A.4 appear in the literature under the names of Iocallyfinite-normal groups, locally normal groups, or periodic FC-groups. Many results on finite groups have been extended to the class of locally finite-normal groups. This corollary is our first example of a result stating that, under suitable conditions, a periodic group is in fact locally finite. Although results of this type will not be our main concern, we shall encounter them at various points as we go along. When studying the properties of an infinite group G (or of a class of groups) there are two main themes. The first is an investigation of the normal structure of the group (or groups) in question, that is an investigation of the properties of series and normal series. The second enquires into the local structure of G, that is, into the properties of certain local systems of G. These two themes are not unrelated, and in practice they rarely occur in a pure state but are usually intertwined with other ideas. Let I be a linearly ordered index set. A family Q = { V,, A , ; z E I } of pairs of subgroups of G is called a series of type I in the group G, if for all indices 1, K E I one has (i) V , is a normal subgroup of A,; (ii) A , E V, if z < K ; (iii)’ G\<1) = U z e l(A,\V,). If all the pairs of the series C 3 in the group G consist of normal subgroups of G , then Q is called a normal series in G. If G is a series in the group G, then the family of quotient groups {A,/V,; z E I} is called thefamily of factors of the series G. (The notion of a series (respectively, normal series) in the group G generalizes the notion of “normal system” (respectively, “invariant system”) in KuroS [l], Vol. 2, p. 171.) The above conditions for a series of type I in the group G imply that for each index z E I one has
V , = U A , and A, K
=
n V,,
K>l
Ub<,
where, if z = 0, an initial element of I, the union Archas to be interpreted as V , = (I), and if z = z, a terminal element of I, the intersection V , has to be interpreted as A, = G. The series G of type I in G is a reJiraement of the series (5’ of type I’ in G if for every index I E I‘ and ( V ,, A ,) E (5’ the subgroups V , and A , appear in
OK,,
CH.
1,
5 A]
I2-FINITE GROUPS, SERIES A N D LOCAL SYSTEMS
5
the series G. The refinement G of the series G‘ in G is aproper rejinement, if there is a subgroup of G belonging to one of the pairs of G, that does not appear in any pair of 6‘. A series G in G which does not admit a proper refinement is called a composition series of G, its factors are called composition factors of G. Clearly, a composition factor of G is a simple group. By definition, simple groups which appear as composition factors of some group, are called absolutely simple. A normal series G of G which does not admit a proper normal refinement is called a chief series of G, its factors being called chief factors of G. Chief factors are characteristically simple, that is, they do not have any proper characteristic subgroup. By the maximum principle of set theory, every series in G has a refinement which is a composition series of G, and every normal series has a refinement which is a chief series of G. It should be pointed out that, in contrast to the case of a finite group G, the sets of isomorphism types of factors of two distinct composition (chief) series of an infinite group may well have empty intersection. Also, a chief factor of an infinite group G need not be a direct product of isomorphic simple groups, a fact first observed in McLain [l], see also P. Hall [4].We now formulate a question without pursuing it any further (for a partial answer see 1.B.3). Question 1.1. Zs every simple, n-Jnite group absolutely simple? Does every characteristically simple, n-jinite group occur as a chieffactor of some n-jinite group?
If the index set Z is well-ordered (respectively, inversely well-ordered), a series G of type Z in G is called an ascending series of type Z (respectively, a descending series of type Z). In this case A , = V,+ and we omit the second group of the pair. The subgroup S of the group G is serial in G, if there is a series G in G, so that S belongs to one of the pairs of G; S is ascendant (respectively, descendant) in G if the series (5 in G can be chosen to be ascending (respectively, descending); if the series G can be chosen so that it consists only of finitely many pairs, then S is subnormal. Let X be any class of groups: by that we understand that (1) E X and that with the group X also every group isomorphic to X belongs to 2.The series C 5 in the group G is an X-series, if all the factors of G belong to 2. The group G is a hyper-X-group if it has an ascending normal X-series. 1.A.5 Lemma. Zf X is a class of groups such that with the group XaIso every homomorphic image of X belongs to X, then the group G is a hyper-X-group if” and only if” every homomorphic image H # <1) of G has a normal Z-subgroup N # (1).
6
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1,
8A
Proof. If G is a hyper-X-group, let (5 = { N z ;I E I } be an ascending Xseries of normal subgroups of G . If u is a homomorphism of G onto the group H, then let 1be the smallest index in the well-ordered set I such that N," # (1). Then the subgroup N," # (1) is a normal %-subgroupof H , for 1 cannot be a limit ordinal. If every homomorphic image H # (1) of G has a normal %subgroup N # (I), then one can define an ascending %-seriesof normal subgroups of G by transfinite induction. Let v be any ordinal of cardinality at most IGI. Assume for all ordinals p < Y the subgroup N,, has N,, . If v is not already been chosen. If v is a limit ordinal, define N , = a limit ordinal, it has a predecessor z, z 1 = v. If GIN, = (I), put N, = N , = G . If GIN, # (l), then GIN, has a normal X-subgroup H / N , # (1). Choose N, = H. In this way, one defines an ascending %-series of normal subgroups of G of order type smaller than the smallest ordinal of cardinality larger then IGI. I
+
up<,
1.A.6 Lemma. If X is a class of groups such that with the group X also every normal subgroup of X belongs to 3,then every normal subgroup K # (1) of the hyper-%group G contains a normal X-subgroup L # (1) of G. Proof. Let 6 = { N L ;I E I> be an ascending X-series of normal subgroups of G . Let 1 be the smallest element of I such that L = K n N , # (1). Clearly L is a normal subgroup # (1) of G. We have to show it is an %group. Obviously the index 1is not a limit index, so it has a predecessor p, p+ 1 = 1, and NJN, is a normal X-subgroup of GIN,,. The intersection L = K n N , is isomorphic to (KN,, n N,),", and so is an X-group. I For certain classes of groups X apparently rather mild assumptions on hyper-%-groupsalready imply rather stringent further properties. This remark is substantiated in the following exercise on the class $,% of hyperfinite groups. Exercise. Prove that the infinite, hyperfinite group G satisfies the following two properties. a. If the centre of the product of all the finite normal subgroups of G is trivial, then ( 1 ) is the intersection of all normal subgroups of finite index in G ; b. If all the elementary abelian 2-subgroups of G are finite, then G has an infinite abelian, normal subgroup. (Observe that the condition in b implies by the Feit-Thompson Theorem that in G there are only finitely many non-abelian minimal normal subgroups.)
CH.
1,
6.41
n-FINITE GROUPS, SERIES A N D LOCAL SYSTEMS
7
If the normal subgroup N of the group G is a hyper-X-group for some class of groups, it is useful to know whether for every homomorphism a of G with N" # < I ) there exists in N" a non-trivial normal X-subgroup of the group G". In general this will not be the case, so that whether this type of result holds will depend either on the particular class X of groups or on the way the normal subgroup N is embedded in G. We shall not push this cursory remark any further. The group L is said to be locally cyclic, if every finitely generated subgroup of L is cyclic.
X
1.A.7 Proposition. Let N be a normal subgroup of the group G such that f o r every homomorphism u of G with N" # (1) there exists in N" a non-trivial, locally cyclic, normal subgroup of G". If A is maximal among the abelian normal subgroups of G contained in N, then A = CNA. Proof. Assume, if possible, that A $ C = CNA. Let B be maximal among the normal subgroups of G satisfying A = B n C. Then
NB/B 2 CBIB N C / ( B n C ) = CIA # (1>. Thus there exists in NB/B a non-trivial, locally cyclic subgroup D/B normal in GIB. By the maximality of B one has C n D = E 2 A. The factor group E/A is locally cyclic, since E/A
=
E/(B n C n D )
N
EB/B
=
( D n CB)/B.
Now E is a normal subgroup of G , and since A% Z E and E/A is locally cyclic, E is abelian. This contradicts the maximality of A. I
1.A.8 Proposition. Let N be a normal subgroup of the group G such that f o r every homomorphism u of G with N" # (1) there exists in N" a non-trivial, abelian, normal subgroup of G". If A is maximal among the abelian normal subgroups of G contained in N, then A is contained in a nilpotent subgroup B of class two of C, A such that B is normal in G and satisJies A = C , B.
Proof. If A = C,A, then put A = B. So assume A # CNA, and let B be maximal among the normal subgroups of G contained in CNA satisfying B' E A. One easily has that B # A. The normal subgroup B of G is nilpotent of class at most two, and by construction, A E ZB. Since A is maximal among the abelian normal subgroups of G contained in N , one has A = ZB = B n C NB. If C NB # A , then let D be a normal subgroup of G maximal with respect to the properties A E D c C N B and D' E A. Then D # A, and A = B n D. Thus the subgroup BD is a normal subgroup of G
8
BASIC METHODS, CONCEPTS AND EXAMPLES
contained in C, A , and it clearly satisfies (BD)' imality of B. This contradiction shows that A
E
=
[CH.
1, $
A
A , contradicting the max-
C,B. I
Exercise. If ( G J I Eis I an ascending series of the group G with locally finite factors prove that G is locally finite. Show that this is false in general if the series is not ascending.
Studying the properties of a group G with respect to some class X of groups, one of the natural objects of interest is the set of all X-subgroups of G. How well does this set represent G? How well does this set cover the group G? The group G is said to be IocaIIy an X-group, or to belong to the class LX, if every finite set of elements of G is contained in some X-subgroup of G. If every finitely generated subgroup of an X-group is again an X-group, then the group G is locally an X-group if and only if every finitely generated subgroup of G is an 2-group. A particular example of such a class LX is the class L s of locally finite groups. For an arbitrary class 5 of groups, it is very difficult to say anything about the class LX, and the class of those groups in which every finitely generated subgroup actually is an X-group seems to be far too restrictive. So one tries to steer some middle course and define a class of locally X-groups which has some flexibility. The set C of subgroups of the group G is a local system of G if G = S and if for every pair S, T E C there is a subgroup U E Z of G with S, T c U . For the general class 2 of groups a subclass of LX, which still is rather general, but on the other hand considerably more flexible than LX, is the class of those groups which have a local system consisting of X-subgroups. Observe that this class of groups contains the class 5. We shall not derive now any formal properties of this class of groups, but shall describe the local systems required whenever we need them. However, for later reference, we record the following characterization of countable, locally finite groups.
USEI
1.A.9 Lemma. The group G is a countable, locallyfinite group ifand only if there is a local system C of G consisting of finite groups and linearly ordered by inclusion. Proof. If the group G is locally finite and countable, then enumerate the elements of G and put S, = ( g i ; 1 5 i S n ) . The system of (distinct) subgroups in this sequence, obviously, is a local system of G consisting of finite groups. For the converse, let C be a local system of the group G consisting of finite
CH.
1, $ A ]
II-FINITE GROUPS, SERIES A N D LOCAL SYSTEMS
9
subgroups of G and linearly ordered by inclusion. Clearly, G is a locally finite group; and since Z is, in fact a well-ordered sequence of finite groups, G is countable. I
Remark. The reader should convince himself of the fact that there exists a group G with a local system consisting of finitely generated subgroups of G and linearly ordered by inclusion, which is uncountable. If Z is a local system of the group G then one can often omit many of the members of C and still have a local system of G . The following easy but very useful lemma describes such a situation. l.A.10 Lemma. If the local system Z of the group G is a union offinitely many subsets, say Z = Ui= Zi,then at least one of these subsets Ziis a local system of G. Proof. If none of the subsets Cj is a local system of G, then for each index i there either exists an element x iE G that does not lie in any membir of Zi or there exist two subgroups S i , TiE Zi such that no subgroup of G containing both belongs to Ci.Let U iE Z be a subgroup containing either the eleBy the definition of a local system ment x ior the two subgroups Si and Ti. there exists a subgroup U E Z containing the subgroups Ui,1 6 i 5 r. By choice of the U i , the subgroup U cannot belong to any of the subsets C i of Z, and so cannot belong to C. This contradiction shows that one of the subsets Zi of C must be a local system of G. I We now give three standard ways in which examples of locally finite groups are constructed. i Let {HJnEN be a sequence of finite groups. Put HI = GI, and define inductively G , + , = H,,, 2 G,, the standard wreath product. In this construction, there is an obvious embedding of the group G,, into G,,,. Thus one may consider the sequence {Gn)nsN as an ascending sequence of finite groups. Clearly, the group G = G, is a locally finite group, and the subgroups {G,) form a local system of G . ii For any infinite set 52 let G be the group of all those permutations of s2 which leave all but finitely many elements of SZ fixed. Clearly, this group G of almost trivial permutations of 52 is a normal subgroup of the group of all permutations of the set 52 (see, for example Scott [I], Chapter 11or Wielandt [2] for further information about that group). The group G is locally finite; it has a local system consisting of finite symmetric groups, one for each finite subset of 52. The group G is usually called the restricted symmetric group on Q. It has a simple subgroup of index two, the alternating group on 52, consisting of all even permutations on Q.
u:=l
10
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1, 8
B
iii If F is an (infinite) algebraic extension field of a finite prime field, then every finite set of elements of F is contained in a finite subfield of F. Thus the field F is locally finite. (Conversely, every locally finite field is an algebraic extension of a finite prime field.) The group GL(n, F) of all invertible n-by-n matrices over F i s locally finite. It has a local system consisting of finite groups GL(n, Fl), one for each finite subfield Fl of F. Section B. Locally soluble chief factors and some characteristic subgroups
This section consists of two parts. In the first we show that the chief factors of locally nilpotent (respectively, locally soluble) groups are central (respectively, abelian). These results, originally due to Mal’cev, are proved here by a method due to McLain [2]. In fact his idea accounts for results 1.B.3 to 1.B.8. In the second part of this section a method is described for transferring information about the subgroups belonging to some local system ,Z of finite subgroups of the locally finite group G, to the group G itself. Starting from certain characteristic subgroups of the groups S E C one defines a characteristic subgroup of G. This method is then given some substance by several important examples. A group is called hypercentral, if it coincides with its hypercentre, which is the terminal member of its (transfinite) upper central series. l.B.l Lemma. Let H be the hypercentre of the group G. If the normal subgroup N of G satisfies N n H # (l), then N n ZG # (1).
uv
Proof. Since N n H # (1) and H = Z,G, there is a least ordinal a such that N n Z,G # (1). Clearly a cannot be a limit ordinal. Thus [G, N n Z,G] c N n Za-l G = (I), and so N n Z,G is a subgroup of the centre of G. 1 1.B.2 Lemma. Every hypercentral group is locally nilpotent. Proof. Let G be a finitely generated hypercentral group and X a finite set of elements generating G. If o is the first infinite ordinal, choose any element a E Zm+ G. There exists a finite ordinal i (depending on a and X)such that [a, x] E Zi G for every element x E X.This means that aZ, G is an element of the centre of the factor group G/Z,G. Hence a E Zi+ G c Z,G, and thus Z,G = G. Since G = Z i G and X is a finite set, there exists a finite ordinalj such that X c ZjG, and so G = ZjG is nilpotent of class at most
Ui<,
j.
I
CH.
1,
5 B]
LOCALLY SOLUBLE CHIEF FACTORS
11
Every proper subgroup of a hypercentral group G is properly contained in its normalizer in G. Groups with this property are said to satisfy the normalizer condition (they are also called N-groups). Plotkin proved that groups satisfying the normalizer condition are locally nilpotent (see for example KuroS [I], Vol. 2,s 63). For some time it had been conjectured that every group satisfying the normalizer condition was hypercentral. That this is not so was recently shown in Heineken and Mohamed [l]. 1.B.3 Proposition. Let M be a minimal normal subgroup of the group G. If for every finitely generated subgroup H of G the intersection H n M is soluble (respectively polycyclic) then M is abelian (respectively elementary abelian). Proof. Suppose that M is not abelian. Then there exists elements a and b of M such that c = [a, b ] # 1. The minimality of Mimplies that M = (c'). Hence there exists a finite subset X of G such that ( a , b ) c (c'). Put H = ( c , X ) . Then c E (c')', so (c") = ( c H ) ' . But by assumption the group H n M 2 (c") is soluble. This is clearly impossible since c # 1, and thus M must be abelian. Now assume that K n M is polycyclic for every finitely generated subgroup K of G. By the first part M is abelian. Suppose that x is an element of M of infinite order. If y = x 2 then the minimality of M implies that M = (y'). Hence there exists a finite subset Y of G such that x E (f). Put K = ( y , Y ) . By assumption K n M is polycyclic and hence is a finitely generated abelian group. But if A = ( x " ) we have x E ( y y ) E A', so A' = A c K n M. This is clearly impossible. Therefore M is periodic and being characteristically simple is obviously elementary abelian. I
1.B.4 Corollary. The locally soluble chief factors of a locally finite group are elementary abelian. 1.B.5 Corollary. The chief factors of a locally soluble group are abelian. 1.B.6 Corollary. The chief factors of a locally polycyclic group are elementary abelian. 1.B.7 Proposition. The chief factors of a locally supersoluble group are cyclic of prime order. Proof. It suffices to prove that a minimal normal subgroup M of a locally supersoluble group G is cyclic of prime order. By 1.B.6 the group M is elementary abelian. Let F be any finite non-trivial subgroup of M. Every
12
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1,
5B
non-identity element of F generates M as a normal subgroup of G, and thus any element of F is a product of conjugates (in G) of any non-identity element of F and their inverses. Therefore, there exists a finitely generated subgroup H of G such that F E H and F c ( x " ) whenever x E F and x # 1.
But then every normal subgroup of H either contains F or has trivial intersection with F. Thus F is isomorphic to a subgroup of some chief factor of H, and hence is cyclic. Therefore M is cyclic. I
l.B.8 Corollary. The chief factors of a IocaIIy nilpotent group are central of prime order. Proof. Let M be a minimal normal subgroup of a locally nilpotent group G. By 1.B.7 the group M is cyclic of prime order. Hence G/CGM is a finite group and so G contains a finitely generated (and hence nilpotent) subgroup N such that M E N and N CGM = G. Clearly M is a minimal normal subgroup of N and so by l.B.l it follows that M is central in N. Hence M is central in G. I About chief factors of locally finite groups in general we know very little. For example we have been unable to show that every chief factor of a locally finite group is a direct product of isomorphic simple groups. The following exercise gives the relative forms of 1.B.7 and 1.B.8 corresponding to 1.B.3.
-
Exercise. N is a normal subgroup of the group G . If for every finitely generated subgroup H of G the group H n N is supersoluble (respectively, nilpotent) prove that every chief factor of G covered by N is cyclic (respectively centralized by N ) . Denote by t a rule which selects for every finite group S a subgroup FS of S. Thus F is (apart from possible set-theoretic problems) a function defined on the class of all finite groups; it may select quite different subgroups in two isomorphic finite groups. The rule t is called a radical (respectively, co-radical) rule, if for every pair S, T of finite groups with S E T,one has
S n t T E FS (respectively, S n FT 2 FS). If G is a locally finite group and Z a local system of G consisting of finite subgroups of G, then for any radical (respectively, co-radical) rule t: defined on the class of all finite groups define a subgroup r(G, C) of G by r(G, C)
=
U ( n (sn E T ) )
SET TEZ SET
(respectively, F(G, C)
=
SET
u
u
( TEZ ( S n FT))= SET FS). SCT
CH.
1, 5
13
LOCALLY SOLUBLE CHIEF FACTORS
B]
Once we have shown that these subgroups t(G, Z) are in fact independent of the particular local system Z of G, we shall put t G = t(G, Z) and thus extend the domain of definition of the rule t from the class of all finite groups to the class of all locally finite groups. 1.B.9 Proposition. Let t be a radical (respectively, co-radical) rule dejined on the class of allfinite groups. For every locallyjnite group G and every local system Z consisting offinite subgroups of G, the subgroup t(G, I)of G defined above does not depend on the particular local system Z of G. Extending the dejinition of the rule t to the class of all locally jinite groups by putting t G = t(G, C) for any local system Z of G consisting offinite subgroups, one obtains a rule (still denoted by t) which is radical (respectively, co-radical).
Proof. Let Z and Z, be any two local systems of the locally finite group G, both consisting of finite subgroups. Observe that if G is finite, then G E Z and G E Z,, and the above definition shows that t ( G , Z) = t G = t(G, Z,).
Thus we may assume that the locally finite group G is infinite. Let S E Z, and Put R = (S n tT) (respectively, R = ( S n tT)).
u
n
T.E__ E
TET SGT
SC T
Since Z, is a local system, since S is finite and since t is a radical (respectively, co-radical) rule on the class of all finite groups, there exists a subgroup U E Zlsuch that for every finite subgroup X of G containing U one has S n t X = R. Thus
R c
VET1 UCV
U nrV
c t(G,
Z,)
(respectively R
u
c
U n t V E t(G, Z,)),
VET1
UEV
and so t(G, Z) E t(G, Z,) (respectively t(G, Z)
c t(G, Z,)).
By symmetry t(G, Z,) E t(G, Z) (respectively t(G, Z,) E t(G, Z)),
and so in either case t(G, Z) = t(G, Z,) = tG, which is therefore independent of the local system used to define it. Now let Z denote the local system of G consisting of all the finite subgroups of G. If H is a subgroup of G, then
14
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1, 5
B
using that t is radical on finite groups, respectively, H n t G = V(HntS)? U t ( H n S ) = t H ) . l SET
SET
If the radical (respectively, co-radical) rule t is defined on the class of all locally finite groups then one has in general only tG
G
t(G, Z) (respectively, t G 2 t(G, Z)),
where Z denotes the local system consisting of all finite subgroups of the locally finite group G. The radical (respectively, co-radical) rule t is smooth on the class of all locally finite groups if t G = t(G, Z).1.B.9 states that for every radical (respectively, co-radical) rule t defined on the class of all finite groups there exists a (unique) smooth extension 1: to the class of all locally finite groups. Further the proof of 1.B.9 shows that for a locally finite group G and a smooth radical (respectively, co-radical) rule t the subgroups R
=
n ( S n FT)
Tfinite SGT
(respectively, R =
u
( S n TT))
SCT T finite
form a local system of finite subgroups of the group tG. Also, for every local system Z, consisting of finite subgroups of G, and every S E Z one has for every smooth radical (respectively, co-radical) rule t defined on the class of all locally finite groups that there exists a subgroup T EZ such that S n t G = SntT. Many of the rules defined on the class of all finite groups, that are useful in the theory of finite groups, have a functorial property: for every isomorphism ct of the finite group S to the finite group T we have (tS)"= tT. Call such a rule functorial. Observe that a functorial rule t associates to every finite group S a characteristic subgroup tS of S and that it is essentially defined on the isomorphism types of finite groups. One easily checks that the smooth extension of the functorial and radical (respectively, co-radical) rule t from the class of all finite groups to the class of all locally finite groups is again a functorial and radical (respectively, co-radical) rule. We give now several examples of functorial and radical (respectively, coradical) rules that occur frequently in finite group theory. These rules will be defined on the class of all finite groups, and their smooth extension to the class of all locally finite groups will usually be described - differently from 1.B.9 in a group-theoretically concrete way. The occasional verification is left to the reader as an Exercise.
CH.
1,
5
B]
LOCALLY SOLUBLE CHIEF FACTORS
15
1 For any fixed set n of primes let 0,be the functorial radical rule that associates to each finite group S its largest normal n-subgroup 0,S (= (1) if n is empty). The smooth extension of 0,to the class of all locally finite groups associates to the locally finite group G its largest normal n-subgroup 0, G. If n is the set of all odd primes, then the rule 0,is often simply denoted by 0,and correspondingly the subgroup 0,G of the locally finite group G by OG.
2 Let 8 be the functorial radical rule that to each finite group S associates its largest (locally) soluble normal subgroup SS. The smooth extension of S to the class of all locally finite groups associates to the locally finite group G its largest locally soluble normal subgroup 8G.
Remark. In an infinite group, in general there is no unique largest locally soluble normal subgroup, see Robinson [3], so that it exists in this case is a particular property of locally finite groups. Since the following property of 0 holds in finite groups, it holds in locally finite groups: @(G/OG) = (1).
3 Let n be the functorial and radical rule that to each finite group S associates its largest (locally) nilpotent normal subgroup nS.The smooth extension of n to the class of all locally finite groups associates to the locally finite group G its largest locally nilpotent normal subgroup nG. Remark. In every group G there is a unique maximal locally nilpotent normal subgroup nG, the Hirsch-Plotkin radical of G, see Robinson [3]. 4 For the fixed natural number n let Z, be the rule that associates to each finite group S the n-th term Z,S of its upper central series. Z, is a functorial and radical rule. Its smooth extension to the class of all locally finite groups associates to every locally finite group G the n-th term Z, G of its ascending central series. 1' For any set n of primes let 0"be the functorial and co-radical rule that associates to each finite group S its smallest normal subgroup 0"ssuch that the factor group S/O"S is a n-group. The smooth extension of 0"to the class of all locally finite groups associates to every finite locally group G its smallest normal subgroup O"G such that G/O"G is a n-group. 2' Let b be the functorial and co-radical rule that to each finite group S associates the smallest normal subgroup b5' such that S/bS is (locally) soluble. The smooth extension of b to the class of all locally finite groups associates to the locally finite group G the unique smallest normal subgroup bG such that G/bG is locally soluble.
16
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1,
8
6
3' Let C be the functorial and co-radical rule that to each finite group S associates the smallest normal subgroup CS such that S/cS is (locally) nilpotent. The smooth extension of c to the class of all locally finite groups associates to the locally finite group G the unique smallest normal subgroup CG such that G/CG is locally nilpotent. Remark. The groups O"G, bG, CGas defined for an arbitrary locally finite group G by the above properties do not exist for arbitrary groups. Their existence is a remarkable property of the class of all locally finite groups. Combination of functorial rules often yields functorial rules; we emphasize this only for example 1 (so the following can be considerably generalized). Let z and z1 be two sets of primes, let 0,.R 1 be the functorial and radical rule which associates to each finite group S the preimage in S of O,,(S/O,,S). The smooth extension of O,,,, to the class of all locally finite groups asG, in G of sociates to every locally finite group G the preimage, o,,(G/O,GI. When studying finite soluble groups S, one often studies the characteristic subgroups Op,,p S for some prime p , and the factor groups S/O,, ,S. (Here p' denotes the set of all primes distinct from p . ) For infinite, periodic, locally soluble groups this approach is not in general available, since for every nonempty set rc of primes the method of P. Hall [4]yields the existence of characteristically simple, locally soluble n-groups, such that every prime p E rc is the order of an element of the group. Exercise. Generalize the concept of O,,,, (resp. On,n1) to arbitrary pairs of functorial radical (resp. co-radical) rules. Show that while this combination of functorial co-radical rules always yields a functorial co-radical rule, the corresponding statement for functorial radical rules is false. The following characterizations of the subgroups nG and r3G of the locally finite group G are direct extensions of the finite case. Their proofs are left to the reader as an Exercise. l.B.10 Proposition. For a subgroup N of the IocaIlyJinite group G the following conditions are equivalent: i N N N N N
ii iii iv v
= nG; = X O,G, the direct product being taken over all primes p ; = O,,,G, the intersection being taken over all primes p ; = Op,PrG,the intersection being taken over allprimesp; is the intersection of the centralizers of all the chief factors of G.
n
0
CH.
1,
8 C]
17
STABILITY GROUPS
l.B.ll Proposition. The subgroup @Gof the locallyjinite group G coincides with the intersection of all the centralizers of all the non-abelian chief factors of G. Section C. Stability groups
Let
(*I
G = Go 2 GI
2
... 2
G, = ( 1)
be a finite chain of subgroups of the group G. The automorphism said to stabilize the chain (*) if one has (xG,)” = xGi
for every x E Gi-l and 1 5 i
CI
of G is
5 r.
Clearly the set of all automorphisms of G stabilizing the chain (*) is a subgroup of the group of all automorphisms of G ; this subgroup is called the stability group of the chain (*). Stability groups often appear during reduction procedures, and the chains stabilized in such situations usually consist of normal subgroups. This is precisely the case that we shall encounter. Hence weshall prove the main result, namely that stability groups are nilpotent, for this case only, and we shall not present any more than a fraction of the large body of results on stability groups (for further results see P. Hall [l], Hall and Hartley [l], B. Hartley [l]). Throughout this section we shall be dealing with a group of automorpihsms A of some group G; we shall consider both as subgroups of the split extension H = G x A .
l.C.l Theorem. Let G = Go 2 G , 2 . . . 2 G, = (1) be ajinite chain of normal subgroups of the group G, and let A be the stability group of this chain. Then the group A is nilpotent of class at most r - 1. This result first appears in Kaluinin [l 1; it has been improved considerably in P. Hall [I] where it is shown that the stability group A of any chain of subgroups of length r (without any normality assumption) is nilpotent of class a t most +r(r- 1). Proof. We remind the reader of P. Hall’s Three-Subgroup Lemma: If G is a group and X , Y , Z are subgroups of G such that the commutator subgroups [Y,Z , X I and [Z, X,Y ] are contained in the normal subgroup
N of G, then also [ X , Y, Z] Huppert [l], 111.1.10).
E
N (see for example Gorenstein [I], 2.2.3 or
18
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1,
5C
>,
For all natural numbers i 2 r put G i = (1 and denote by A j the j-th term of the lower central series of A . Clearly, each of the groups Ci is normal in the group H = C x A , and by assumption [ G i ,A ] E G i , for each natural number i. Suppose that for each natural number i one has [ G i ,A j ] G G i + j . Then [A,Gi, A j l IGi+17 Ajl Gi+j+l and [Gi7Aj7 A ] E [Gi+j7 A1 G i + j + l . By the Three-Subgroup Lemma [Aj7A7 Gil =
rAj+l,
Gil = [Gi7 A j + l I E G i + j + l .
Therefore by induction on j one obtains [Go, A,] E C, = (1). But A is a group of automorphism' of G , so A , = ( 1 ) and A is nilpotent of class at most r-1. I Now we consider just the special case of a group of automorphisms stabilizing the chain G 2 N 2 (1 where N is a normal subgroup of G . By l.C.l it is clear that the group A is abelian.
>
1.C.2 Proposition. Let N be a normal subgroup of the group G , and let A be the stability group of the chain G 2 N 2 (1). For any pair g E G and a E A one has a the commutator g-'ga = [g, a ] is contained in the centre of N , b the map 0, c1 I-, [g, a ] is a homorphism of A into the centre of N ; c the map ya : g I+[g, a ] is a derivation of G into the centre of N, and the kernel of ya contains N. If the image of G under ya is contained in the centre of G , then ya is actually a homomorphism.
Proof. a For the element n E N one has gn-'g-' E N, since N is normal in G. As the automorphism a centralizes N, we obtain gn-lg-l
=
(gn-lg-l)a
=
gan-lg-a.
Hence g-'ga centralizes n. Since this holds for every element n E N , it follows that g-'ga is an element of the centre of N . I b If a, /? E A , then since A centralizes N and the elements ueg and c If x, y E C , then (XY>'"
= [XY,
.I
peg
commute by a. I
= [x, a-y[y, a] = (X'9yY'3,
CH.
1,
5 D]
19
MAXIMAL p-SUBGROUPS
and ym is a derivation. If the image of G under y a is in the centre of G, then clearly ya is a homomorphism. I Using these mappings one can obtain some more information.
1.C.3 Proposition. Let N be a normal subgroup of the group G, and let A be the stability group of the chain G -Z N 2 (1). Suppose that X be a set of elements of G such that G = N(X), and denote the centre of N by 2. Then a the map 6 : a H ( ~ 1 is~ an~ embedding ) ~ ~ ~of A into the Cartesian product of 1x1 copies of 2; b the map y : ct H ya is an embedding of A into the commutative group Der(G, 2) of all derivations of G into Z, where 2 is a G-module via conjugation. If the subgroup Z is contained in the centre of G, then y determines an embedding of A into Hom(G/N, 2). Proof. a Since for each element x E X the map Ox is a homomorphism of A , so is the map 8. If ' a = 1, then [x,a] = 1 for every element x E X. But [ N , a] = (I), and G = N ( X ) . Hence a is the identity automorphism of G. Consequently, the map 6 is an injection of A into the Cartesian product of ( X I copies of Z . I b IfgEGanda,pEA,then gY"B
= (app =
(a")(p")
=
pp.
Hence y is a homomorphism. If ay = 1, then g-'g' = gya= 1 for all elements g E G, and consequently a is the identity automorphism of G. Thus y is an injection of A into the commutative group Der(G, 2).If the subgroup Z is contained in the centre of G, then Der(G, 2) = Hom(G, Z). Since for every element a E A the normal subgroup N is contained in the kernel of yo, the injection y induces an embedding of A into Hom(GIN, 2). I Section D. Maximal p-subgroups If p is a prime, for any group G denote by Max, G the set of maximal p-subgroups of G . By the maximum principle of set theory, everyp-subgroup of the group G is contained in at least one maximal p-subgroup of G, so the set Max, G is not empty. From the point of view of finite groups it seems natural to ask whether all the elements of the set Max, G are conjugate in G, but that this need not be so, even in a locally finite group G, has been pointed out in the Introduction. We shall see the details of a further example of this rather common occurrance in 3.4. In the literature the elements of Max, G have often been called Sylow p-subgroups of G (see, for example KuroG [11,
20
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1, 8
D
9 54 and Q 85, Schenkman [l], p. 125; Scott [l], Chapter 6). We shall abstain
from this usage because of the general absence of a satisfactory theory for Max, G along the lines of the theory of Sylowp-subgroups of finite groups, and shall reserve the name of Sylow p-subgroups of G to a rather special class of elements of Max, G which will appear on the scene in Chapter 3. The question of the conjugacy of the elements of Max, G has stimulated considerable research activity; the note at the end of this section is a short guide to the literature. We shall use this problem mainly to underline again the importance of local systems consisting of finite subgroups for the general structure theory of locally finite groups. Let C be a local system consisting of finite subgroups of the (locally finite) group G. We shall say that the p-subgroup P of G reduces into the local system C if for every subgroup S E C the intersection S n P is a Sylowp-subgroup of the finite group S.
l.D.l Lemma. The p-subgroup P of the locallyfinite group G is a maximal p-subgroup of G i f there is a local system C of G consisting offinite subgroups such that P reduces into C.
Proof. Suppose ,Z is a local system of G consisting of finite subgroups, such that P reduces into C, and assume, if possible, that there is a p-subgroup P I of G containing P properly. Choose any element g E P,\P. The element g lies in some subgroup S E C, and (9, S n P) is a p-subgroup of S, properly containing the intersection S n P. But, since P reduces into C, the intersection S n P is a Sylow p-subgroup of S, and thus cannot be properly contained in anyp-subgroup of S . This contradiction shows that there cannot be any p-subgroup P, of G properly containing P. I Since, in a way, the elements of Max, G that reduce into the local system C of G are controlled by C, one wonders for which local systems C are there elements of Max, G reducing into C. That this can only be so for local systems C not containing “too many” of the finite subgroups of G, is made clear by the following lemma. 1.D.2 Lemma. The locally finite group G is p-closedfor the prime p, if and only if there is ap-subgroup P of G reducing into the local system C consisting of allfinite subgroups of G. Proof. Given P we show that all the subgroups S E ,Z are p-closed. If this were not so, then there would be a finite subgroup S of G that is not p-closed and hence has two distinct Sylow p-subgroups, R and S n P. But, by the assumption on C,one has R E C, and since P reduces into Z, the intersection
CH.
1,
5 D]
MAXIMAL p-SU0GROUPS
21
R n P is a Sylow p-subgroup of the p-subgroup R. Thus one obtains R c S n P,contradicting the assumption that R and S n P are two distinct
Sylow p-subgroups of S. Hence every subgroup S E Z is p-closed. If on the other hand the group G is p-closed, then there is but one element P in Max, G, and this clearly reduces into every local system Z of G which consists only of finite subgroups of G. 1 If the locally finite group G is countable, however, one can always assert the existence of a local system Z of finite subgroups of G such that some element of Max, G reduces into Z. This is an indication of how weak this vague notion of “control”, of the properties of p-subgroups of G reducing into Z by the properties of Z, actually is.
1.D.3 Proposition. ff the local system Z of the group G consists ofjinite subgroups only and is linearly ordered by inclusion, then there is a p-subgroup P of G which reduces into C. Proof. By assumption Z = {Sn}nE,with S, E S,,,. For each natural number n choose a Sylow p-subgroup P, of the finite group S,. Inductively, choose elements g, E S,, such that the set {P,““} is linearly ordered by inclusion (put gI = 1 and assume that such a choice has been made for all natural numbers S k; then by Sylow’s Theorem there exists an element g k + E s k + such that Pp c P,&+t). But then the subgroup P = Uns,P,”” is ap-subgroup of G and reduces into the local system Z. 1 Observe that every finitep-subgroup of G is conjugate to a subgroup of the maximal p-subgroup P of G so constructed.
,
1.D.4 Lemma. If K is a normal subgroup of the locally jinite group G such that the factor group GIK is a countablep-group for some prime p then there is a p-subgroup P of G with KP = G. Proof. Since the factor group GIK is countable, there is an ascending sequence {G,Jns, of subgroups G, of G containing K and such that the factor groups G J K are finite and G = G,. Now choose any finite subgroup S, of GI such that S,K = G , , and inductively, if S, is already chosen, choose as S,,, any finite subgroup containing S, such that S,, , K = G,+,. Put S = S,. Every maximal subgroup P of the subgroup S that reduces into the local system (S,},,, of S will meet our requirements. By 1.D.3 there are such subgroups. I The type of argument used in the proofs of 1.D.3 and 1.D.4 has many direct applications for the structure of countable, locally finite groups, see the following Exercise.
UnsN
uns,
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Exercise. Show that in the countable, locally finite-soluble group G for every set n of primes (with complementary prime set 7~') there is a maximal n-subgroup P and a maximal d-subgroup Q of G such that G = PQ. If the local system Z, of the group G consists of finite subgroups only, then denote by Max, C the set of all those p-subgroups of G reducing into C. By 1.D.1, one has Max, C E Max, G . In this setting the following problem may be worth recording, although the following discussion will scarcely shed any light on it. Question 1.2. If the local system C of the group G consists offinite subgroups only, are then any two p-subgroups of G that reduce into Z isomorphic? This problem is recorded in Rae [l] for the special case that for any S, T E C, the inclusion S E T implies that S is a subnormal subgroup of T; in this situation Max, G = Max, C. Examples in Rae [ I ] and Wehrfritz [8] show that if such an isomorphism exists, it is in general not induced by an automorphism of G . The situation described in 1.D.3 contains information about the cardinal of IMax, CI. 1.D.5 Lemma. If the local system Z = of the group G consists of finite subgroups only and is linearly ordered by inclusion, and i f for infinitely many natural numbers n there are two subgroups P,,, Qn E Max, C such that
Snn Pn = Sn n Q n , but sn+, n Pn # then IMax,C]
=
sn+,
n Qn,
2'O.
Proof. Clear. I This easy observation has as an immediate corollary. 1.D.6 Lemma. If the local system C of the group G consists of finite subgroups only and is linearly ordered by inclusion, and i f furthermore IMax, CI < 2'O, then for almost all natural numbers n the subgroup P E Max, Z is the unique element of Max, C intersecting the finite group Snin S,, n P.
Despite their inherent simplicity, the last two statements give us the tools to prove our next result. 1.D.7 Proposition. If the local system of the group G consists of finite subgroups only and is linearly ordered by inclusion, then the following three properties of the set Max, C are equivalent: a the elements of Max, C are conjugate in G; b [Max,CI < 2'"; c the set Max, C is finite.
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Proof. If the elements of Max, C are conjugate in G, then /Max, Cl 5 N: , since the group G is countable. Thus a implies b, and obviously c implies b. Thus, we shall assume b and prove a; the finiteness of the set Max, C will be obtained en route. By 1.D.6, for almost all natural numbers n, every Sylow p-subgroup P,, of the finite group S, determines a unique subgroup P E Max, C with P, = S,, n P. If for such a natural number n and subgroup Q E Max, C one considers the intersection S,, n Q, then there is an element s E S,, such that P,, = S,,n Qs,and thus P = @. The number of different elements of Max, C is therefore the index IS,, :N,,P,,/. 1 The following result of Asar [I] further exhibits the connection between the number of maximal p-subgroups of a locally finite group and their conjugacy. See also B. Hartley [ 6 ] . 1.D.8 Theorem. I f for someprime p, every countable subgroup of the locally finite group G contains only countably many maximal p-subgroups, then the maximal p-subgroups of G are all conjugate. Proof. Suppose that P and Q are two non-conjugate maximal p-subgroups of G. We show firstly that at least one of the groups P and Q has the property (*) that each of its finite subgroups lies in at least two maximalp-subgroups of G. Suppose that X is any finite subgroup of P and that Y is a finite subgroup of Q such that Q is the only maximal p-subgroup of G containing Y. SyIow's Theorem applied to the finite group (1, Y ) shows that for some element g of G the subgroup ( X , Yg)is a p-group. By the assumption on Y it follows that ( X , Y g ) c Qg.Now P # Qg as P and Q are not conjugate, and hence X lies in the two maximal p-subgroups P and Qgof G. Thus if Q does not satisfy (*) then P does, and we may assume that P satisfies (*). If X is any finite subgroup of P there exists a maximal p-subgroup R # P of G containing X. Let x E P R\ and put T = P n ( X , x ) . Since ( X , x) is a finite p-subgroup of R there exists an element t of ( X , x)\T normalizing T. If t normalizes P, then since t is ap-element, the maximality of P would imply that t IZ P, and this is false by construction. Thus (P, P ' ) is not a p-group and so there exists a finite subgroup X o of P containing X such that if X , = X t , then ( X o ,X , ) is not ap-group. Note that X # X o since ( X , X ' ) c T, ap-group. We may repeat the above construction with X replaced by X o (and hence also by its conjugate X,) and thus construct X o o , Xol, X,, and X,, . We then repeat the construction with each of the X i j . Doing this infinitely often we construct sequences
x
c xi, c xi,i2c
. . . c X i , i 2 . . . i nc . . .
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D
of finite p-subgroups of G, where each i, is either 0 or 1, and for n E N the group <Xilil
...ini
3
X i l i z . . .i,j>
is not ap-group whenever i # j. Hence the unions of two distinct such chains are countable p-subgroups of G that are not both contained in any single p-subgroup of G. It follows that the countable subgroup of G generated by the countably many finite subgroups X i l i 2 . , of G, contains 2"O maximal p-subgroups. This contradiction shows that no such subgroups P and Q can exist and completes the proof. I The above proof in fact shows that if for some prime p, every countable subgroup of the locally finite group G contains less than 2*O maximal psubgroups then the maximal p-subgroups of G are all conjugate. The above theorem has a number of corollaries. If S is any subgroup of the group G, then every maximal p-subgroup of S is contained in at least one maximal p-subgroup of G and no two distinct elements of Max, S can lie in a single element of Max, G. Thus IMax, SI 5 JMax, GI, and the following result (of Asar [l] and B. Hartley [4])is an immediate consequence of the proof of 1.D.8. 1.D.9 Corollary. If p is a prime and G is a countable locally finite group, then the maximal p-subgroups of G are all conjugate if and only if JMax, GI < 2O '. l.D.10 Corollary. If the group G satisfies the hypotheses of 1.D.8, then in every factor of G the maximal p-subgroups are all conjugate. Proof. Trivially every subgroup of G satisfies the hypotheses of 1.D.8. Hence it sufficesto show for any normal subgroup N of G, that GIN satisfies the hypotheses of 1.D.8. It follows from 1.D.4 that for any countable subgroup H of G
]Max, H N / N J= /Max, H/(H n N ) J S IMax, HI. The corollary follows from this remark. I
niSN
Notice that if G denotes the Cartesian product Tiof countably many copies of the non-abelian group of order six, then any two maximal 2-subgroups of G are conjugate, while its subgroup G corresponding to the direct product of the Tidoes not have its maximal 2-subgroups all conjugate. The point is that G is a countable subgroup of G with O 2' maximal 2-subgroups.
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Question 1.3. What can be said about the structure of a group G such that for the (resp. every) primep and everyfactor S of G, the maximalp-subgroups of S are all conjugate in S? A rather complete discussion of these questions for locally soluble groups G can be found in B. Hartley [4] and [ 5 ] .
l.D.ll Lemma. If the countable locally finite group G contains a normal subgroup H of finite index such that H has only countably many maximal psubgroupsfor someprime p, then G also contains only countably many maximal p-subgroups. Proof. Suppose that G has uncountably many maximalp-subgroups. Then there exists a maximal p-subgroup Q of H such that W = { P E Max, G; P n H E Q } is uncountable. For each P E W there exists a finite subgroup Po such that P = ( P n H)P,. But G contains only countably many finite subgroups, and hence there exist two distinct subgroups P and R in W for which Po = R,. But then ( P n H, R n H, P o ) is a p-subgroup of G containing both P and R (since
1.D.12 Lemma. If the IocallyJinite group G contains a$nite maximalp-subgroup P,then the maximal p-subgroups of G are all conjugate to P. Proof. If Q is any finite p-subgroup of G then Sylow’s Theorem applied to the finite group (P,Q ) yields that some conjugate of Q lies in P. Thus every p-subgroup of G is finite and if Q is actually a maximal p-subgroup of G, then P and Q are conjugate. I
26
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Appendix. Conventional Sylow theory and related ideas
A considerable amount of work has been done to extend the standard theorems on Sylow subgroups and Hall subgroups of finite groups to various classes of infinite (almost invariably locally finite) groups. The sort of result we have in mind here, in contrast to the type of result to be proved in the earlier part of Chapter 3, is for example: in such-and-such a group the maximal p-subgroups are all conjugate. In order to obtain such a result heavy restrictions have to be imposed upon the group and a large number of nasty examples of every shade and hue have been amassed. See for example K O V ~ CNeumann S, and de Vries [l 1, B. Hartley [7], and Wehrfritz [7]; see also 3.3 and Heineken’s example given in the Introduction. Because of this the usefulness of such results seems to be virtually nil, at least to date, and this topic forms no natural part of our narrative. Indeed the “unconventional” Sylow theory of Chapter 3, although very useful in that context, is applicable really only to very limited situations. However rather than ignore completely such a large body of results, we propose here to give a short survey of the state of (our) knowledge at present. We have already proved (1.D.4) the theorem of Asar [l] and B. Hartley [4] that the maximal p-subgroups of a locally finite containing only countably many maximal p-subgroups are all conjugate. A similar type of result that has been known for a long time (see Kurog [l], 9 54) is that a group G has a single conjugacy class of maximal p-subgroups whenever it has a finite such class. We shall give as 3.4 a result of Zalesskii [ l ] and Wehrfritz [l] which implies that if G is a locally finite group, such that the lattice of centralizers and the collection of p-subgroups of G both satisfy the minimal condition, then the maximal p-subgroups of G are all conjugate. An immediate corollary, due originally to Baer [5], is that the maximal p-subgroups of a locally finite group satisfying the minimal condition on subgroups are all conjugate, a result that is now an easy consequence of the structure theorems of Chapter 5. The proof of 3.4 that we give is essentially Zalesskii’s, Wehrfritz [l] takes a different approach. He proves there, and also in Wehrfritz [2], results of the following kind. If in sufficiently many subgroups of the group G the maximal p-subgroups are all conjugate, then the maximal p-subgroups of G are all conjugate, or at least nearly so. The result that leads to 3.4 is the following. If G is a locally finite group whosep-subgroups all satisfy the minimal condition on subgroups, and if in the centralizer of every non-central p element of G the maximal p-subgroups are all conjugate, then the maximal
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MAXIMAL P-SUBGROUPS
21
p-subgroups of G are all conjugate. I n other results of this type the distinguished subgroups are normalizers rather than centralizers. A further corollary of 3.4 is that the maximalp-subgroups are all conjugate in a periodic linear group over a field of characteristic other than p. This is true without the latter restriction, that is, for every prime p the maximal p-subgroups of a periodic linear group are all conjugate (Platonov [I] and Wehrfritz [4]). Wehrfritz [2] also proves an abstract generalization of this theorem for the case when p is the characteristic of the ground field. A number of papers contain generalizations of the Cunihin-Hall, SchurZassenhaus and Wielandt theorems concehing the conjugacy of Hall subgroups in finite groups. See Dietzmann, Kurog and Uzkov [l], Dixon [I], Gol’berg [2] and [4], B. Hartley [4] and [6], KazaEkov [I], [4] and [6], Plotkin [l], TygkeviC [1] and Wehrfritz [l], [2], [3] and [4]. Twenty years ago Gol’berg [2] extended P. Hall’s theory of finite soluble groups to soluble groups satisfying the minimal condition on subgroups, and this type of enquiry has been conducted since. Stonehewer [ I ] extends Hall’s theory to the class of periodic locally soluble groups containing a locally nilpotent subgroup of finite index. He includes a study of Carter subgroups of groups in this class and in Stonehewer [3] develops a local formation theory for them. Platonov [ l ] and Wehrfritz [5] extend Hall’s theory to the class of soluble periodic linear groups, the latter paper in facts works with the wider class of all (locally) soluble homomorphic images of periodic linear groups and contains an account of their Carter subgroups. McDougall [2] proves similar results for metabelian groups satisfying the minimal condition on normal subgroups. Gardiner, Hartley and Tomkinson [ l ] extend the above results as follows. They consider the largest subgroup-closed class U of poly-locally-nilpotent, locally finite groups G such that for every set 7c of primes the maximal n-subgroups of G are all conjugate*. They show that a U-group G has Sylow bases, and that the Sylow bases and basis normalizers of G have the expected properties. Furthermore they develop the full apparatus of a formation theory for the class U. B. Hartley [3] and [8] takes this further by considering 3-normalizers and 3-abnormal subroups of U-groups. Apart from the counter examples mentioned above there are some positive results which indicate that to insist that the maximalp-subgroups of a group be conjugate, is to impose a severe restriction on that group. For example the class U is much smaller than might at first be apparent and its previously studied subclasses mentioned above are really quite reflective of its complex* This can be weakened somewhat, see B. Hartley [4].
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ity. B. Hartley [4] proves that a U-group G contains a locally nilpotent normal subgroup N such that GIN is a finite extension of a countable metabelian group. A further striking result of this nature, again due to B. Hartley [ 5 ] , is that a countable locally soluble group isp-separable (that is, contains a series of finite length whose factors are eitherp-groups orp’-groups) if in each of its subgroups the maximal p-subgroups are all conjugate and p 2 3. Analogous Sylow, Hall and formation theories exist for the class of locally finite-normal groups, except that local conjugacy replaces conjugacy throughout. We discuss this Sylow theory briefly in Section K. For further results see Baer [I], Gol’berg [l], Stonehew& [2] and [4] and Tomkinson [I]. P. Hall [2] gives a characterization of the class of countable locally finite-normal groups, some of his ideas being developed further in GorCakov [I] and [3]. Kargarpolov [3] studies semi-simple locally finite-normal groups. Rae [I ] and [2] considers a generalization of this class of groups, some of the ideas in section D playing a role here. Wehrfritz [6] extends to the class of periodic linear groups many of the results concerning the existence of normal complements of Sylow subgroups and Hall subgroups in finite groups that are usually associated with transfer theory. To the best of our knowledge this is the only successful attempt to generalize this type of result to infinite groups. The number of papers which could be mentioned in the above few paragraphs is very large indeed and we make no pretence to doing more than drawing the reader’s attention to some of them. We have included additional references in the bibliography. The interested reader seeking further guidance to the literature is recommended to read sections 83.7, 84.6, 84.7 and 85 of the third edition of KuroS [I].
Section E. The minimal condition on subgroups An important class of groups is the class of those groups that satisfy the minimal condition on subgroups, that is, the class of all those groups G with the property that every non-empty set of subgroups of G, partially ordered by inclusion, has a minimal member. The importance of this class of groups derives from two facts. For groups satisfying the minimal condition on subgroups as well as certain further conditions one may prove rather strong structural properties, and in the study of other classes of groups - linear groups, for example - groups satisfying the minimal condition appear naturally. From this conjunction a considerable amount of research has originated. Consequently, the study of the class of groups satisfying the minimal con-
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THE MINIMAL CONDITION ON SUBGROUPS
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dition on subgroups will - implicitely or explicitely - occupy a considerable part of this text. The present section (and its immediate successors) will record only the most elementary facts about this class and a few related classes of groups. We shall abbreviate the expression “the minimal condition on subgroups” to “min”; and we shall denote the class of all groups satisfying rnin by Min. Groups satisfying min have also been termed Artinian, after E. Artin (1896-1962) who pointed out the importance of the minimal condition (for left ideals) in ring theory. It is an elementary exercise on partially ordered sets that the group G satisfies rnin if and only if G satisfies the descending chain condition for subgroups, that is, if and only if every properly descending chain of subgroups of G has finite length.
l.E.l Lemma. r f G E Min, then every subgroup and every homomorphic image of G belongs to Min. If G, H E Min, then every extension E of G by H satisjies min.
Proof. The first claim is clear from the definition of Min. Let N be a normal subgroup of the group E such that both N a n d E/N satisfy min and let &‘ be any non-empty set of subgroups of E. Since E/N E Min, the set (ANIN; A E d >contains a minimal member BNIN, say, where B E d. Let g = (A E d ;AN = BN}. Since N E Min, the set {A n N; A E g}has a minimal member C n N, say, with C E 97. Clearly, C E d .Let A be an element of d such that A G C. By the choice of B, one has AN = CN = BN, and hence A E 39. Thus, by the choice of C one has A n N = C n N. Therefore, by Dedekind‘s modular law, C = AN n C = A ( C n N ) = A, and consequently, C is a minimal member of d.1 Clearly, the infinite cyclic group does not satisfy rnin, and so l.E.l entails the following observation.
1.E.2 Lemma. Every group satisfying rnin is periodic. By considering an infinite, elementary abelian p-group, one realises that Min # LMin (otherwise, of course, the class L s of all locally finite groups would be contained in Mm). But fortunately, a slightly restricted local theorem still holds.
1.E.3 Lemma. If every countable subgroup of the group G satisjes min, then G E Min.
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Proof. If this were not so, then in the group G there would be an infinite, strictly descending chain {Gi}, Gi 3 G i + l , of subgroups. For each natural number i choose an element gi E G,\G,+ . Put H = ( g i , i E N). This subgroup H is countable and therefore satisfies min by assumption. But by construction one has a strictly descending chain { H n Gi} of subgroups of H. This contradiction shows that the assumption, that such a chain {G,} exists, cannot be maintained. I Examples of groups in Min are easily found: every finite group satisfies min. There is one further basic class of groups satisfying min, namely the class of Prufer groups. Let p be any prime. For each natural number i the cyclic group Cpi+lof order pi+’ contains a unique copy of the cyclic group Cpjof order pi.Choosing an embedding of Cpiinto Cpi+ I for each i we obtain a direct system. The direct limit of this system is called a Cp,-group or a Prufer group, after H. Prufer (1896-,1934). It is easily seen that such a group is uniquely determined up to isomorphism by the prime p, that is, it is independent of the choice of the embeddings. Alternative definitions of a Prufer p-group describe it as an infinite, locally cyclicp-group (this is essentially the above definition) or as the p-primary component of the factor group Q/Z; the exponential map shows that the group Q/Z is isomorphic to the group of complex roots of unity. Every known group satisfying min has a finite sequence of subgroups
,
(1)
=
.
G o E G, G . . E G,
=
G with G i 4 G i + l for 0 S i < n
such that each factor Gi+i / G i either is finite or a Prufer p-group for some prime p. Notice that according to 1.A.2 all these examples are locally finite groups. We shall see below that every such group is even abelian-by-finite, being an extension of a direct product of finitely many Prufer groups by a finite group. Do these examples exhaust the class Min? Question 1.5. Is every group G E Min abelian-by-finite (and hence, in particular, locally finite)? A. Tarski is said to have suggested that there might exist an infinite group G such that every proper subgroup of G has prime order. Such a group obviously would have to be simple, and it would satisfy min as well as the maximal condition for subgroups (every chain of subgroups in fact, consists of three terms at most). Since any two involutions would generate a dihedral subgroup, it is clear that such a group could not contain involutions. The existence of such “Tarski monsters” G is still an enigma, and it seems hard to imagine how one might tackle this problem, even in the case in which the group G has prime exponent.
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In view of Cernikov's prolonged concern with Question 1.5 and his partial solutions to this question, we shall call a group G a C'ernikou group if it is abelian-by-finite and satisfies min. (This slightly extends the use of the name in KuroS [I], 9 64, where only primary groups with this property are called Cernikov groups.) For a wide class of groups an affirmative answer to Question 1.5 is given in 1.E.5. That locally finite groups satisfying min are Cernikov groups has been shown in Sunkov [12] and in Kegel and Wehrfritz [I 1. We shall obtain this result in a considerably more general context in Chapter 5 (see 5.8 or 5.10).
1.E.4 Proposition. If the set of all normal subgroups of finite index of the group G satisfies the descending chain condition, then the intersection JG of all subgroups offinite index in G has itselffinite index in G. Proof. If H is a subgroup of finite index in the group G, then the intersection H galso has finite index in G. Thus the intersection JG of all subgroups of finite index in G coincides with the intersection of all normal subgroups of finite index in G . By the descending chain condition on normal subgroups of finite index G has a minimal normal subgroup of finite index. Let Kand L be two such minimal normal subgroups of finite index in G, then the intersection K n L is also of finite index in G. Thus K = K n L = L, and JG is of finite index in G. I Now we shall discuss locally soluble and locally nilpotent groups satisfying min. The following statement is an immediate consequence of 1.B.4 and 1.B.8.
nsEti
1.E.5 Lemma. Let G be a group such that the set of all normal subgroups If G is locally soluble, then it is hyperabelian; i f G is locally nilpotent, then it is hypercentral. of G satisJies the descending chain condition.
After these formalities we come to the first (and for almost twenty years only) general result pointing in the direction indicated by Question 1.5. It appears in Cernikov [14].
1.E.6 Theorem. If in the locally soluble group G the set of subnormal subgroups satisjies the descending chain condition, then G is a Cernikov group; in particular G is soluble. Proof. By 1.E.4 (and l.E.l) we may assume that G is infinite and does not contain any proper subgroup of finite index. If H is any normal subgroup of G then the commutator quotient group HIH' satisfies min. Hence it is periodic (1.E.2) and only finitely many of its primary components are nontrivial; also every elementary abelian subgroup of H/H' is finite, and so each
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primary component of the abelian group H/H’ has finite rank. Therefore HIH’ is the direct product of some finite abelian group and a finite number of Prufer groups. In particular, if H/H‘ # <1) then it contains a non-trivial finite characteristic subgroup. Now, by 1.E.5 the group G contains a non-trivial, abelian normal subgroup, and by the preceding paragraph this subgroup contains a non-trivial, finite characteristic subgroup F, say. Since G / C , F is isomorphic to the group of automorphisms of F induced by elements of the group G, the index IG : C,FI is finite. Thus F is central in G, since the group G does not have any proper subgroup of finite index. Let Z denote the centre of G. If Z = G , the proof is complete. Thus suppose that G / Z # (1). Then as above we find that the centre of G / Z is non-trivial, and so we pick an element c Z # Z of the centre of the group G/Z. Since this centre is periodic, the element c has finite order r, say. Then the group (c, 2 ) is an abelian normal subgroup of G and hence satisfies min. Thus the subgroup H = (x E ( c , 2 ) ; xr = I} is a finite characteristic subgroup of ( c , 2 ) and hence normal in G. But then we find that G = CGH, and so c E 2,contrary to our choice of the element c. Therefore G = 2, and G is abelian and satisfies min. I
Remarks. The preceding proof, in fact uses only the descending chain condition on the set of subnormal subgroups of defect two of G. (For more on this theme, see Robinson [3].) Moreover the condition that the group G be locally soluble can be slightly weakened by assuming only that G contains a normal series with abelian factors (that is, G is an SI-group in the sense of Kurog [I], 9 57). 1.E.7 Lemma. If the group G has afinite series the factors of which are either finite groups or Prufer groups, then G is a Cernikov group. Proof. By l.E.l the group G satisfies min. By 1.E.4 and a simple induction argument G has a finite normal series
...
( l } = H o d K o ~ H l e K l d dHrQKr=G where each factor H i / K i - is abelian and each factor K J H , is finite. Now for each i the centralizer C,(Ki/Hi)has finite index in G. Therefore the minimal subgroup of finite index in G is soluble, and so it is abelian by 1.E.6. Thus G is a Cernikov group. I A locally soluble group satisfying the minimal condition on norma1 subgroups need not be soluble and hence it need not satisfy min. Thus the minimal condition on normal subgroups is substantially weaker than the
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minimal condition on subnormal subgroups (of defect 2), let alone min. This more complex situation will be discussed later in Section H. We have already seen (1.E.5) that a locally nilpotent group is hypercentral if it satisfies min, and by 1.E.6 that it is a Cernikov group and thus an extension of an abelian group by a finite nilpotent group. Just as a warning, let us point out that such a group need not be nilpotent. Let A be a Priifer 2-group and x the automorphism of A that maps each element of A onto its inverse. Let G be the split extension of A by (x). The group G is a locally finite 2-group, consequently it is locally nilpotent. Also G satisfies min by l.E.l. If a E A , then the group A contains an element b such that b2 = a. Thus one has [x,b ] = (x-'b-'x)b = b2 = a, and so [ A , GI = A . Thus the descending central series of G becomes stationary at A # (l), and G is not nilpotent. The following lemma is a triviality. 1.E.8 Lemma. A group satisfying rnin contains no proper subgroup isomorphic to itselJ:
Section F. Periodic groups of automorphisms of ternikov groups The principal aim of this section is to show that periodic groups of automorphisms of Cernikov groups are themselves Cernikov groups. This result enables one to prove that a group G satisfying min is a Cernikov group, if G does not involve any infinite simple group. The reduction principle obtained in this way will be used in the proof of 0. Ju. Smidt's result: Every 2-group satisfying rnin is a Cernikov group. We prepare the way with two lemmas. l.F.l Lemma. Let N be an abelian, radicable normal subgroup of the group G . V H is a subgroup of G such that [ N , ,H] = (1) for some natural number r, and if HIH' is a periodic group, then [N, H ] = (1). In particular, if the group G is periodic and nilpotent, then every abelian, radicable normal subgroup of G is contained in the centre ZG.
Proof. For every natural number i with 0 I i 5 r put N i= [N, 3 1 . The subgroup No = N is radicable. Suppose that the subgroup N ihas already been shown to be radicable. If x E H , then [Ni, x ] = N:-' is a homomorphic image of the abelian group Ni.Thus the group [Ni, x ] is radicable. Now the group Ni+'is abelian, being a subgroup of N , and it is generated
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by all the groups [Ni, x ] for x E H. Hence the group Ni+ is radicable. Thus we have that each one of the groups N , , 0 S i 5 r, is a radicable abelian normal subgroup of NH. By assumption, there exists a least integer n such that N , + , = (1). Suppose, if possible, that n 2 1. The subgroup H of N H stabilizes the series (1) c N, c N,-l (via conjugation), and thus there exists by 1.C.3. a homomorphism cp of H into the additive abelian group D = Hom (Nn- ,NJ. So one has H E ker cp = C H N n - ., Since the group Nn-l is radicable the group D is torsion-free, and because HIH' was assumed to be periodic, this implies that Hq = ( 1 ) . But then H = CHNn-l, and thus N, = (1). This contradicts the choice of n, so n = 0, and [N, H ] = ( 1 ) . Choosing H = G, the second statement of l.F.l follows from the first. I
,
1.F.2 Lemma. Let a be an automorphism of finite order of the periodic, radicable abelian group A. If the centralizer C A a contains every element of A the order of which is the square of aprime, then A = CAa,that is a is the identity automorphism of A. Proof. Since every automorphism of A leaves every primary component of A invariant, we may assume that A is a radicable abelian p-group for some primep. Since a has finite order, we may assume that a has order q, a prime, if a is not the identity. Put A i = ( a E A ; I(a)l = pi). Suppose that i 2 2 is a natural number such that for every x E A i one has x" = x. If now a E A i + z , then one has (aa)Pz =
(aP2>"= U p 2 .
That is, (a-1aa)P2= 1, and so the automorphism a stabilizes the series (1) G A , E A i + 2 . By 1.C.3 the stability group of this series is ap-group. Hence i f p # q, then a" = a for all a in A i f 2 . Suppose that p = q. Since the mappings 8, and ya of 1.C.2 are homomorphisms, one has
[up,a ] = [a, a l p = [a, a"] = 1. Thus a" = a for every a in A i + l (since A i + l = (Ai+$'), and induction now completes the proof. I
Exercise. If the automorphism a of finite order of the periodic radicable abelian group A not containing any involutions leaves fixed every element of square-free order of A , then a is the identity mapping. (See, for example, Robinson [I], 2.36 for a solution).
CH.
1, 8 F]
AUTOMORPHISMS OF CERNIKOV GROUPS
35
The following result is due to Cernikov [17]; it is the central result of this section.
1.F.3 Theorem. If A is a periodic group of automorphisms of the C'ernikov group G, then A is also a cernikov group; i f the C'ernikov group G is nilpotent, then the automorphism group A isjnite. It is shown in Rae and Roseblade [ l ] that the possibilities of embedding a Cernikov group into any group as an ascendant subgroup are rather limited. Proof. Let N be the minimal subgroup of finite index of G. Since N is a characteristic subgroup of G, it is left invariant by A . The subgroup N is radicable and abelian. If No denotes the subgroup of N generated by all the elements of cube-free order of N , then No is a finite group. By 1.F.2 one has C A N = C AN o . Consequently, the subgroup U = C A N n C,(G/N) is a normal subgroup of finite index in A that stabilizes the sequence (1) E N E G. Since the group GIN trivially is finitely generated, and since N is an abelian Cernikov group, 1.C.3a yields that the abelian group U is also a Cernikov group. Thus the periodic group A of automorphisms of the Cernikov group G is a Cernikov group. Assume now that G is nilpotent. By l.F.l the subgroup N lies in the centre of G. Hence by 1.C.3b the group U of automorphisms of G is isomorphic to a subgroup of Hom (GIN, N ) . But clearly, there are only finitely many homomorphisms of the finite group GIN into the abelian Cernikov group N . Therefore U is a finite group, and so is A . I We shall use 1.F.3 to prove the following reduction result. Although more general and elaborate reduction principles will extend and supersede it (see 3.32), this result should give the reader a good idea of a type of argument and proposition that will frequently occur throughout the book.
1.F.4 Theorem. If the group G satisfying min is not a C'ernikov group, then in G there exists a perfect subgroup H such that every proper subgroup of H is a Cernikov group and the factor group HIZH is an injnite simple group. Further, the subgroup H is either finitely generated or locally finite.
Proof. Since the group G satisfies min, the set of those subgroups of G that are not Cernikov groups (is non-empty and so) contains a minimal member H, say. By this choice the subgroup H i s not a Cernikov group, but every proper subgroup of H is a Cernikov group. Let N be a proper normal subgroup of H. Then N is a Cernikov group, and so by 1.F.3 the normal subgroup CHN of H is such that H / C H N is a Cernikov group. But then
36
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1,
5
P
1.E.7 yields that H is a Cernikov group if H # C, N. Since this cannot be, one has H = C,N, and so every proper normal subgroup of H is central. Thus HIZH is an infinite simple group. If H # H', then H is nilpotent of class two and 1.E.6 would make H a Cernikov group, which it is not. Finally, if H is not finitely generated, then every finitely generated subgroup of H is a proper subgroup and so a Cernikov group, that is, every finitely generated subgroup of H is finite. Thus H is either finitely generated or locally finite. I There is an immediate consequence of 1.F.4 which it sometimes is quite convenient to have.
1.F.5 Corollary. The group G is a Cernikov group if (and only if) it satisfies min and no infinite simple group is involved in G. In view of 1.B.5 this result provides an alternative proof of Cernikov's structure theorem for locally soluble groups satisfying min, which, in slightly stronger form, appears as 1.E.6. As an illustration of how 1.F.4 can be put to use, we shall prove the result of 0. Ju. Smidt [3].
1.F.6 Theorem. If the 2-group G satisfies min, then it is a Cernikov group and thus IocalIy finite.
Proof. Assume, if possible, that the statement of 1.F.6 is false. Then there exist counter-examples to the assertion. By 1.F.4 there exists an infinite simple 2-group G such that every proper subgroup of G is a Cernikov group. The group G is finitely generated, for otherwise it would be locally finite and thus locally nilpotent and then by 1.E.6 a Cernikov group. In G, clearly, the maximal subgroups are just the maximal locally finite subgroups of G. The main step of the proof is to show that in any such group G the intersection of any two distinct maximal subgroups is trivial. Choose the maximal subgroup M of G and suppose, if possible, that there are maximal subgroups K # M of G such that the intersection K n M is not trivial and has finite index in M. Among these choose K so that the index IM :K n MI is minimal. Put D = K n M and consider the normalizer N, D. This is clearly a proper subgroup of the simple group G and so it is contained in some maximal locally finite subgroup R of G. Since by l.E.5 the subgroups K and M of G are hypercentral, one has that N,D 2 D
5 N,D.
Thus R # M , and the maximal subgroup R of G has an intersection with
CH.
1, 8 F ]
AUTOMORPHISMS OF
CERNIKOV GROUPS
37
M properly containing D. Thus we have a contradiction to the choice of K, and no such maximal subgroup K can exist. Suppose, if possible, that there are maximal subgroups K of G such that the intersection K n M = D is non-trivial and has exponent larger than e = IM :JMI, where JM denotes the radicable abelian subgroup of finite index of M , Then obviously the subgroup
D , = (d", d E D ) is a non-trivial, abelian, characteristic subgroup of D . Since NKDI2D E NMDI
7
the normalizer N, D , is contained in some maximal locally finite subgroup R # M of G . But since D , E JM, one has J M E R,and so IM : M n RI is finite, which cannot be. Thus the intersections of M with other maximal subgroups of G have bounded exponent, and hence finite order bounded in terms of the structure of M . Suppose now, if possible, that there are maximal subgroups K # M of G such that the finite intersection K n M is non-trivial. Since the orders of these intersections are bounded, we may choose K such that the order the intersection D = K n M is maximal. Again, one has
NKD
2D
NMD,
and thus the normalizer N , D is contained in some maximal locally finite subgroup R # M of G. But since
Rnbf?NMDsD, one has arrived at a contradiction to the choice of K. Hence any two distinct maximal subgroups of G have trivial intersection. Now let M and N be any two distinct maximal subgroups of G, and let i E M and j E N be elements of order two (involutions).Then the subgroup ( i , j } intersects the subgroups M and N of G non-trivially, and so (i, j ) cannot be contained in any maximal subgroup of G. But the element ij has finite order (a power of 2) and it generates a normal subgroup of index two in ( i , j } . Thus the subgroup (i, j ) of G is finite, and hence it is contained in some maximal locally finite subgroup R of G. This is a contradiction. Hence there cannot be such a simple group G and thus 1.F.6 must hold true for all 2- groups. I
Remark. The initial reduction to the situation in which any two maximal subgroups of the simple group have trivial intersection in fact only uses that
38
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH. 1 ,
5G
every proper subgroup of G is hypercentral. But something like the dihedral property of involutions is needed to show that such a simple group cannot exist. Exercise. The group G is a locally nilpotent Cernikov group if and only if G satisfies min and any two elements of G generate a nilpotent subgroup. That this latter property together with periodicity is not enough to force local finiteness is made evident by the groups constructed in Golod [l]. Section G. The minimal condition on abelian subgroups We shall say that a group satisfies the minimal condition on abelian subgroups, min-(21 for short, or that it belongs to the class Min-(21if each of its abelian subgroups satisfies min. In what circumstances does min-'2l imply min? That this is not so in general has been shown recently by Novikov and Adjan [l] who have produced infinite groups of finite exponent that are finitely generated and in which every maximal abelian subgroup is finite. These groups are not locally finite, nor do they satisfy min. In Chapter 3 we shall prove that locally soluble and certain related types of Min-%groups satisfy min, and in Chapter 5 we shall show that for locally finite groups in general the minimal condition on abelian subgroups does indeed imply the minimal condition on subgroups (see 5.8). In this section we shall prove some very elementary results involvingsoluble and locally nilpotent groups. Since essentiallythe same arguments yield slightly more general results, we shall work with somewhat weaker conditions than min-U.
l.G.l Lemma. Let the cernikov group N be a normal subgroup of the periodic group G . r f every abelian (respectively, ascendant abelian, subnormal abelian) subgroup of G satisfies min, then also every abelian (respectively, ascendant abelian, subnormal abelian) subgroup of GIN satisfies min. Proof. The factor group G/CGNis by 1.F.3 a Cernikov group. Thus we may assume that G = CGN. Let BIN be any abelian subgroup of GIN and A a maximal abelian subgroup of B. Then the subgroup A contains Nand is normal in the nilpotent group B. Furthermore C,A = A . If the subgroup A satisfies min, then by 1.F.3 the factor group BIA is finite, and so B and BIN satisfy rnin too. If the abelian subgroup BIN is ascendant (respectively, subnormal) in GIN, then the maximal abelian subgroup A of B is ascendant (respectively, subnormal) in G, and the result follows from the preceding considerations. 1
CH.
1,
5 GI
THE MINIMAL CONDITION ON ABELIAN SUBGROUPS
39
1.G.2 Proposition. If G is a periodic hyperabelian group such that every abelian subnormal subgroup of G satisfies min, then G is a Cernikov group. Proof. By 1.A.8 the hyperabelian group G has a subgroup N that is nilpotent of class at most two and satisfies C , N = ZN. By assumption every abelian subnormal subgroup of N satisfies min, and SO by l.G.l the abelian group N / Z N satisfies min. Hence by 1.E.7 the normal subgroup N of G is a Cernikov group. By l.F.3 the group G / Z N , which essentially is the group of automorphisms of Ninduced by elements of G, is a Cernikov group. Thus a further application of 1.E.7 yields that G is a Cernikov group. I The same argument yields:
1.G.3 Proposition. If G is a periodic hypercyclic group such that every abelian normal subgroup of G satisjies min, then G is a Cernikov group. Proof. By 1.A.7 every hypercyclic group G contains an abelian normal subgroup A with C,A = A. By assumption A satisfies min, and by 1.F.3 the factor group G/A is finite. 1
1.6.4 Proposition. The locally nilpotent group G is a Cernikov group if and only if it satisjies min-M.
A locally finite p-group, for some prime p , clearly is locally nilpotent, and an abelian p-group satisfies min if and only if it has finite rank. Thus 1.6.4 implies that the p-subgroups of a locally finite group satisfy min if and only if the abelian p-subgroups of that given group all have finite rank, or equivalently, if its elementary abelian p-subgroups are all finite. This special case of 1.G.4 has important repercussions in most chapters of this text. One obtains 1.G.4 immediately from its countable case, for if every countable subgroup of a group G satisfies min, then by 1.E.3 the group G itself satisfies min and by 1.E.6 the locally nilpotent group G satisfying min is a Cernikov group. In the countable case, however, we can prove a little more. 1.G.5 Theorem. The countable, locally nilpotent group G is a Cernikov group if and only if every abelian ascendant subgroup of G satisfies min. Proof. If G is a Cernikov group, then clearly every subgroup of G satisfies min. Now consider the countable group G and enumerate its elements G = { g i, i E N]. Put Gi = ( g l , g 2 , . . ., g i ) . Then each of the subgroups G iof G is nilpotent and G = UisNGi. Since Gi 5 G i + l , it is clear that Gi is subnormal in Gi+ 1. Thus each of the subgroups Gi of G is an ascendant subgroup of G. Since the abelian ascendant subgroup G , = ( g , ) satisfies
40
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1, 8
G
min, the element g1has finite order. Since the enumeration of the elements of G could have started with any element, it is clear that the group G is periodic. In view of 1.G.3 it suffices to show that G is hypercentral. If this is not so, then l.G.l shows that one may assume that the hypercentre of G is trivial. We shall find a contradiction to this assumption. Denote by Zi the subgroup of the centre ZGi generated by the elements of square-free order. Put H i =
H a HG,
Q
a
..
. . . u HGi = G 03
.4a
HGia
Q
HGi+, a
Q
i=l
shows that the abelian group H is ascendant in G. But then H satisfies rnin by hypothesis, and thus H i s finite. Consequently, H = Hk for some k, and so if k c j < i one has Zi
_c
Hin ZiG Z j .
Put Z = n i , k z i . Then since each of the subgroups Zi is finite and nontrivial, we have Z # 1, and the subgroup Z is contained in the centre of U i z k G i = G. This is the desired contradiction. I It should be pointed out, that in 1.G.5 one cannot weaken the assumptions by replacing ,,ascendant” by ,,subnormal”, for in P. Hall [4]a construction is given of a countably infinite, locally finite p-group, for any prime p, in which (1) is the only abelian subnormal subgroup. Since there exist periodic locally nilpotent groups not satisfying min the nilpotent ascendant subgroups of which all satisfy min (see 2.8), the assumption of countability cannot be deleted from l.G.5. Finally, since there exist countable, periodic, locally soluble groups not satisfying min, the locally nilpotent ascendant subgroups of which all satisfy min (see Wehrfritz [3]), local solubility cannot replace local nilpotency in the hypotheses of 1.G.5. The following useful generalization of 1.G.4 can be found in Blackburn [l] and, at least implicitly, in the works of Cernikov. 1.6.6 Proposition. If a locally Jinite p-group G contains a maximal elementary abelian subgroup E which is$nite, then G is a Cernikov group.
CH.
1, 8
GI
THE MINIMAL CONDITION ON ABELIAN SUBGROUPS
41
Proof. Consider for the moment a vector space V over a field of characteristic p and a finite elementary abelian p-group E of automorphisms of V. We claim that C,E has dimension at least [IEI-' dim V ] (trivially it is a subspace). For if x E E\
dim C,E 2 [IEI-' dim V ] . We now consider the group G of the proposition. If Z denotes the hypercentre of G, then by l.A.I the group Z contains an abelian normal subgroup A of G such that A = C,A. E acts on the vector space Q1A , so by the previous paragraph Jz,A is finite (trivially C,,AE E E). Hence A satisfies rnin and consequently 1.F.3 implies that Z / C , A = Z / A is finite. Therefore Z is a Cernikov group. Clearly the maximality of E implies that Jz, C , E c E, so every abelian subgroup of CGE satisfies rnin. Thus by 1.6.4 the centralizer CGE satisfies min. Let H / Z be an elementary abelian subgroup of G / Z containing EZIZ. The group EZ is a Cernikov group, so H/C,(EZ) satisfies rnin by 1.F.3. But trivially C,(EZ) lies in the Cernikov group CHE. Therefore H satisfies rnin by l.E.l and H / Z is finite. It follows that G / Z contains a finite, maximal elementary abelian subgroup and that we may assume that Z = (1). Suppose that G # (1). In this situation, the centre of G is trivial but E is not. Hence for each e in E\(1> there exists an element x , of G such that [e, x,] # 1. Put X = ( E , x,; e E E (\ 1)). The group X is a finitep-group and hence there exists a non-trivial central element x of X of order p . Clearly the group ( E , x ) is elementary abelian, so by the maximality of E we have x E E. But by construction E intersects the centre of X trivially and this contradicts x # 1. Therefore G = (l), which is a Cernikov group. I In the group G , the element g satisfies a (left) Engel condition with respect to the element h if all but finitely many terms of the sequence { g n } inductively defined by
h
=
90,
gn+l = Cgn,gl
are trivial. This is a considerable generalization of the condition that g centralizes h. The element g is an Engel element of G if it satisfies the Engel condition with respect to every element of G. The group G is an Engel group if it is generated by its Engel elements. Obvious examples of Engel groups
42
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1, 8 H
are the locally nilpotent groups, but Golod [ I ] shows that there are periodic Engel groups which are not locally nilpotent. In his Ph. D. Thesis (University of London, 1969) John Martin proves the following result, which we state without proof. See Martin and Pamphilon [11. 1.6.7 Theorem. For the group G the following two conditions are equivalent: a G is a hypercentral Cernikov group; b G is an Engel group, and the set of all the Engel subgroups of G satisfies the descending chain condition.
Section H. The minimal condition on normal subgroups Since there are infinite simple groups, the minimal condition on the set of normal subgroups of the group G will not have enough impact to force the group G to be a Cernikov group. One might reasonably cherish such a hope only if G has many normal subgroups. In this section we shall show that this hope is justified for locally nilpotent groups, while even for soluble groups this is not so in general. l.H.l. Lemma. Let M be a locally soluble minimal normal subgroup of the group G . If G/MC,M is locally finite, then M is elementary abelian (and in particular locally finite).
Proof. Let H be any finitely generated subgroup of G. If M H n M is abelian. If not, then M n C,M = (1 and
>
H nM
H nM
c H n MC,M c
G
C,M, then
M, where the bar indicates modulo C , M . Now H / ( H n M C , M ) is finite since G/MC,M is locally finite and thus H n MC,M is finitely generated. The local solubility of M now implies that H n M is soluble. Therefore M is abelian by 1.B.3. Replacing G by M x (G/C,M) if necessary we may assume that M = C,M. But then, clearly, H n M is a finitely generated abelian group and 1.B.3 again implies that M is elementary abelian. I This result is the key to our next theorem. N
N
1.H.2 Theorem. If the soluble group G satisfies the minimal condition for normal subgroups, then G is locally Jinite.
Proof. By induction assume the theorem proved for all soluble groups of derived length less than n. Let G be a soluble group of derived length n satisfying the minimal condition on normal subgroups. Suppose that A is
CH.
1,
8 H]
THE MINIMAL CONDITION ON NORMAL SUBGROUPS
43
the last non-trivial term of the derived series of G. By the induction hypothesis, the factor group G / A is locally finite. The minimal condition on normal subgroups implies that there exists a (transfinitely) ascending series { A , } of normal subgroups A , of G contained in A such that < I ) = A , , A , E A v f l and there is no proper G-invariant subgroup of A , + J A , , for limit ordinals A one has A , = U V < A A ,and , there is an ordinal z with A , = A . Now l.H.l yields that each one of the factors A , + , / A , of the preceding series is locally finite. Thus the abelian normal subgroup A of G is the union of periodic subgroups, and so it is periodic. By 1.A.2 the group G then is locally finite. I That we should not raise our hopes too high even for soluble groups is made clear by an example due to Carin [I]. (See Roseblade and Wilson [I] for related examples.)
1.H.3 Example. There exists a metabelian group G satisfying the minimal condition on normal subgroups that is not a C'ernikou group. Proof. Let p and q be distinct primes, GF(p) the field of p elements and K the field obtained from GF(p) by adjoining a qi-th root of unity for each natural number i. Denote by A the additive group of K and by H the multiplicative group of q-power roots of unity in K. Then A is an infinite elementary abelian p-group, and H is a Prufer q-group. Also A is an H-module, where the scalar product of an element of A with an element of H is simply their product in K.Denote by G the split extension of A by H. Let B be any non-trivial H-submodule of A , and choose an element b E B\{O}. Then in K the element b-' admits an expression
b-' = alhl
+ . . . +a,,h,,
for some ai E GF(p) and hi E H. Now one has bH E B, and so
al(bh,)+
. . . -I-a,,(bh,,)= bb-I
= 1 E B.
But by the construction the elements of H generate A as an abelian group, and so A = B. Suppose that N is any non-trivial normal subgroup of G . Since C,A = A , we have N n A # (1). But then this intersection is an H submodule of A , thus N 2 A , and A is the unique minimal normal subgroup of G. Since G / A ci H satisfies min, the group G satisfies the minimal condition on normal subgroups. But G contains an infinite elementary abelian subgroup so it cannot satisfy min. I Observe also that the group G constructed above does not contain any proper subgroup of finite index. In McDougall [l] it is shown that metabelian groups satisfying the mini-
44
BASIC METHODS, CONCEPTS A N D EXAMPLES
[CH.
1,
5H
ma1 condition for normal subgroups that are not Cernikov groups do not difier too much from the example 1.H.3. However, for locally nilpotent groups no such example as 1.H.3 can exist.
1.H.4 Theorem. If the locally nilpotent group G satisfies the minimal condition on normal subgroups, then G is a Cernikou group; in particular, G is soluble and hypercentral. Proof. Let N be the minimal subgroup of finite index of G (see 1.E.4) and Z the centre of N. By 1.E.5 the group G is hypercentral, thus Z # 1 (assuming only that G is not finite). We show first that Z satisfies min. If a E 2 has infinite order, then - since IG : C,al is finite - the subgroup A = ( a G ) is a finitely generated infinite abelian group. But then A
3
A2
3
A4
3
...
is a strictly descending chain of normal subgroups of G and this contradicts the hypothesis on G. Therefore the subgroup Z is periodic. Clearly only finitely many of the primary components of the abelian group Z can be nontrivial. It also follows that G is periodic. Let p be a prime and put P = ( x E Z ; x p = 1). Suppose the subgroup P is infinite. Clearly P is normal in G. By the minimal condition on normal subgroups P contains a minimal infinite normal subgroup Q of G. The subgroup Q is an infinite elementary abelian p-group and so it contains a subgroup R of index p . Since the centralizer of R in G contains N , the subgroup R has only finitely many conjugates in G, and each of these has index p in Q. Hence the intersection S = n s e c R g is an infinite normal subgroup of G properly contained (and of finite index) in Q. This contradiction to the minimal choice of Q proves that P is finite, and so 2 satisfies min. In exactly the same way the centre Z,/Z of N / Z also satisfies min. Hence the subgroup Z , satisfies min. But then by 1.E.6 and 1.F.3 the factor group G / C , Z , is finite, and so N = C , Z , and Z = Z, . Thus N = Z is an abelian normal subgroup of G, and 1.H.4 follows. I
1.H.5 Corollary. If the nilpotent group G satisfies the minimal condition on normal subgroups, then G is a Cernikov group and its centre hasfinite index in G. Proof. By 1.H.4 the group G is a Cernikov group, and by l.F.l the minimal subgroup of finite index of G is contained in the centre of G. I
1.H.6 Corollary. If the locally supersoluble group G satisfies the minimal condition on normal subgroups then G is a hypercyclic Cernikou group.
CH.
1,
$11
2-GROUPS OF SMALL DEPTH
45
Proof. By 1.B.7 and the minimal condition on normal subgroups G is hypercyclic. If N denotes the minimal subgroup of G of finite index (see 1.E.4) then N centralizes every cyclic normal factor of G, since the automorphism group of a cyclic group is finite. Therefore N is a hypercentral group and the result follows as in the proof of 1.H.4. 1 McLain 131contains a generalization of l.H.6. In the same paper McLain gives an example of a locally soluble group satisfying the minimal condition for normal subgroups that is not soluble. By 1.B.5 such a group is necessarily hyperabelian, but McLain’s example is also locally finite, and in view of 1.H.2 the following is an obvious question. Question 1.6. Is a locally soluble group satisfying the minimal condition for normal subgroups necessarily locally finite? ( N o ! See Final Comments.)
The following result is an immediate consequence of 1.B.6, 1.A.2 and comments.
1.H.7 Corollary. A locally polycyclic group satisfying the minimal condition f o r normal subgroups is locally jinite. Exercise. An abelian-by-finite group is a Cernikov group if and only if it satisfies the minimal condition on normal subgroups. Exercise. A hyperfinite group is a Cernikov group if an only if it satisfies the minimal condition on normal subgroups. Section I. 2-Groups of small depth
If a subgroup S of an arbitrary group G can be generated by two elements x and y , then in general there is little beyond this fact that one can say about the structure of S, unless both x and y are involutions (elements of order two). Definition. The group D is called a dihedral group if it can be generated by two distinct involutions. A dihedral group is abelian exactly if it is elementary abelian of order four, otherwise its order will be larger than four. For dihedral groups one has a very precise structural result. It is this result which often makes the presence of many involutions in a group G a very precious occurrence, since then one obtains some information about the structure of many of the subgroups of G.
46
BASIC METHODS, CONCEPTS AND EXAMPLES
[CH.
1, $ 1
1.1.1. Proposition. Let the dihedral group G be generated by the two involutions x and y . Set c = x y and C = (c>. Then one has: a The subgroup C is normal and of index 2 in G. b If the dihedral group G is non-abelian, then C i s a characteristic subgroup of G. c Every element of G \ C inverts every element of C, that is, g E G \ C implies C' = c- l . d Every element of G \ C is an involution. e The set G \ C is a single conjugacy class if and only if the order ICI is finite artd odd; it is the union of two conjugacy classes otherwise.
Proof. a From c
= xy
one has
cx = x - l c x = x(xy)x = y x = y - ' x - l
=
c-1 .
Similarly cy = c-'. Hence C = (c) is a cyclic normal subgroup of G = ( x , y). Clearly, C n ( x ) = (1) and C(x> = G, SO IG : CI = 2.1 b If G is non-abelian, then [CI > 2. Once we have estalished d, then the uniqueness of C is clear. I c Since G = C ( x ) , it follows that every element g E C G \ has the form g = c l x , c1 E C. Clearly one has cg = cx = c-'. I d The element g E G\C has the form g = clx. Hence g 2 = c l x c l x = c1(c1)-' = 1, and g is an involution. I e For any integer r one has by c (C2rX)=
and (C2'-1X)cr
=
=
C-rC2rXCr
c-rc2r-lxcr
=
=
Cr(XCrX)X
Cr-l(XCrX)X
=
x,
=c
- 1x
= y.
Since the set G C\ is invariant under the inner automorphisms of G it must consist of at most two classes of conjugate elements. It will consist of a single conjugacy class if and only if the involutions x and y are conjugate. An arbitrary conjugate of x is XXCs
=
XCs
=
c-2s X
Thus the involutions x and y are conjugate in G if and only if there exists a positive integer s such that cZs = c, that is, if and only if the order ICI of C is finite and odd. 1 Clearly the finite dihedral group whose cyclic normal subgroup C has order n can be defined by
CH.
1,
5 I]
2-GROUPS OF SMALL DEPTH
47
D, = ( x , y ; x2 = y 2 = (xY)” = 1). The group G is called IocaIIy dihedraI if every finite subset of G is contained in some dihedral subgroup of G. From 1.1.1 one obtains rather precise information about this class of groups. We only note the following result. 1.1.2 Proposition. The IocaIIy dihedral group G has a IocaIIy cyclic normaI subgroup of index 2, and every etement of G \ C is an invoIution that inverts every element of C . In view of 1.1.1 this needs proof only if the group G is not finitely generated.
Proof. If the group G has order larger than four, then there are non-involutory elements in G. Let C be the subgroup of G generated by these elements. If X is any finite subset of C , then there exists a finite set Y of noninvolutory elements of G such that X E ( Y ) . By our assumption on the order of G the locally dihedral group G contains at least two non-commuting involutions i andj, say. Let g be any element of G. Then there exists a dihedral subgroup D of G containing the subset { i , j , g } v Y. Denote the unique cyclic normal subgroup of D of index 2 by C,. By 1.1.1 one clearly has X E ( Y ) E ( d E D ; d 2 # 1) = C D E C. Since the subgroup C, is cyclic, the subgroup C is a locally cyclic normal subgroup of G. If i E C,, then ij = j i , which we had assumed not to be the case. Thus for the element g one has g E D = C D V ~ C D EC v i C .
Since g is an arbitrary element of G this implies that G = C u iC. The group G is non-abelian; thus IG : CI = 2. Also if x E X , then x E C,, and so xi = x-‘. Therefore i inverts every element of C , and hence every element of the coset iC = G\C inverts every element of the abelian subgroup C . Finally, if c E C, (ic)’ = (ici). = 1. Thus every element of iC is an involution. I Up to isomorphism there exists only one infinite locally cyclic 2-group namely the Prufer 2-group C Z m .Hence, up to isomorphism, there exists only one infinite locally dihedral 2-group, namely,
D,,
=
N
CZm, cic
=
1 = i2 for every c E C>.
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[CH.
1,
$1
1.1.3 Lemma. The group DzoDcontains a single conjugacy class of non-central involutions.
Proof. In the above notation, D,,\C element d E C such that dZ = c. Then
=
iC. If c E C, then there exists an
id = j(id-'j)d = id2 = jc.
Thus all the elements of D2,\C are conjugate in D z m . The subgroup C being locally cyclic, contains only one involution, z say, and z is central in D Z msince z' = z-l = z. 1 The (generalized) quaternion group Q,,, is closely related to the dihedral group D,,.It is defined by
Qzn= (x, y ; x2
=
y 2 = (xy)'" = z, where z 2
=
1).
The group Q2 is the ordinary quaternion group of order eight. Clearly, the element z of the (generalized) quaternion group QZn is in the centre of Q2.. Thus one visibly has the isomorphism D,, N Q2./(z>.One easily shows that the element z is the only involution of the (generalized) quaternion Q,.. As in 1.1.1 one shows that the (generalized) quaternion group Q,, contains a normal cyclic subgroup of index 2, which is characteristic if n > I, and every element of Q",C \ is of order four and inverts every element of C. The group G is called locally quaternion if every finite subset of G is contained in some (generalized) quaternion subgroup of G. Clearly, for locally quaternion groups one has an analogue to 1.1.2. Since a locally quaternion group is, by definition, a 2-group, one finds that, up to isomorphism, there exists only one infinite locally quaternion group, namely
QZm=
N
CZm, cxc
=
1 = x4 for every c E C, and x 2 = z the involution of C>.
In Gorenstein [l] p. 193 a survey is given of all finitep-groups containing a cyclic normal subgroup of index p. For p = 2 besides the dihedral and (generalized) quaternion groups two more types occur. We shall comment only briefly on one of these types, namely the quasi-dihedral groups, since these groups play a special role in characterizations of certain finite and locally finite simple groups (see 4.20 and 4.22 or Alperin, Brauer and Gorenstein [I]). The quasi-dihedral group Q-D,n is defined by Q-D,, = ( x , y ; x2" = y 2 = 1, x y = x-'+'"-'
) for n 2 3.
CH.
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2-GROKPS OF SMALL DEPTH
49
In Theorem 5.4.3 of Gorenstein [ I ] p. 191, some properties of the groups Q-D2. are given. In particular, the maximal subgroups of Q-D,,, are cyclic, (generalized) quaternion, or dihedral. Since the (generalized) quaternion and dihedral groups do not contain quasi-dihedral subgroups, one sees that nothing is gained by introducing the class of locally quasi-dihedral groups, since these groups must be quasi-dihedral and hence finite. Very sharp information has been obtained in Gorenstein and Walter [2] on the structure of finite groups with dihedral Sylow 2-subgroups. These results extend to locally finite groups (see 4.19). Brauer and Suzuki [I] show that a finite group cannot be non-abelian and simple if its Sylow 2-subgroups are (generalized) quaternion. This result has since been considerably extended by Glauberman [l]. Both these results extend to locally finite groups. Let us first show this for the more general result of Glauberman [l]. 1.1.4 Theorem. If the locally jinite group G contains an involution i such that every maximal 2-subgroup of G contains at most one conjugate of i, then the centre of the factor group G / O G contains the involution iOG, and thus is non-trivial. Proof. Since every 2-subgroup of G is contained in some maximal 2subgroup of G, the hypothesis of 1.1.4 implies (and is equivalent to) the fact that every 2-subgroup of G contains at most one conjugate of the involution i. If Z, denotes the local system of all finite subgroups of G containing the involution i, then the hypothesis of 1.1.4 holds in each SEZ. But then Glauberman’s Theorem yields that [S, i ] E 0s. By 1.B.9 and comments there is a subgroup T E C containing S such that S n OT = S n O G . Thus one obtains [S, i ] E S n [T, i] E S n OT G OG.
But this implies that [G, i ] G O G , as contended. I The extension of the theorem of Brauer and Suzuki [I] is an immediate corollary, for in a locally cyclic or in a locally quaternion group there is at most one involution. 1.1.5 Corollary. If every 2-subgroup of the locallyjinite group G is either locally cyclic or locally quaternion, then either G = OG or the centre of the factor group G / O G contains precisely one involution.
Remark. Since one does not know very much about the set of all maximal 2-subgroups of a locally finite group G (see Section D), it seems difficult to
50
BASIC METHODS, CONCEPTS AND EXAMPLES
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1, 5
J
check the conditions of 1.1.4 in any given group. To lessen that task it would seem desirable to obtain, if possible, a similar result under weaker assumptions involving only certain maximal 2-subgroups. Section J. Frobenius groups
A Frobenius group (after G. Frobenius 1849-1917) is a group G that contains a subgroup H with G 2 H 2 (1) such that H n H g = (1) for every element g E G \ H . Such a subgroup H of the Frobenius group G is usually called a Frobenius complement of G , although in general the subgroup H need not complement any subgroup of G. If in the Frobenius group G with Frobenius complement H the subset K = (G\ U g E G H 9u) (1) is a subgroup, then the subgroup K is called the Frobenius kernel of G (with respect to H). If the Frobenius kernel K exists in the Frobenius group G (with respect to the Frobenius complement H ) , then one often has G = HK. It is this situation from which the names derive. The classical theorem of Frobenius asserts that for a finite Frobenius group G the set of Frobenius complements in G consists of a single conjugacy class of subgroups and that the Frobenius kernel of G (with respect to any of these subgroups) exists and is a characteristic subgroup of G. It is our aim in this section to extend these and some further properties of finite Frobenius groups to locally finite Frobenius groups. We thus obtain a rather satisfactory survey of the structural properties of locally finite Frobenius groups. These results essentially appear in Kegel [1] and Busarkin and Starostin [l]. For a discussion of finite Frobenius groups see Huppert [I], 6 V.8 or Gorenstein [I], 5 2.7, 0 4.5, and 5 10.3. The first result holds for Frobenius groups in general; it just gives a description of Frobenius groups in the language of permutation groups. It is in this context that one often encounters Frobenius groups. 1.J.l Proposition. The group G is a Frobenius group i f and only if it has a faithful, transitive permutation representation on a set 52 with 1521 > 1 such that every element g # 1 of G satisfies wg = w f o r at most one w E SZ, and i f furthermore there is a non-trivial element of G leaving an element of 52 Jixed. Proof. If G is a Frobenius group and H a Frobenius complement in G, then take as 52 the set of all conjugates of H in G with G operating via conjugation. From the definition of a Frobenius complement it is clear that G operates in the desired way on the set SZ. If on the other hand, the group G has such a permutation representation on a set 52, then choose any element
CH.
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FROBENIUS GROUPS
51
w E 52 and put H = G, = ( g E G ; og= o}.Since G acts transitively on Q one has H # ( l ) , and since no non-identity element of G has more than one fixed-point in 0,H has trivial intersection with every conjugate. Since 1521 # 1, the subgroup H i s not the whole of G. Thus G is a Frobenius group with Frobenius complement H. I If G is the free group on two generators x and y , and if H = ( [ x , y ] ) , then it is easy to check that for every g E G \ H the subgroup ( H , H g ) is not cyclic. Hence the group ( H , H g ) is a non-abelian free subgroup of G and its centre is trivial. But then, since the intersection H n Hg lies in the centre of ( H , H g ) , one has H n H g = ( 1 ) for every g E a H , and the group G is a Frobenius group with Frobenius complement H. Suppose K is any normal subgroup of G with HK = G . Since H is abelian, we have G' -c K. But H c G', and so K = G. Hence the subgroup H of G does not have any proper normal supplement in G. Further it is easy to check that every element of R = (UgeGHg)\(l) has length at least four, so x-'y-' and x y are elements of K = C\R whose product visibly does not lie in K. Thus H does not determine a Frobenius kernel of G either. For locally finite groups the situation is very much better. 1.5.2 Theorem. Let H be a Frobenius complement of the locally finite Frobenius group G. Then G contains a unique Frobenius kernel K , and K is a normal complement of H in G. The subgroup K is nilpotent, and there exists a set rc of primes such that G is rc-closed and K = 0, G. Any Frobenius complement of G is conjugate to H under an element of K and every abelian subgroup of H is locally cyclic. If S is any non-trivial subgroup of G such that K n S = (l), then ZS # (1) and S is conjugate to a subgroup of H.
Proof. If G is a finite group then the theorem is well known to be true. The uninitiated reader should consult Gorenstein [I], Q 10.3 or Huppert [ I 1, 0 V.8. The full theorem is deduced from the finite case. Put K = (G\UgEG H 9 ) u { 1) and let Z denote the set of finite subgroups F of G satisfying ( 1 ) F n H 2 F. This set is clearly a local system of G such that each of the subgroups F E Z is a finite Frobenius group with Frobenius complement F n H. If g is any element of G such that (1) F n H g $ F where F E Z, then F n H g is also a Frobenius complement of F. A finite Frobenius group has a unique Frobenius kernel and this kernel is a normal complement of every Frobenius complement of the group. Hence for each F E Z, it follows that F n K is the Frobenius kernel of F and that F = ( F n H)(F n K). Therefore K is a Frobenius kernel of G, K is unique, and G = HK.
52
BASIC METHODS, CONCEPTS AND EXAMPLES
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5J
For each F E Z the kernel F n K is a nilpotent Hall subgroup of F. Hence K is locally nilpotent and K = 0 , G for some set x of primes such that G is n-closed. Further, since every non-trivial element of H induces on K an automorphism of finite order that leaves only the identity element fixed, a theorem of G. Higman [l ] yields that K is even nilpotent (of class bounded by a function of the smallest prime dividing the order of an element of H). By a result of Burnside for each F E Z any subgroup of F n H of order p q , where p and q are (possibly equal) primes, is cyclic. In particular the abelian subgroups of H are locally cyclic and the p-subgroups of H are locally cyclic or (if p = 2) locally quaternion. Suppose that H contains an involution i. Then i acts fixed-point freely on K since H n H k = (1) for all k E K, and so k' = k - for each k E K. But this automorphism lies in the centre of the automorphism group of K. Hence i lies in the centre of H and is the unique involution of H. Let S be any non-trivial subgroup of G such that K n S = (1). The subgroup S is isomorphic to a subgroup of H. Hence if S contains an involution, this is unique and is contained in the centre ZS of S. Suppose that S does not contain any involutions. Then every primary subgroup of S is locally cyclic, and S is metabelian, S' and SJS' are locally cyclic and the elements of S' and SJS' have coprime orders (Huppert IV.2.11). If S' # (1) let s and t be elements of prime orders p and q respectively, with s E S' and t E S\S'. Now (s) is clearly a normal subgroup of S, so the subgroup (s, t ) has order p q and thus is abelian. Hence t centralizes s. In view of the structure of the automorphism groups of the cyclic and Pruferp-groups (Scott [ l ] 5.7.12; KuroS [I] Vol. 1, p. 155) the element t centralizes the maximal p-subgroup of S'. This holds for each primary component of S'. Hence the element t centralizes S', and so ( t ) is a characteristic subgroup of the abelian normal subgroup (S', t>of S. As such ( t ) is normal in S. Further ( t ) n S' = (I}, so t is central in S. We have now shown that S always contains a non-trivial central element, z say. It follows from HK = G and the definition of K that for some element k E K we have z E Hk. Thus if x E S, then z E H k n Hkx,and so S c NGH k = H k , from the definition of Frobenius complement. Finally if H, is a second Frobenius complement of G then the uniqueness of K implies that G = H , K and H , n K = (1). But then H I G Hkfor some element k E K and so H k = H,(Hk n K ) = H,. I The reader has probably guessed by now that the following characterization of locally finite Frobenius groups holds.
CH.
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FROBENIUS GROUPS
53
1.5.3 Proposition. If G is a locallyfinite group then the following three statements are equivalent. 1. G is a Frobenius group. 2. G has aproper normal subgroup N such that for every non-trivial element n o f N the centralizer CGnis contained in N. 3. G has a local system Z consisting offinite Frobenius groups. Proof. If G is a locally finite Frobenius group with Frobenius complement H let N be the Frobenius kernel of G. For any n EN\( 1) and any g E G, we have H gn C G n E H g n Hg" = (1). Hence CGn E N, and thus 1 implies 2. Suppose 2 holds and let C be the set of finite subgroups F of G such that (1) # F n N # F. It follows at once from Scott [I], 12.6.1 that C satisfies condition 3. Now assume that C is a local system of G consisting of finite Frobenius groups. Let S E C, and denote by C any (fixed) Frobenius complement of S. If T E C is such that S E T, then there exists a unique Frobenius complement C , of T containing C, for the Frobenius kernel of T is nilpotent, S is not nilpotent, and any Frobenius complement of T intersecting S properly is a Frobenius complement of S and thus is conjugate to C. Put
H =
c C,,
T E Z).
If g E q H and x E H n H g , there exists a subgroup X E C such that { g , x, S ) -c X. Then H n X = Cx and Cxn C,B = (1). Hence H n H g = {I), and G is a Frobenius group with Frobenius complement H. ! Remark. It can be proved more generally that the existence of a local system C of Frobenius subgroups of a group G implies that G itself is a Frobenius group, assuming only some coherence propertyfor C (which is automatically satisfied whenever the members of C are finite, see Kegel [4]). We shall call a locally finite Frobenius group G (sharply) doubly transitive, if the inner automorphisms of G induced by the Frobenius complement H permute the non-identity elements of the Frobenius kernel K transitively, the language taken from the associated permutation representation discussed in l.J.l. If G is doubly transitive then K is elementary abelian. Question 1.7. Does every locally finite, doubly transitive Frobenius group possess a local system offinite, doubly transitive Frobenius groups? If the answer to this question were positive, then there would be some hope of extending the classification of the finite doubly transitive Frobenius
54
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[CH.
1, 8
K
groups (near-fields) given by Zassenhaus [l] to locally finite doubly transitive Frobenius groups. On his way to a complete description of the finite doubly transitive Frobenius groups Zassenhaus [ 1 ] obtains many very incisive properties of Frobenius complements of finite Frobenius groups in general. He finds that if the Frobenius complement H of the finite Frobenius group G is soluble, then one has that H has derived length four at most; if H is not soluble, then H has a normal subgroup H, of index at most 2 such that Ho is a direct product of the group SL(2, 5 ) with a metacyclic group M of order relatively prime to 30. (For a readable - and accurate - account of these results see Passman [I] 9 18). These results extend to locally finite groups. In this case, however, one has one additional fact namely that the Frobenius complement of any locally finite Frobenius group is countable. In contrast to this a Frobenius kernel may have arbitrary cardinality. In Wielandt [I] a generalization of Frobenius’s Theorem is given. This generalization extends to locally finite groups.
1.5.4 Theorem. Let H be a proper subgroup of the locally finite group G. If N is a normal subgroup of H such that H n H g E N for every element g E G \ H , then there is a normal subgroup K of G such that H K = G and HnK=N. We leave it as an Exercise for the reader to provide a proof of this statement similar to the proof of the first part of 1.5.2. Section I<. Inverse limit arguments
In this section we discuss the notion of inverse limit. This is a standard set-theoretical tool, which at several instances in this book will be of considerable importance. Let I be a non-empty partially ordered index set, and let {S,, 1 E I} be a family of non-empty sets indexed by I . Suppose that for every pair I , K E I of indices with I 5 K one is given a map ( K --f I ) : S, + S , such that (i) ( I -, 1 ) is the identity map on S, for every z E I, and 1 one has (ii) for every triple z, IC, I E I of indices with z 5 K
(a --+
-+ 1 ) =
(a + l ) .
2); I , K E I, I 5 K > of sets and The inverse limit of the system { S , , ( K mappings is defined to be the subset S of the Cartesian product C = flIEISI given by --f
S = ((s,) E C;if z 5
K,
then
(IC -+ z)
:s, H s,].
CH.
1 , 8 K]
INVERSE LIMIT ARGUhENTS
55
Often this inverse limit is simply w r i t t e n h S , , the remainder of the information being understood from the context. An inverse limit is in fact a solution of a particular universal mapping problem, but this aspect of inverse limits will not be of use to us here. The set I is called directed, if it is partially ordered, and if for any two elements z, K E I there is an element A E I satisfying 1 5 , Iand K 5 A. If I is (K + I); I , JC E I, I 6 JC}is called an inverse system. directed, the system {SL, Inverse systems occur rather naturally in many contexts. Let G and H be two groups. Is G isomorphic to a subgroup of H? One way to test this is to consider some local system Z of subgroups of G. Assume that each subgroup S E C of G is isomorphic to a subgroup of H , and denote by I, the set of all monomorphisms of S into H. If T E Z contains S, then every monomorphism of T into H induces one of s;thus restriction from T to S defines a m a p (T+ S) : IT -+ I,. Clearly, this system {I,,(T+ S); S, T EZ, S s T } is an inverse system. If i = (is) is an element of the inverse limit ]im I, then i defines a monomorphism of G = uss,S into H . Thus it is important to know whether the inverse limit can be empty. It can! In the previous example let G and H be vector spaces over the prime field F = GF(p),and assume dim, H = No, dim, G = 2"O. Then clearly G cannot be isomorphic to a subgroup of H. Let Z be the local system of all the finite-dimensional subspaces of G. Then the set I, of all embeddings of S E Z into H is non-empty, and furthermore, for each T E Z containing S, the restriction from T to S is a surjection of IT onto I,; and yet the inverse limit in^ I, is empty. For our purposes the fundamental result on inverse limits is the following.
l.K.l Theorem. The inverse limit of an inverse system of non-empty Jinite sets is non-empty. This result suggests that, if in the above problem we want to embed G into H, we must find a local system Z of G such that the set of suitably restricted embeddings of S into H is finite. Before giving a proof of l.K.1 we illustrate further the applicability of this result. The first application is a special case of a rather general principle, and so allows many variations. 1.K.2 Proposition. Let 32 be a class of groups such that for groups X 2 S 2 Y with X , Y EX the finiteness of either of the indices IX : S I and (S : YI
implies that S beIongs to %. If the group G has a local system Z consisting of Jinitely generated subgroups, such that each subgroup L E Z has a subgroup of index at most n which beiongs to X then G has a subgroup of index at most n which has a local system consisting of finitely generated %-subgroups.
56
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[CH.
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5K
Proof. For each L E Z denote by I, the set of all X-subgroups of L of index at most n. By assumption, this set is not empty. Since the group L is finitely generated, it has only finitely many subgroups of index at most n, and these groups are also finitely generated. Let M E Z contain the subgroup L, and let X E I,. Then the intersection L n X has index at most n in L. Is it an X-subgroup ofL? Yes it is; for the intersection Y of all subgroups of index at most n in L is an X-subgroup of L, since it is contained with finite index in some X-subgroup of index at most n in L and contains the h u b group (1) of L. Hence the intersection S = L n X contains Y as X-subgroup of finite index, and so S is an X-subgroup of L. The intersection Y n L determines a map ( M -,L ) :IM --t IL.Obviously, the system {I,, ( M + L); M E Z, L G M ) is an inverse system of finite sets. By l . K . l the inverse limit ?!' I, is not empty. Let (XL)E 'k' I,, where X, E I,, and put X = X,. Clearly X n L = X, for L E Z and the subgroups {X,, L E Z} form a local system of finitely generated X-subgroups of X . We now show that IG : XI 5 n.
ULEr
If this were not so, then there would exist at least n+ 1 cosets of X in G. Let x l , . ., x , , ~ be representatives of n+ 1 distinct left cosets. There is a subgroup L E Z with ( x l , . . ., x,,, c L and these elements represent n 1 distinct left cosets of L n X = X , . But this is a contradiction, since IL : X,l 5 n by assumption. This contradiction shows that f G : XI 5 n, and thus proves the proposition. I
.
+
Remark. The preceding proof shows a little more, namely, that if in the situation of 1.K.2 each of the subgroups L E Z of G has a normal %-subgroup of index at most n, then G has a normal subgroup of index at most n which has a local system of finitely generated %-groups. The following application of l.K.l was given in Baer [ I ] and first raised hopes for some general, though weakened form of Sylow theory for locally finite groups. We now know that such general hopes were unfounded. 1.K.3 Proposition. Ifthe group G is generated by itsjinite normafsubgroups, then for every prime p and any two maximal p-subgroups P and Q of G there is ( a IocaIIy inner) automorphism a of G with Pa = Q. Observe that every element of the group G is contained in some finite normal subgroup of G, so G is a locally finite-normal group, see 1.A.4 and comments. Proof. If N is any finite normal subgroup of G, then the intersection N nP
CH.
1,
0 K]
INVERSE LIMIT ARGUMENTS
57
is a Sylowp-subgroup of Nfor every maximalp-subgroupP of G. For (NP:PI is finite, so the subgroup Z = nseNpPg = nnENPn has finite index in NP. Since the normal p-subgroup I of NP is contained in the maximalp-subgroup P of NP, the factor group P/I is a maximal p-subgroup of the finite group NPIZ. Thus the Sylowp-subgroup PIZ of NPII intersects the normal subgroup NIII in a Sylowp-subgroup of NZ/I. Hence P intersects the finite normal subgroup N in a Sylow p-subgroup of N. Now let Z be a local system of G of finite normal subgroups of G. For P and Q the two given maximalp-subgroups of G denote by I, the set of automorphisms of S E C which are induced by inner automorphisms of G and map the Sylow p-subgroup P n S onto Q n S. The set I, is not empty by Sylow's Theorem. Since the normal subgroup S E C of G is finite, the set Is of automorphisms of S is finite. Let T E C be a finite normal subgroup of G containing S, and let ~ E Z Then ~ . B- being induced by an inner automorphism of G - leaves the normal subgroup S of G invariant. Thus
( S n P)# = S n ( T n P)# = S n (T n Q ) = S n Q . Hence restriction of B E IT to S defines an element of Z, and thus we have a well-defined map (T-+S ) of ZT into I,. By l.K.l the inverse limit '? I, is non-empty. Let (Ps) E ' y Is with Ps E Z, for S E Z. Then (&) defines an automorphism a of G which, restricted to S E Z coincides with ps. This automorphism of G maps (Pn S) onto Q n S for every S E C. Since {P n S ; S E C} is a local system of P and {Q n S; S E Z} is a local system of Q, one has that Pa = ( P n S>. = (Q n S ) = Q.
u
,€I
u
SEZ
Clearly for every finite subset of G there is an element of G so that on that subset, a and the inner automorphism of G induced by that element coincide. I We now turn to a proof of l.K.l. Proof of l.K.l. Let {S,, (K --+ 1 ) ; z, K E I, z 5 K } be an inverse system of non-empty finite sets. Denote by 9the set of all families {TI;I E I } where for each z E Z the set T, is a non-empty subset of S , such that for z, K E I with z 5 K one has TF-") E T,. Clearly 9 is not empty. Observe that each of these families together with the appropriately restricted maps (K z) is again an inverse system. Define a partial order on S by putting {T,}< (U,}whenever TI 2 U , for every z E I. --f
a. The partially ordered set .Fpossesses a maximal member { M I ; z
EI}.
58
[CH.
BASIC METHODS, CONCEPTS AND EXAMPLES
1,
5
K
Proof. Let 9-= {{Tf, z E I} a E A } be any linearly ordered tower in 9. Consider { U,, I E I} with U , = naeAT;.Since each of the sets T: is finite and non-empty, the set U , is a finite, non-empty subset of S , . Also for z 5 K one has
ur-0
=(
n ~ y -c)n aeA
aEA
( p - l ) )
E
n 71"
=
U€A
u,.
Hence the family { U,,I E I} belongs to 9and for every {Tf} E Y one has {T;} i { U J . Consequently, by the maximal principle of set theory, the partially ordered set 9has maximal elements. I b. For each z
E
I, the intersection ( ) , I K E r M y iis) non-empty.
Proof. Suppose otherwise. Then there exists an index E E I such that for each element x E M , there exists an index J C E~ I with 1 5 IC, such that the image does not contain the element x. Since the index set I is directed and because the set M , is finite, there exists an index 1E I such that K, =< I for every element x E M,. But then for every element x E M , one has
4~
=, ~ y 0 - % ) h + =, ) MY-+,)
2 -~
and thus the image of the non-empty set M , is empty. This absurdity shows that the assumption that b is false is untenable. 1 c. For each I E I the set M , contains only a single element m,.
Note that the proof of c will also complete the proof of l.K.l, since manifestly the element (m,) of C = S , lies in the inverse limit '? S , .
Proof. Let z be any index. By b there exists an element m = m,E M , such that the subset dK+*)-' of M, is non-empty for every index K 2 1. For every index A E I put Ia = {K E I; I , 1 5 K}. Since the index set I is a directed set, the set I, is non-empty for every 1 E I. Put N - m(~+~)-l(~-+i) a -
n
KEIA
Clearly N , = {m}. We shall prove that every N , , A E I, is a non-empty set. Suppose that for the index /z E I, the set Nl is empty. Then, since S, is a finite set, there exists a finite subset A of I such that the intersection ( ) K E n m f K - r ) - l ( Kis- aempty. ) Since I is directed, there exists an index p with K 5 p for every K E A. But then m(P+o-l(P+A)
for every index K
E A,
- m ( ~ - ~ ) - l ( ~ + ~ ) - l ( 8 + ~ )(~~ m,( ) ~-i)-l(~+a) Thus the set m(lr+c)-1(8-*a) is empty. But this is pos-
CH.
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sible only if the set m(p+r)-'is empty, which contradicts the choice of the element m = m, E M , . Thus the set N, is non-empty for every index I E I. For K, 2 E I with K I one has N ~ + K )
=
(
n
m(v-+O-l(v+A)
vsrn
1( 2 - t i ) ,-= (I
m(V+o-l(V-tK)
VEI*
Since I is directed, for every v E I, there exists an index v' Then m(V'*I)-l(V'-ti)
-
and thus Nil'")
c
m(V-")-l(V+K) Vdn.
,(V~I)-l(V'+V)-l(V'+V)(V--tK)
,= -
n
vsr,
m ( ~ ' + ~ ) - l ( ~ ' + ~ )
-
n vsr,
EZ ,
with v S v'.
m(V+l)-l(V*K)
-
m ( ~ - + ~ ) - l ( ~ + ~ )
NK
Thus we have shown that the family { N 2 ,A E I } belongs to the set 9. Since one has N , c M Afor all I E I, one has ( M , } < {N,}, and by the maximality of the family {M,} one obtains { M , } = { N , ) ; that is, N , = M A for all indices 2 E I. In particular we get {m> = N , = M , for the particular index i with respect to which the preceding construction was carried out. Since this index z E 1was chosen arbitrarily, we have established c in general. I Note. The non-emptiness of the inverse limit of an inverse system of sets and maps, that is to say l.K.l, can be established more generally for compact Hausdorff spaces and continuous maps. In this context a shorter proof of l.K.l is available based on Tychonov's Theorem that the Cartesian product of compact spaces is compact. More detailed information on the subject of inverse (or projective) limits may be found in Bourbaki's treatise. No doubt, l.K.2 has been well-known for a long time, but it seems to appear in print only in 1963; see Baer [5]. Section L. Linear groups
In the final section of this chapter we collect a few facts about linear groups that we shall need to refer to. Here by a linear group we mean any subgroup of the group GL(n, F ) = GL( V )of units of the ring End, V of all linear transformations of the n-dimensional vector space V over F into itself, for any natural number n and any field I;. (Abusing this terminology somewhat we shall occasionally consider groups that are isomorphic to groups of linear transformations as linear groups.) The linear transformation x E End, V is called unipotent if all its eigenvalues are 1E F (or equivalently, if ( x - 1)" = 0 where 0, 1 E End, V ) . The subgroup G of GL(n, F )
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is called unipotent if each of its elements is unipotent. Such a group G is necessarily nilpotent : by a theorem of Wedderburn (see Curtis and Reiner [ l ] ,27.27) the F-subalgebra R of End, Vgenerated by the set { g - 1 ; g E G} is nilpotent (notice that ( x - l ) ( y - 1 ) = ( x y - 1 ) - ( x - 1 ) - ( y - 1 ) ) ; by construction the group G stabilizes the series V 2 V R 3 ... 2 V R n = (0)
of subspaces of V, and it is nilpotent by l.C.l. Observe that if the field F is locally finite, i.e. if F is an absolutely algebraic field of finite characteristic, then the group GL(n, F) is locally finite. This is obvious if the elements of GL(n, F ) are interpreted as matrices, for then the entries of finitely many matrices generate a finite subfield of F, so that every finitely generated subgroup of GL(n, F ) is a linear group over a finite field, and thus is finite. That the restriction to locally finite fields is unnecessary for periodic groups is the content of the following result which essentially is due to Schur [I]. (See Curtis and Reiner [ l ] ,36.2 for a quite different proof).
l.L.l Theorem. A periodic linear group is locallyfinite.
Proof. Let G be a finitely generated periodic subgroup of GL(n, F) for the field F, which we may (and hence shall) assume algebraically closed. Denote by A the algebraic closure in F of the prime subfield of I;. We want to show that G is finite. For this purpose, we regard G as a group of matrices and denote by R the A-subalgebra of F generated by the entries of the matrices belonging to G. Since the group G is finitely generated, R is a finitely generated (commutative) A-algebra. If M is a maximal ideal of R, the field RIM is a finite dimensional A-algebra. Thus R/M is a finite algebraic extension of A. Since A is algebraically closed, A N RIM. Let v be the natural A-algebra homomorphism of R onto RIM( N A). Then v clearly induces a homomorphism of the free R-module M of rank n onto the vector space V of dimension n over A and an R-algebra homomorphism of End,M onto End, V. Consequently v induces a group homomorphism V of G into GL( V ) . Since G is a torsion group, the eigenvalues of each g E G all lie in A, and they are preserved under this homomorphism V. But then the kernel K of the homomorphism i j is a unipotent and hence nilpotent group. If G" is finite, then by 1.A.2 and comments G is locally finite and hence finite. Denote by S the subring of A generated by all the entries of the matrices belonging to H = G', and let L be the quotient field of S. Since H i s finitely
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61
generated, S is a finitely generated ring and L is a finite algebraic extension of the prime subfield of A . If the characteristic of A is positive, then the field L is finite; hence GL(n, L ) is finite, which forces the finiteness of H (see also the remark preceding the theorem). Thus to complete the argument, we may assume that A has characteristic zero. For every element h E H the eigenvalues of h are roots of unity and also roots of the characteristic polynomial of h. This polynomial has degree n and its coefficients belong to the field L. Thus the eigenvalues of h have (multiplicative) order dividing t = 1.c.m. {r E N;4 ( r ) 5 n(L : Ol}. Here 4 ( r ) is the Euler function, and it is easily seen that 4 ( r ) -+ co as r -+ co. Thus t is finite. The trace tr h of h can take at most t” different values. Let k denote the product of all the distinct non-zero values of (tr h ) -n actually taken as h ranges over H. The element k E S is non-zero; hence there is a maximal ideal N of S with k 4 N (for the Jacobson radical of a finitely generated integral domain is zero). The factor ring T = S/N is a finite field, and the natural map p : S + T induces a homomorphism p of H into the finite group GL(n, T ) . If for h E Hone has tr h # n, then, since k # 0, tr hp = (tr h)” # (tr I>”= n”. Thus tr h = n for all the elements of the kernel C of p. But since the eigenvalues of h are roots of unity, the equation t i h = n implies that all the eigenvalues of h actually are 1 (see Curtis and Reiner 30.1 1). Thus the kernel C of ji is unipotent and hence nilpotent. Again 1.A.2 applies and so H is finite. * I In Winter [l] it is shown that periodic linear groups have a rather special structure.
1.L.2 Theorem. A periodic linear group has a unipotent normal subgroup of countable index. In particular, every simple periodic linear group is countable.
Proof. Let G be a periodic subgroup of GL(n, F). Clearly, we may assume that F is algebraically closed. The n-dimensional vector space V over F has a composition series of length r with 0 < r 5 n as an FG-module. Choose such a composition series (0) c
v, c . . . c vrT1c v, = v,
and let p i denote the homomorphism of G into the group GL(Vi/Vi-,) for
*
One even has C = < 1 > since in characteristic zero unipotent groups are torsion-free.
62
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I r defined by the action induced by G on the composition factor
Vi/Vi-l of V. Clearly the intersection K = flr=l ker p i is unipotent. Since Gpi, it suffices to the factor group GIK embeds into the direct product show that each of the irreducible groups G p iis countable, in order to deduce that the group GIK is countable. Let H be a periodic subgroup of GL(n, F ) which is irreducible. By a theorem of Burnside (Curtis and Reiner [l], 27.4) there are n2 elements in H which are linearly independent over F (as elements of End,V). By l.L.l the periodic linear group H is locally finite. Let D be any finite subgroup of H containing n2 linearly independent (over F ) elements. Let A be the algebraic closure (in F ) of the prime field contained in F. If E is a finite group and if zl,. . ., z, is a maximal set of pairwise non-equivalent irreducible representations of E over A , then the extended representations 7 7 , . ., $ are a maximal set of pairwise non-equivalent irreducible representations of E over F (Curtis and Reiner [l], 29.21). Consequently, if z is any irreducible representation of E over F, then z is equivalent to one of the representations 7.: So there exists an element x E GL(n, F ) such that D" C GL(n, A ) . For every element g E H, the subgroup
of H is finite, and hence there is an element y = y ( g ) such that (D,g)' c GL(n, A ) . If now [ D ] denotes the ring of matrices generated by D,i.e. its additive closure, then clearly A [D"] = A [PI= A , , the ring of n-by-n matrices over A . But gy E A [ D Y ] and , thus g E A [ D ] .Therefore g" E A[D"] = A, for every g E G, and so G" c GL(n, A ) . Since the field A is countable, the group GL(n, A ) is countable, and so is G. I For special classes of groups one can say quite a bit more.
n:=
.
1.L.3 Lemma. For every prime p other than the characteristic of the field F each p-subgroup P of GL(n, F ) is a Cernikov group. Proof. Let P be anyp-subgroup of GL (n,F ) and A any finite elementary abelian subgroup of P. Clearly we may assume that F is algebraically closed. By Maschke's Theorem (see Gorenstein [l] 3.3.1, or Curtis and Reiner [l] 10.8) the subgroup A is completely reducible. It is a simple consequence of Schur's Lemma that an abelian irreducible linear group over F has degree one (see Gorenstein [ l ] 3.2.4). Hence GL(n, F) contains an element x such that every element of A" has diagonal form. Clearly the group of all n-by-n diagonal matrices over F of finite order is isomorphic to the direct product of n copies of the group of roots of unity in F. Thus IAl 5 p". By l.L.l the group P is locally finite, and so 1.6.4 yields that P is a Cernikov group. I
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A result of a similar nature (due originally to Schur [l]) is that any periodic linear group over a field of characteristic zero has an abelian subgroup of finite index. This result is a simple consequence of the theorem of Jordan (see Curtis and Reiner [ l ] 36.13) and 1.K.2. A recent extension of Jordan’s Theorem to fields of positive characteristicin Brauer and Feit [I] allows something to be said also about periodic linear groups over fields of positive characteristic.
1.L.4 Theorem. There exists an integer-valued function f(n, p , IPI)of three variables such that any periodic linear group G of degree n over afield F of characteristic p 2 0, that contains a finite maximal p-subgroup P, has an abelian normalp’-subgroup A offinite index bounded by f ( n , p , /PI). I f p = 0, then we interpret P as (1); thus 1.L.4 gives the theorem of Schur mentioned above. Proof. Let Z denote the local system of all those finite subgroups of G that contain the subgroup P . If p # 0, then P clearly is a Sylow p-subgroup of every member of Z.By Jordan’s Theorem if p = 0 and by Brauer and Feit [ l ] if p > 0, there exists a function f (n, p , [PI)such that every member S of Z has an abelian normal p’wbgroup of index bounded by f(n, p , IPI). Applying now 1.K.2 to G with respect to this local system one obtains that G, too, has an abelian normalp’-subgroup of index bounded byf(n, p , IPI).I This result has a slightly weaker corollary that has the advantage of not mentioning characteristic, degree or bound.
1.L.5 Corollary. If every primary subgroup of the periodic linear group G satisfies min, then G has an abelian normal subgroup offinite index. Proof. Since G is linear, there exists an integer n and a field F of characteristic p 2 0 such that G is isomorphic to a subgroup of GL(n, F ) . Suppose the characteristic p of F is not zero, and let P be a maximal p-subgroup o f G. If we can show that the group P is finite, then the corollary follows at once from 1.L.4. The group P satisfies min, and by l.L.l it is locally finite. Hence P is a Cernikov group. But P is also unipotent, so if x E P then 0 = (X-l>” = (X-1)P”
= XP-1,
sincep is the characteristic of F. Thus P has finite exponent and so is finite. I One more remark on linear groups in general: in a linear group G c GL(n, F ) ascending (and descending) chains of centralizers ( o f subsets of End, V ) have lengths bounded by n2, since the centralizer in End, Y of any subset of End, V is an F-subalgebra of End, V. Thus to any chain of central-
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izers in G there corresponds a chain of centralizer F-subalgebras of End, V. Since dim,End, V = n2, no chain can be longer. (In fact this bound can be considerably improved.) We shall use this observation to motivate a more general form of 1.L.5 where linearity is replaced by chain conditions on the set of centralizers in the group, c.f. 3.22 and 5.10. The condition that the p-subgroups of the linear group G over the field F of characteristic p be finite cannot be deleted in l.L.4, since for any infinite, locally finite field F and every natural number n 2 2 the group PSL(n, F) is a simple, locally finite group (see, for example Scott [l], 10.8.4). The following problem on linear groups is related to questions treated in Chapter 2 and in Chapter 4, Section A, but we have not been able to handle it. Question 1.8. If G is an infinite simple, periodic linear group, does every element of G have an infinite centralizer in G? For thorough discussions of the properties of periodic linear groups in comparison with the properties of finite groups see Curtis and Reiner [l], 9 36, Platonov [l], and Wehrfritz [4], [ 5 ] , and [6]. Appendix. Mal’cev’s representation theorem by ultraproducts In this short appendix to Section 1.L, we outline a proof (of a slight generalization) of a theorem of Mal’cev [ l ] that we shall have occasion to use at several instances in this book. The method of proof sketched here might also serve as an illustration of the applicability of the ultraproduct construction to algebric questions of a certain type. Perhaps the best reference at present to this area of mathematics is J. L. Bell and A. B. Slomson: Models and Ultraproducts, An Introduction, 1969, North-Holland, Amsterdam. The relevance of the notion of ultraproducts to (universal) algebra is also clearly brought to the fore in the relevant sections of P. M. Cohn: Universal Algebra, 1965, Harper and Row, New York, and of B. I. Plotkin [2]. 1. Filters and ultrafilters on a set. - If I is a set, then the non-empty family 9of subsets of Z is afilter on Z if a the empty subset !Zi of Z does not belong to 9, b X , Y E 9implies X n Y E 9, and c X c Y c Zand X E 9 implies Y E 9. The filter 9 is an ultrafilter on Z if furthermore it satisfies d for each subset X of Z one has either X E 9or Z\XE 9. It is a simple consequence of the maximum principle of set theory that every filter on Z actually is a subfamily of some ultrafilter on Z. The obvious example of ultrafilters that comes to mind is the family of all subsets of the non-empty set r which contain a fixed element i E Z. Such
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an ultrafilter is called principal. On a finite set I every ultrafilter is principal. If the set I is infinite, the family of complements of finite subsets of I is filter on I, called the co-jinite$lter on I. An ultrafilter on the infinite set I is non-principal (and hence interesting) if and only if it contains (as a subfilter) the co-finite filter on I. 2. Reduced products and ultraproducts. - For every element i of the index set I let a non-empty set S , be given, and consider the Cartesian product niErSi.If 9is a filter on the index set I, define 9-equivalence for elements of n [ i s r s i by
(ai) F(bi) if and only if (i E I; ai
=
bi} E 9.
The set of 9-equivalence classes of elements of nierSi is denoted by ( H i ES , i ) / 9and called the product of the Sireduced relative to 9, or simply the 9-reduced product. If the filter 9 is an ultrafilter on the index set I, then (njEISi)/9 is usually called an ultraproduct of the Sirelative to 9).
3. Additional structure. - We shall not try to find out what sort of alalgebraic (and other) structure on the Si survives the formation of reduced products (or ultraproducts). We shall only be concerned with easy examples that we shall state concisely, the proofs being left as an Exercise. For each element i E Ilet R, be a ring and Si a left R,-module (respectively with n generators at most); further let G i be a group of automorphisms of the R,-module S i .Let Fbe any filter on the index set I, put R = (niErR i ) / 9 , S = ( n i E r S i ) / and 9 G = (n[iErGi)/9. Then R has a natural ring structure, and S is in a natural way a left R-module (respectively with n generators at most), while G is a group of automorphisms of the R-module S. If the rings Ri happen to be (skew) fields, then R = ( n i e r R i ) / 9 is also a (skew) field, provided that 9is an ultrafilter. If the element (ai)E niErRi is not 9-equivalent to the element (0,)then { i E I; ai # O,} E 9.Obviously (a,)(a;) = (aiai) is 9-equivalent to the element ( l i ) if one puts a, =
,
(1a;'
if a, if ai
=
Oi
z Oi '
and so the image of ( a i )in the ring R (with 1) has an inverse. For the applications of the preceding elementary observations that we want to make, we need the notions of linear and projective representations of a group. The map p : G GL(n, F ) is a linear representation of G if it is a homomorphism, it is aprojective representation if the map : G + PGL(n, F ) obtained by composing p with the natural projection of GL(n, F ) onto --f
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PGL(n, F ) = GL (n, F ) / Z ( G L (n, 8’))is a homomorphism. The linear (respectively projective) representation p of the group G is faithful if ( 1 ) is the kernel of p (respectively, p).
1.L.6. If f o r every element i of the non-empty index set I there is given a group Gi together with a faithful linear (respectively projective) representation p i : Gi+ GL(n, F i ) offixed degree n over somejield F i y then f o r every ultrafilter 9on I, the group G = ( n i E I G r ) / S has a faithful linear (respectively projective) representation
p =(
of degree n over thefield F =
n ieZ
p i ) / 9:G
-P
GL(n, F )
(niEI Fi)/9.
In view of the preceding remarks on ultraproducts of modules, oiily the corresponding statement for projective spaces needs verification. 4. Applications of special ultrajilters. - As a further application of ultra-
products we prove
1.L.7. If C is a local system of subgroups of the group G, then G is isomorf o r a suitable ultrafilter 9 phic to a subgroup of the ultraproduct (nserS)/9 on the set C. Proof. Denote by C the Cartesian product C = nsezS. There is an injection of the set G into C given by
For every S E C put Ps = ( T EC; S E T } . For X , Y EC there is an element Z of the local system Z, with X , Y E Z , hence Px n P, 2 Pz. The set {Ps; S E C} of subsets of C generates a filter Fz on C; let 9be any ultraas a subfilter. filter on C which contains Fz By choice of the ultrafilter 9one has gv = ( g s ) ( 1 ) if g # 1; hence the map V of G into the 9-equivalence classes of C induced by v is still injective. It also is a homomorphism: if g , h E G, there is an element S E Z with g, h. E S. For every T EPs one has g T h , = (gh)= that is to say gvhv g(gh)’. I 1.L.8. r f Z is a system of normal subgroups of the group G such that for every jinite set S of elements of G there is a normal subgroup N E C with g N # hN for any two distinct elements g, h E S, then G is isomorphic to a
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subgroup of the ultraproduct ( n N G z G / N ) /for F a suitable ultrafilter 9 on the set C.
Proof. Denote by C the Cartesian product C homomorphism of G into C defined by g
I+ ( g N ) N d .
=
nNEZ GIN. There is a
nNEI
That this is a monomorphism follows from the fact that N = ( 1 ) . For every finite set S of elements of G denote by Ps = ( N E C; g N # hN for any g, h E S, g # h). By assumption on C the set Ps is not empty. Let S and T be any two finite sets of elements of G . Then
B # psvT c PsnP,. Thus the collection of all these subsets of C generates a filter Fz on C. Let 9be any ultrafilter containing Fx. Then clearly for any two distinct elements g, h E G, one has g y J h". Hence the homomorphism 9 of G into C l 9 induced by v is in fact an injection. I For many purposes it is useful to combine the two last statements. Putting the last three results together gives us an important result of Mal'cev [l]. 1.L.9. Let C be a local system of the group G such that for every subgroup S E C there is a system Cs of normal subgroups of S such that for every finite set X of elements of S there is a normal subgroup N E Cs with x N # y N i f x , y are distinct elements of X.If for every N E Cs there is a faithful linear (respectively projective) representation pN of SIN of jixed degree n (over suitable, possibly different fields) then there is afield F such that the group G has a faithful linear (respectively projective) representation of degree n over F. It should be remarked that the preceding type of result also holds for many other algebraic and non-algebraic structures.
CHAPTER 2
Centralizers in Locally Finite Groups
Since the study of infinite groups began, it has been asked whether every infinite group contains an infinite abelian subgroup. The answer to this question is negative; the counter examples must, of course, be periodic groups. In Novikov and Adjan [3], it is shown that for every odd natural number n larger than 4381 every finitely generated, non-cyclic free group of exponent n is infinite, but each of its maximal abelian subgroups is finite. Every infinite, locally finite group, however, does contain infinite abelian subgroups. This positive result, due to Kargapolov [ 6 ] and P. Hall and Kulatilaka [l], is, as far as the remainder of the book is concerned, the most important result of this chapter. A question closely related to the preceding one is known, particularly in the Russian literature, as Smidt’s Problem. Under what conditions does an infinite group contain proper infinite subgroups? Are the Prufer groups the only infinite groups every proper subgroup of which is finite? The solution of Smidt’s Problem for locally finite groups is an easy consequence, as we shall see, of the existence of infinite abelian subgroups in infinite, locally finite groups. We shall present these results in a slightly more general context by asking for conditions under which a group of cardinal N contains a proper (respectively, abelian) subgroup of cardinal N . The “generalized Smidt Problem” can be solved in the class of locally finite groups for regular cardinals N ,but the existence of very large abelian subgroups in a locally finite group is shown to be a sharp restriction on the group. An intermediate version of the preceding problems asks, whether there exists a non-trivial element g in the group G of cardinal N such that the centralizer Ccg satisfies lCGgl= X. We shall show that this is so for locally finite groups of regular cardinal X . Thus, in every infinite, locally finite group there are elements with large centralizers. It is via this result that we approach the preceding problems. Since locally finite groups have many elements with large centralizers, 68
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69
it seems reasonable to expect that the assumption that many elements of the infinite, locally finite group G have very small centralizers will have rather strigent structural consequences for G. The final part of this chapter throws some light on this situation: there we assume that the centralizers of many of the involutions of G are finite. However, a more detailed discussion will be left to Chapter 3. Unlike the later chapters of this book, this chapter will not be rigorously confined to locally finite groups; in fact, most of the statements will concern certain periodic groups which may very well not be locally finite. But with the exception of the closing paragraphs these results are presented only in this greater generality since their proofs do not require extensive extra work. A cardinal N is called regular if it is infinite and if IUrrEIS,,( < tz whenever the sets I and S,, for each n E I are of cardinal less than N . For example x 0 and N, are regular. More generally, any infinite cardinal N , with a non-limit suffix 01 is regular. 2.1 Theorem. Let x be a regular cardinal and G a periodic group of cardinal at least X. If G contains an involution i such that lCGij < Et then: a G contains an element g such that lCcgl 2 x and g i = 9 - l ; further g can be chosen to be either an involution or a non-central element of G of odd order; b G contains an infinite abelian subgroup A such that either ai = a - f o r every a E A , or A is a 2-group and a' = a f o r every a E A . Notice that if in 2.1 we have N = x o , then we are assuming that the centralizer C G i is finite, and hence the second possibility for A cannot arise. If K > N~ then it most decidedly can arise as easy examples show.
Proof. a. If cGicontains an involution g such that IcGgl 2 N then g satisfies the requirements of a. Suppose therefore that the cardinal of the centralizer in G of any involution z of C G i is less than x . Since K is regular the set S = C G z has cardinal less than N , where the union is taken over all the involutions z E C , i. Let j be an involution of G such that ij has even order. Then the dihedral group ( i , j ) contains a central involution, and so j E S. Clearly i has at least N distinct conjugates in G. Hence, if T denotes the set of all non-trivial elements i j of G of odd order, where j ranges over the involutions of G, then (TI 2 N. We fix an element a of T. Then a = ik for some involution k # i of G, and i = k"' for some element a , E ( a > . For any x in G, if ik" has even order, then k" E S. Hence the elements x of G for which ik" has even order lie in fewer than N right cosets of CGk.Since i and k are conjugate the set of all
u,
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[CH.
2
these x E G has cardinal less than N and if M = {x E T : ik" has odd order}, then IMI 2 N. Let c E M and put s = ik'. Since s has odd order, there exists an element s1 of (s) such that i = (kc)sl.But then i = k"' = k'", so h = csl a; E CGk . Now j = ci is an involution of G and
'
since c and s1 lie in T. Therefore, for every c E M there exists an element h E C G ksuch that the involutionj = ci conjugates ha, onto its inverse. Since k is conjugate to i we have that lCGkl < N. The regularity of N now implies that there exists an element r of C , k and a subset N of M of cardinal at least N such that ci conjugates y = ra, onto its inverse for every c E N. Clearly cd-' E C G yfor all c, d E N , and SO lCGyl 2 N. If C G y = G , then r E C G k implies that k = k"' = i, a contradiction which shows that y is not central in G . Let c E N . Since the element c has odd order, there exists an element c1 E "= i. Put g = y"'. Then g is a non-central element of G such that ICGgl 2 N and gi = g - ' . Finally, the centralizer of an involution of C G ihas cardinal less than N and IcGgl 2 N, so g must have odd order. b. By a there exists a non-trivial element g 1 E G such that g: = g;' and IC,gll 2 N. Clearly the group
.
An easy inductive argument leads to the existence of elements g n ,n E N, . ., of G such that H = (gn; n E N,. . .>is an infinite hypercyclic group and g,gt
.
,g 2 , . .,g,-,>
E
4
H
for all n > I.
The group (i, H ) is also hypercyclic, and therefore by 1.A.7 the group ( i , H ) contains a self-centralizing abelian normal subgroup A , and A will also be infinite. A is a direct product of a 2-group A , and a 2'-group A,,, at least one of which is infinite. Clearly i acts fixed-point freely on A,, and ai = a- for every a E A , , . Thus we may assume that A , is infinite. The map a H d-' is an endomorphism of A , with kernel CA2iand image [ A z , il, so at least one of the subgroups CA2iand [ A , , i] is infinite. Finally (ai-l)i = which implies that ai = a - 1 for every a E [ A , , i ] , and the result follows. I
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2.2 Corollary. Let G be a periodic group and x a regular cardinal such that IG( 2 X . I f G contains an involution, then G contains a non-trivial element g such that lcGgl 2 K.
2.2 is an immediate consequence of 2.la. Our next major objective is to obtain the conclusion of 2.2 for an arbitrary locally finite group. A major part of the argument is isolated into the following lemma.
2.3 Lemma. Let G be a group, p a prime and R an elementary abelian psubgroup of G or orderp2. Suppose that A?isa set ofjnite subgroups of G each containing R such that 1-41 = x is a regular cardinal. Ifevery M E & contains a non-trivial abelian normal subgroup N M ,then there exists an element g # 1 of some M E -4 such that (C,gl 2 N. Proof. Suppose that lcGgl < N for every g E M\(1> and every M E A. Put C = UgaR\
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[CH.
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Now a finite group each of whose Sylow subgroups is cyclic, must be metacyclic (Gorenstein [ l ] 7.6.2). Thus G is locally metacyclic and thus is metabelian. But then G contains a non-trivial abelian normal subgroup, which, in view of the structure of the prime power subgroups of G is Iocally cyclic and so contains a non-trivial cyclic characteristic subgroup S. Hence C,S has finite index in G and therefore ICGSI 2 N . This contradiction of our initial assumption completes the proof. I In the above proof we have not exploited in full the local finiteness of G. We have only used in fact that every 4-generator subgroup of G is finite. One can squeeze a little more out of this proof. If N is a regular cardinal and G is a group such that any three elements of G of prime order generate aJinite subgroup, then G contains a non-trivial element g such that [C,gl 2 N. For if one assumes that no such element g exists then clearly every nontrivial element of G has at least N distinct conjugates. Just as in the above proof one then obtains that every subgroup of G generated by three elements of prime order is metacyclic. Since by assumption G is a counter example to the contention, the subgroup H of G generated by all the elements of G of prime order has cardinal at least N and is not abelian. Thus G contains two non-commuting elements, x and y say, of prime order. If R = (x, y ) then for every element z of G of prime order, the group ( R , z ) is metacyclic and so R‘ is normal in (R, z ) . But then R‘ is a non-trivial finite normal subgroup of H and hence IC, R’I 2 N. I With a different approach Strunkov [2] and [3] has shown that in 2.4 it is only necessary to assume that every 2-generator subgroup of G is finite. (Whether this is also too much is tied up with the existence or otherwise of the “Tarski monsters” mentioned in $ 1.E.). The appropriate generalizations of 2.5,2.6 and 2.7 below follow without trouble once the generalization of 2.4 is proved, see Strunkov [2], [3]. 2.5 Corollary. I f the injinite group G is either locally Jinite or a 2-group,
then G contains an injnite abelian subgroup. Proof. By 2.2 or 2.4 the group G contains a non-trivial element g1 such that G, = C,g, is infinite. G,/(g,) also satisfies the hypothesis of 2.5, so in the same way G,\(g,) contains an element g2 such that G2 = CG, ( ( g l , g 2 ) / ( g l ) ) is infinite. Repeating this choice countably often we obtain a sequence {g,; n E N) such that the group M = (g,; n E N ) is infinite and hypercentral. By 1.A.7 the group H contains a self-centralizing abelian normal subgroup A . If A is finite then H / A = H/C, A is also finite, which contradicts the infiniteness of H. Therefore A is an infinite abelian subgroup of G. I
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2.6 Theorem. Let K be a regular cardinal and G a group of cardinal at least that is either locally j n i t e or a 2-group. Then, except f o r the trivial case where N = K and G is a Priifer group, G contains a proper subgroup of cardinal N .
K
Proof. Suppose that every proper subgroup of G has cardinal less than Et . If N is any proper normal subgroup of G and x E N , then x has fewer than N conjugates in G. Thus IC,xl 2 N, and consequently C , x = G. It follows that G/ZG is a simple group. If G is not abelian, then lZGl < K and SO IGjZGI 2 N. But then 2.2 and 2.4 imply that the factor group G/ZG contains a non-trivial element whose centralizer has cardinal at least s. Since GjZG has trivial centre and G contains no proper subgroup of cardinal at least X , it follows that G is abelian. If G is decomposable, that is if G = H x K for some proper subgroups H and K of G, then both H and K have cardinal less than s and so /GI < N. Therefore G is indecomposable. An indecomposable periodic abelian group is either cyclic or a Prufer group (KuroS [l], Vol. 1, p. 181). Hence G is a Prufer group. Since IGI 2 X, which is infinite, it follows that K = N o . I 2.7 Corollary. A group satisfying min is countable jinite or a 2-group.
if it is either locally
Proof. Let G be an uncountable group satisfying rnin and suppose that G is either locally finite or a 2-group. G contains a minimal uncountable subgroup H. Since N is regular H contains by 2.6 a proper uncountable subgroup. This contradiction demonstrates the point. I Observe that by 1.F.6 a 2-group satisfying min is a Cernikov group, and hence countable. If one uses the deep result 5.8 that every locally finite group G satisfying min is a Cernikov group, the countability of G is clear. The point of 2.7 however, is that this result can be obtained in a rather elementary way. Exercise. The group G satisfies min and contains involutions. Use 2.1a to show that G is countable if the centralizer of each involution of G is countable. Contrary to what might be suggested by 2.4 and 2.6 it is not possible to give a version of 2.5 involving an arbitrary regular cardinal.
2.8 Theorem. For any uncountable regular cardinal N and any prime p there exists a locally finite p-group of cardinal N whose abelian subgroups all have cardinal less than K .
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For the construction of such groups we need the following simple fact concerning wreath products. Let W = X t Y be the standard (restricted) wreath product of the groups X and Y and let S be a non-empty subset of Y. If B denotes the base group of W then N,S = N y S . C,S. Further if S is infinite then C,S = (1) and N,,S = NyS. If S = (1) all this is obvious, so suppose otherwise. Clearly N , S . C E S E N,S. Let z be an element of N,S. Then z = yb for some y E Y and b E B. For each s E S there exists an element b, of B such that sz = sYbsE Y n Yb,.
We have b, = 1 and sz = sy. Therefore y E N,S. Also [sy, b] = (s')-~s'
=
1.
But Sy = S, and consequently b E C,S. Suppose now that S is infinite and let b E C , S. The element b is a mapping of Y into X with finite support (the set of elements of Y not mapped to 1). For any y E Y and s E S we have that ( y ) b = (Y)bS= (V)h
so b is tonstant on the right cosets of S in Y. Since S is infinite and b has finite support, b maps Y onto (I); that is b = 1, and thus C,S = (1). 1 Proof of 2.8. Let p be the first ordinal of cardinal N. We define a group G, for every ordinal CJ 5 p as follows. Put Go = (l), G,,, = X u + , t G, where Xu+ is a cyclic group of order p , and for limit ordinals A set Gz = I G, . Clearly G = G,, is a locally finite p-group of cardinal 8 . Let A be any infinite abelian subgroup of G. The subgroup A contains a countably infinite subgroup S. Since p is a limit ordinal of uncountable regular cardinal, S c G, for some 0 < p. Let a E A . If z is the first ordinal such that a E G,, then z is not a limit ordinal and z < p. Suppose that IS < 7. G, = X , ? G,- I . Also S c G, E G,so by the above remarks a E N,, S c G, - ,. This contradicts the minimality of z and so z 5 CJ.Since this is so for every a E A it follows that A E G,. But CT < p, and therefore IAl 5 IGul < K. I The above argument in fact shows that the normalizer in G of any infinite subgroup of G of cardinal less than N has cardinal less than N and, more generally, a subgroup of G has cardinal less than Et if (and only if) it contains an infinite ascendant subgroup of cardinal less than x (we could shorten the above proof slightly by relying more heavily on the commutativ-
u,<
CH.
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ity of A). It is not difficult to derive from this that every SN*-subgroup (that is, a subgroup with an ascending series with abelian factors) of G has cardinal less than K. A different type of uncountable locally finite group, the locally nilpotent subgroups of which are all countable, is exhibited in Baer [lo]. The group G constructed in 2.8 has other noteworthy properties. For example, although G is a locally finite p-group, the only nilpotent ascendant subgroup of G is the trivial one. Thus G is an example of a periodic locally nilpotent group that satisfies the minimal condition on nilpotent ascendant subgroups, and which visibly does not satisfy min. Thus 1.G.4 cannot be extended along the lines of 1.G.5 (see remarks after 1.G.5 in this context).
Exercise. If Q is the quaternion group then Q has an automorphism a of order 3 such that whenever x is an element of Q of order 4 then [x, x"] # 1. For i E N, let Q iN Q and put A4 = YLEI Qi, where the centres of the Q iare amalgamated. The automorphism a of Q determines an automorphism ai of M of order 3 that fixes Q j for i # j and operates on Q ias 01. If E is the subgroup of the automorphism group of M generated by all the aiand G is the split extension of M by E, prove that G is a periodic soluble group of derived length 3 that is infinite but which contains no infinite abelian normal subgroups. We come now to the final body of results in this chapter. They concern the structure of periodic groups that are not 2'-groups and whose involutions have small centralizers. The following result is basic.
2.9 Theorem. Let G be an infinite periodic group containing an involution i such that the centralizer in G of every involution of G centralizing i isjnite. Then G is a locallyjnite group and all the involutions of G are conjugate. Proof. The proof of 2.9 will be accomplished in a series of steps. For brevity we shall say that the involution j of G inverts the subset S of G if sj = s-' for every s E S. By 2.lb the group G contains an infinite abelian subgroup A inverted by i. Notice that A is necessarily a 2'-group since otherwise every involution of A would have infinite centralizer in G and yet would be centralized by i. a All
the involutions of G are conjugate.
Proof. Let j be any involution of G; we prove that j is conjugate to i. If the product ij has odd order then i and j are conjugate. Suppose therefore that i j has even order. Then the group (i, j ) contains a central involution, z
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2
say. Since z centralizes i the order of C G zis finite and z has infinitely many conjugates in G. If k is any involution of G such that ik has even order then k lies in the centralizer of some involution centralizing i, and by assumption the union of all these centralizers is a finite set. Hence there exists only a finite number of such involutions k of G and thus izx has odd order for at least one element x of G. But then i and zx are conjugate. Therefore the centralizer of every involution of G centralizing z is finite and in the same way z and j are conjugate. Thus in either case i a n d j are conjugate. I
b If a is an element of G of odd order inverted by i and A is an infinite abelian 2'-subgroup of G inverted by i then C A ahasjkite index in A . Proof. Now a = ik for some involution k of G, and thereexists an element a , of ( a ) such that i = k"'. Notice that a: = ki. As in the proof of 2.la there exists an element r of C,k and an infinite subset N of A such that for each c E N , the involution ci inverts ra, . Let c and d be distinct elements of N and put g = c-'d. Since gi = 9-l it follows that the involution i normalizes the centralizer C,g. Clearly ra, centralizes g and therefore [i, (ra,)-'] € C G g .But
[i, ( r u , ) - ' ] = ira, ial'r-'
=
iru?ir-'
=
irkr-I
=
ik = a ,
since r E C G k .Hence g is a non-trivial element of A centralized by a. Also i inverts g, so k also inverts g. (a, i, A ) c H = N,CAa. Either CAa is finite (when clearly H/CAu satisfies the hypotheses of the theorem), or for any involutionj of H the subgroup (j, C , U ) satisfies the hypotheses of the theorem. In the latter case a implies that all the involutions of ( j , C A a ) are conjugate and the Frattini argument yields that c H ( < j , CA a>/CAa ) = CA a
. cH
j.
Hence either H/CAais finite or it also satisfies the hypotheses of the theorem. If CAa has infinite index in A we may apply the above result to H/CAato conclude that A contains an element g such that g $ CAa,while ggk E CAu. It follows that k acts fixed-point freely on the subgroup B = (9, C , a), and hence k inverts B. But then both i and k invert B, so a = ik centralizes B, and in particular g. This contradiction to the choice of g shows that the index of CAain A is finite. 1 c If A is an infinite abelian 2'-subgroup of G inverted by i and k is any intlolution of G, then k inverts some subgroup offinite index in A .
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Proof. In view of a the centralizer in G of any involution centralizing k is finite, and hence there exists at most a finite number of involutionsj for which j k has even order. For each x E A the element ix is an involution of G inverting A . Hence there exists some element x E A such that a = ixk has odd order. The involutions i and ix are conjugate, so by b the subgroup C = C , a has finite index in A . Finally ix inverts C and thus k also inverts
c. I
Let R denote the subgroup of G generated by all the involutions of G. Clearly R is a normal subgroup of G .
d R is locaIly finite.
...,
Proof. Let j , ,j , , j,, be involutions of G and put L = ( j , ,j , , . . .,j,,). We have to prove that L is a finite group. By 2.lb and c above there exists an infinite abelian 2'-subgroup A of G that is inverted by each of the n involutions { j , , j n } .We may choose A to be a maximal such subgroup of G. Suppose that A L / A is an infinite group and let N = NGA. By an argument we have seen in the proof of b the group N/A-satisfies the hypotheses of the theorem. Therefore by 2.lb and c, again there exists a subgroup B of N such that B / A is an infinite 2'-group inverted by each of the involutions j , . But then each j , acts fixed-point freely upon B, and thus B is abelian and inverted by each of the j,. This contradicts the maximal choice of A and consequently ALIA is a finite group. Now 1.A.2 implies that A L is a locally finite group and therefore L is finite. 1
. ..,
e G / R isfinite and G is a locally finite group. Proof. R satisfies the hypotheses of the theorem and so a also implies that all the involutions of R are conjugate in R . The Frattini argument then yields that G = RCGi, and so G / R is a finite group. The local finiteness of G follows from this, d and 1.A.2. This completes the proof of both e and 2.9. 1 In the situation of 2.9 much more can be said. In fact once the local finiteness of G is established one only needs the existence of a single involution i of the group G with finite centralizer in G, in order to prove quite strong results. This is more conveniently done towards the end of Chapter 3. We content ourselves here by merely stating the result under the stronger hypothesis of 2.9 and referring the reader forward. The following corollary is an immediate consequence of 2.9, 3.33 and 3.36. (The reader may like to find a direct proof for himself.) 2.10 Corollary, I f the injnite periodic group G contains an involution i such
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[CH.
2
that the centralizer in G of every involution of G centralizing i isfinite, then G contains a soluble normal subgroup offinite index.
It is not difficult then to show that the group G of 2.10 contains a soluble normal 2'-subgroup S such that G/S is a finite group with a unique involution. The results 2.9, 2.10 and the results of Chapter 3 seem to point in the direction of a positive answer to the following question. Question 11.1. Is a periodic group G necessarily soluble-by-finite if it contains an involution i such that C , i is jinite? In particular, must the group G be locally finite? Notice that 2.lb implies that such a group G contains infinite abelian 2'-subgroups, so something at least can be said. Obviously one can generalize the question by introducing an arbitrary regular cardinal. Question 11.2. If N is a regular cardinal and G is aperiodic group containing an involution i such that IC,i( < K , does G contain a locally soluble normal subgroup whose index in G is less than N ? Trivially G need not be locally finite if K > xo (as is also the case in the more special situation of the exercise below). Exercise. If N is a regular cardinal and G is a periodic group containing an involution i such that IC,zl < N for every involution z of G centralizing i, prove that all the involutions of G are conjugate.
In 2.8 we have seen that it is possible that no maximal abelian subgroup of the infinite, locally finite group G has the same cardinal as G . Hence the condition that in the infinite, locally finite group there is a large abelian subgroup is a restriction on the structure of G. Unfortunately, however, this restriction does not seem to lend itself to further investigation. Let us tighten up this situation. Question 11.3. What can be said about an infinite locally finite group G such that for every infinite factor S of G and for every (regular) cardinal N5I S1 every maximal abelian subgroup M of S satisfies K S IMI? In particular, can such a group be simple? In the context of 2.10 and 3.33 we see that the existence of an involution i in the infinite, locally finite group G such that the centralizer C,i is finite, forces the existence of a proper normal subgroup of finite index in G. Thus if the infinite, locally finite group G is, in particular simple, then every involution has infinite centralizer. Question 11.4. Is it true that in every infinite, locally jinite simple group G one has lC,gl = [GI for every element g in G?
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Throughout this chapter our methods, that is the pigeon hole principle, have forced us to restrict our attention to regular cardinals. Does this restriction depend on the method or does it reflect some of the inherent difficulties of the situations considered? Question 11.5. Are any of the results of this chapter involving a regular cardinal still true for arbitrary injinite cardinals?
Note. The existence of infinite abelian subgroups in infinite, locally finite groups was shown (independently) in Kargapolov [6] and P. Hall and Kulatilaka [l]. Further proofs were given in Sunkov [2] and [6], see also Strunkov [3]. All these proofs make use of the Feit-Thompson Theorem stating that a finite group of odd order is soluble. However, special cases of this result have been known for some time, for example the locally soluble case is dealt with in Cernikov [17]. The 2-group case of 2.5 appears in Held [3 1, Sunkov [6] and Strunkov [ l ] and [3]; see also Kegel 151. We have taken 2.1 from Sunkov [8] (for the special case N = N ~ ) a, preliminary version appearing in Sunkov [6]. Also 2.9 and 2.10 can be extracted from Sunkov [8]. The idea of generalizing these various results by considering regular cardinals instead of N~ first appears in print, it seems, in Strunkov [2] and [3], although these papers are more concerned with properties of the class of 2-finite groups. The connection between Smidt’s Problem and the other topics of this chapter had long been realized, of course, and thus many of the papers referred to above derive the corresponding case of 2.6 from their respective results. Specifically this connection is mentioned in Kargapolov [6], Strunkov [l], [2], and [3], and in Sunkov [2], [5], and [6]. The locally finite case of 2.7 appears in Sunkov [5].
CHAPTER 3
Locally Finite Groups with min-p
A group satisfies min-p for the prime p if each of its p-subgroups satisfies min. This is the case in a number of important situations. Thus certain properties of locally finite groups with min-p play an important role during reduction arguments leading to the characterizations of the groups PSL(2, F ) over certain locally finite fields F (see Chapter 5). These results in turn provide the key to the solution of the structure problem for locally finite groups satisfying min (see Question 1.5 and comments). The object of the present chapter is to study the relatively general notion of min-p is some depth. Although Sylow’s Theorem does not extend to locally finite groups satisfying min-p, such groups do contain p-subgroups whose behaviour reflects various important properties of the Sylow p-subgroups of a finite group. In general the maximal p-subgroups of a locally finite group G with min-p will not be conjugate, nor even isomorphic. But among the maximalp-subgroups of G there are some which contain an isonlorphic copy of all the others, and these p-subgroups of G are isomorphic. We call these subgroups the Sylow p-subgroups of G. They are well behaved under joins and intersections with normal subgroups of G, but they need not be conjugate. Thus it would seem that the Frattini argument applied to the Sylow p-subgroups of a normal subgroup is not available. However, something can be salvaged since every finite p-subgroup of G is conjugate to some subgroup of any Sylow p-subgroup of G, and is contained in some finite p-subgroup of G, whose conjugates are permuted transitively by the automorphisms of G. This substitute for a Sylow theory is expounded in Section A . We put it to use in Section B. In that section we obtain the structure of a locally finite group G satisfying min-p if G has many normal subgroups; in this case Opt,G has finite index in G. Of course some restriction of the normal structure of G is needed, since PSL(n, F ) satisfies min-p for every prime p distinct from the characteristic of F. A suitable such condition can be phrased in terms of Sunkov’s local notion of the range of an element (see 3.29). The section concludes with some applications of these results (promised at the end of Chapter 2), which show 80
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81
that the centralizer of an involution in an infinite locally finite group is very often infinite. For example a special case of 3.33 is that in an infinite, simple, locally finite group the centralizer of any involution is infinite. Section A. A Sylow theory for locally finite groups with min-p As shown in the Introduction maximal p-subgroups of a locally finite group need not all be conjugate (or isomorphic), even if they are Cernikov groups. For this situation we discuss an example in some detail. If all the psubgroups of the locally finite group G are Cernikov groups then their sizes are bounded, and those of largest size are isomorphic. Since these maximal p-subgroups enjoy many of the properties of the Sylow p-subgroups of a finite group, we shall call them Sylowp-subgroups of G. In this section we shall study some of the properties of these Sylow p-subgroups. The main result will be that these subgroups, or rather their size, lend themselves to treatment by induction, and so in the applications in Section B induction on the size of the Sylow p-subgroups will appear, generally in the guise of “least criminal” arguments.
3.1 Lemma. The locally finite group G satisJes min-p if and only if some maximal elementary abelian p-subgroup E of G is finite. Proof. If the group G satisfies min-p, then every elementary abelian p subgroup of G is finite. Let E be a finite maximal elementary abelianp-subgroup of G and suppose that Q is a p-subgroup of G. By 1.E.3 the group Q satisfies rnin if and only if all its countable subgroups satisfy min. Let R be an arbitrary countable subgroup of Q, and consider the subgroup H = ( E , R ) of G. By 1.A.9 the countable, locally finite group H has a local system Z consisting of finite subgroups that is linearly ordered by inclusion, and we may assume that the finite subgroup E of H is contained in the smallest member of Z.By 1.D.3 there exists a p-subgroup P E Max, H which reduces into Z and contains E. Now 1.6.6 applied to P yields that P is a Cernikov group, and so there is an abelian normal subgroup A of finite rank r and finite index IP : A [ in P. Clearly, every finite subgroup of P has an abelian normal subgroup of rank at most r and of index at most IP : A [ . Since every finite p-subgroup of H is conjugate to a subgroup of P,this last statement also holds for all finite subgroups of R. But then 1.K.2. yields that R, too, has an abelian normal subgroup of rank at most r and of index at most IP :A ] . Thus R is a Cernikov group, and Q satisfies rnin. I This result may be slightly rephrased.
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[CH.
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A
3.2 Corollary. If the locally finite group G does contain elements of order p, then G satisfies min-p if and only if there is an element of order p in G such that the centralizer CGgsatisJies min-p. Proof. The necessity of this condition is obvious. Its sufficiency follows from 3.1, since every maximal elementary abelian p-subgroup of G containing the element g is contained in the centralizer CGg and so is finite. I Exercise. Let p be a prime, H a locally finite p'-group, and K a locally finite group of automorphisms of H such that the set { C H X ;X E K}, partially ordered by inclusion, satisfies the descending chain condition. Prove that Ksatisfies min-p. (Effectively a special case of this result is that a linear group over a field of positive characteristic other than p satisfies min-p; see 1.L.3.) For a proof, consult the proof of 4.6 in B. Hartley [4].
Even if the locally finite group G does satisfy min-p the elements of Max,G need not be isomorphic, let alone conjugate in G. This can be seen by choosing the factors suitably in the universal counter example given in the Introduction. We shall now look in detail at a very similar example.
3.3 Example. For any pair (p, q) of distinct primes there exists a countable metabelian {p, 4)-group G satidying min-p such that G contains non-isomorphic maximal p-subgroups. Proof. Let A = (ai; a; = a i - l , a,, = 1, i E N) be a Pruferp-group, and B = (b) a cyclic group of order p. For the prime q # p let X = (x) be a cyclic group of order q. Denote by M the set of all mappings of the direct product A x B into X which take a value different from 1 on finitely many elements of A x B only, and consider M a s a right A x B-module in the usual way. Put G = X ? ( A x B), the split extension of M by A x B. Clearly M is an elementary abelian q-group, and G is a countable metabelian {p, q)-group satisfying min-p. Put H = MA. For each natural number n define the mapping p, E M by
For every eIement g E A x B one thus has g>Pn = (g>P?-'* Hence as elements of G the elements p, and a,- commute. Put c, = a : ' ' * . P n and C = (c,; n E N). The order of the element c, is p* and (g)Pu, =
cl =
(an-1
(az)B1P2...Bn
=
,
(a"Pn
)PI 1
...Pn-1 -
CH.
3,
5 A]
SYLOW THEORY
83
Thus C is a p-subgroup of G. Clearly, one has H = MC, and C is isomorphic to A. We prove that C is a maximal p-subgroup of G. Since A x B is trivially a maximal p-subgroup of G, and since A x B and C clearly are not isomorphic, this will complete the proof. Cis a maximalp-subgroup of H , the index IG : HI = p , and thep-subgroups of G are abelian. Thus if C is not a maximal p-subgroup of G , then G = M C , C . In particular, there exists a mapping p E M such that the element pb-' centralizes C. Thus, for each natural number n one has a;~...~n~b-'
= a;'"'"".
Since the element b centralizes A, the mapping?, = p1 . . . pnp((pl . . .p n ) - l ) b , qua element of G, centralizes the element a, and thus also the element an-l = a,". But ( l ) Y , = (1)p * x-,, and (1)yF-l = ( a , - , ) p x-'. Hence one has (a,,-l)p . X"-' = (I)p for all n E N. Consequently for infinitely many a E A the value (a)p # 1,which contradicts the definition of M. Therefore no such mapping p can exist, and hence C is its own centralizer in G. That is, C is a maximal p-subgroup of G. 1 Using the above method it is possible to construct for each cardinal ct = 1, 2, . . ., No a countable metabelian {p, 9)-group satisfying min-p whose maximal p-subgroups fall into exactly ct isomorphism classes. Further, if p is any infinite cardinal or a finite cardinal prime top, then there exists a metabelian {p, q}-group G satisfying min-p such that the set of maximal p-subgroups of G consists of one isomorphism class but precisely p conjugacy classes; see Wehrfritz [7], Theorems 3, 4, and 5. Under relatively mild additional assumptions on the locally finite group G satisfying min-p, the maximal p-subgroups of G are all conjugate. This general and vague statement is illustrated by the following result, taken from Zalesskii [I] and Wehrfritz [I] and [7]. 3.4 Theorem. If in the Iocally Jinite group G satisfying min-p for euery
properly ascending chain {A,}6N of abelian Jinite p-subgroups the descending chain of centralizers {C, An}neNbecomes stationary after finitely many steps, then the maximal p-subgroups of G are conjugate. Notice that if a group satisfies the hypothesis of the theorem, then so does each of its subgroups.
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mfn-p
[CH.
3,
A
Proof. Suppose the theorem is false and let H be a counter example. If for every subgroup X of H that is also a counter example to 3.4 there is a finite non-central abelian p-subgroup A , such that the centralizer C,A, is also a counter example to 3.4, then we shall derive a contradiction. Put X , = H , and choose inductively I,,,, = CxnAxn.If we put A, = (Ax,; j = 1, 2,.
. ., n>,
,
then A,, is a finite abelian p-subgroup of H and X,,, = C , A,,. The sequence {A,,}noNof finite, abelian p-subgroups is properly ascending, and - by construction - the sequence {X,,},,eN of centralizers is properly descending, which contradicts our general assumption. Hence the group H contains a subgroup G that is a counter example to 3.4 and such that for every finite, non-central abelian p-subgroup A of G the maximal p-subgroups of C,A are conjugate in CGA. Let P be any maximal p-subgroup of G and A its minimal subgroup of finite index; let B be any maximal radicable, abelian p-subgroup of G and Q any maximalp-subgroup of G containing B. By 1.E.6 the group A is radicable and abelian, and B is the minimal subgroup of finite index in Q. To obtain a contradiction - and hence the non-existence of H - it suffices to prove that the subgroups P and Q are conjugate. Put A,, = { a E A ; uPn= l}. By hypothesis one has CGA = C , A,, = C , A j for some natural number j . Hence A contains a finite subgroup A* (= A j + , ) such that C,A = CG(A;). In the same way B contains a finite subgroup B* such that C , B = CG(B,P).By Sylow’s Theorem there exists an element g in the finite sub-group (A,&) of G such that the subgroup S = ( A * , BZ) is a finite p-group. Suppose that S contains an element z such that z E Z, SIZ, S and z pE Z, S. The map x w [x, z ] is a homomorphism of S into Z S with kernel C,z. Moreover, since z p E Z, S, the group [S, z ] has exponent p and S p c C,Z. Consequently
nnsN
,
Z
E C,(A$) n C,((BS,)’) =
CGA n C,(B9).
Hence A u B9 c C,z. But z is not central in S, hence not in G, and so the maximal p-subgroups of C G z are all conjugate. The maximality of B and min-p ensure that some conjugate of B contains A . If S is abelian but not central in G, then S
G
CG(A$) n C,((BB,)P) = CGA n cG(Bg).
Hence A u Bg c C , S. Since the maximal p-subgroups of C , S are conjugate,
CH.
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A]
SYLOW THEORY
85
we again get that some conjugate of B contains A . Finally if S is central in G , then C G A = G = CG(Bg),and thus A 5 3. We have now shown that there always exists an element h in G such that A c Bh. The factor group P / A is a finite maximal p-subgroup of N,A/A, and every maximal p-subgroup of N , A contains A . Thus by 1.D.12 the maximal p-subgroups of N , A are all conjugate. Since Bh c NG A , some conjugate of Bh lies in P . Hence A and B have the same rank, and A = Bh. But then Qhc NG A , and so P and Qhare conjugate, which is the desired contradiction. I As an immediate consequence of 3.4 we obtain a result of Baer [ 5 ] . It should be observed, however, that this result is obvious, once one knows that the answer to Question 1.5 is positive, at least for locally finite groups, see 5.8.
3.5 Corollary. If the locally$nite group G satisfies min, then for each prime p the maximal p-subgroups of G are all conjugate. Many theorems on the conjugacy of the maximal p-subgroups of a locally finite groupare known; see Chapter 1, Section D (in particular the Appendix) for comments on this kind of result. In spite of Example 3.3 and the counterexamples of Wehrfritz [7] mentioned above, there does exist a “Sylow theory” which is just satisfactory for the class of locally finite groups satisfying min-p. Definition. Let p be a prime and G any group. Then the subgroup P is a Sylowp-subgroup of G if P is a maximal p-subgroup of G and contains an isomorphic copy of every p-subgrofip of G .
Clearly, the set Syl, G of all Sylow p-subgroups of the group G is a subset of Max, G which remains invariant under the action of every automorphism of G. If the maximal p-subgroups of G are all isomorphic or even conjugate, then trivially they are all Sylow p-subgroups of G. Thus the Sylow p-subgroups of a finite group are just its maximal p-subgroups, and our terminology is consistent with the ordinary terminology for finite groups. Unfortunately, however, a locally finite group need not contain Sylow p-subgroups for a given prime p . The reader should convince himself of this fact in the following exercise. Exercise. Let A be an uncountable abelian p-subgroup and B an uncountable locally finitep-subgroup such that every abelian subgroup of B is countable (that such a group B exists was seen in 2.8). Then the locally finite group G = ( A * B ) / K K 4 does not contain any Sylow p-subgroups. Here K denotes the kernel of the projection of the free product A*B onto the direct
86
LOCALLY FINITE GROUPS WITH
n7in-p
[CH.
3, 0 A
product A x B, and q is an arbitrary prime # p . (The same construction, repeatedly applied for different primes, allows one to produce a locally finite group not containing a Sylow p-subgroup for any p . ) In view of this fact, it seems natural to ask: What conditions on the locally finite group G assure the existence of Sylow p-subgroups? We shall see that min-p is such a condition. The main tool for this result is the following lemma. 3.6 Lemma. If P and Q are Cernikov groups such that Q contains an isomorphic copy of every finite subgroup of P, then Q contains a subgroup isomorphic to P. Proof. Let A (respectively B ) be the minimal subgroup of finite index in
P (respectively Q ) . For every natural number i put A,
=
=
I for some prime p > .
Choose a fixed subgroup H of P such that P = HA, and put m = J Q : BI. Denote by Si the set of all pairs (a,z) where a is a monomorphism of A i into B and z is a homomorphism of'H into QfB and both are induced by the same monomorphism p of H A , into Q. Since H A , and QlB are finite groups and since B has finite rank, the set Siis finite. If p is a monomorphism of H A i + , into Q, which exists by assumption, then one has A , E B, and so the restriction of p to H A , defines an element of S , . Hence the set Siis not empty. If i, j are natural numbers with i j , then there is a mapping ( j + i ) : S j + Siwhich is defined by ( j -+ i ) : (a,z)
t-, (oIA,,
z)
for (a, z)
E Sj.
One immediately checks that the collection (Si,( j + i ) ; i 5 j , i, j E N} is an inverse system of finite non-empty sets. Consequently (see l . K . l ) the inverse limit !ir~ S , of this system is non-empty. Denote by x i the projection of Siinto the finite set Si.Then for 1 E lim Siand j 2 i, one has ( j + i) : /iff'I+
P.
If P i = (ai, z,), and if this pair is induced by the monomorphism p i : H A i + Q, put C = A:'. Then the map cp: A + C given by qIA,= a, is a well-defined isomorphism. If /HI = n, then H n A E A,. Put R = ( H P " ,C ) . Let x be any element of P and suppose x = ha = k b for h, k E H , and a, b E A , then k - l h = ba-' E H n A E A , . Consequently ( k - ' / z P = (ba- ')", that is, (hP'*)(aV)= (kP")(b").
CH.
3,
5 A]
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SYLOW THEORY
Now define a map p :P R by 9 = (hP")(a") for x ceding lines, the map is well defined. Also -+
(ah)P = (hah)P = (hp")(ah)' = (hpn)(ah)pi =
(hpn)(aPi)hPi
=
ha E P. By the pre-
ifaEAi
since zi = z,, = (hPn)(a")hpn, = (a")(hPn)= aPhP. It follows that the map p is a homomorphism. If (hPn)(a")= 1, then the order of the element a" divides n. Hence a E A , , and so a' = a',. Since p , is a monomorphism, one has ha = 1. Thus, the mapping p :P R E Q is in fact a monomorphism. (The image of P under p is in fact R,since -+
(W u A") E I m p
c_
R = < H P ~ , A"}.)
I
3.7 Theorem. If G is a locallyfinite group satisfying min-p, then G contains Sylow p-subgroups. Further, every finite p-subgroup of G is contained in a Sylow p-subgroup of G. Proof. There exist only countable many isomorphism types of finite groups. Let be a countable collection of finite p-subgroups of G such that every finite p-subgroup of G is isomorphic to one of the Pi.Put H, = ( P i ; 1 5 i 5 n). Set Q , = P1,and if Q1 c Q , c . . E Q,, are p-subgroups of G such that Qi is a Sylow p-subgroup of Hifor 1 5 i 6 n, then let Qnfl be any Sylow p-subgroup of H,,, containing Q,. Denote by Q any maximal p-subgroup of G containing Q,,. Since by Sylow's Theorem some conjugate of Pi lies in Q iand hence in Q, the maximal p-subgroup Q of G contains by 3.6 an isomorphic copy of every p-subgroup of G. Thus Q is a Sylow p-subgroup of G containing PI.I Now we need some information on the relationship between two Sylow p-subgroups of a locally finite group with min-p.
.
uzl
3.8 Lemma. Let G be a locally jinite group satisfying min-p. r f P is a psubgroup of G and if Q is a Sylow p-subgroup of G then thefollowingproperties of P are equivalent: a P is a Sylow p-subgroup of G; b P contains an isomorphic copy of every finite p-subgroup of G; c P and Q are isomorphic p-subgroups of G. Proof. That a implies b is clear from the definition of Sylow p-subgroup, and that c implies b is also trivial. If the p-subgroup P of G satisfies b, and
88
LOCALLY FINITE GROUPS WITH min-p
[CH.
3,
5
A
if R is any maximal p-subgroup of G containing P,then P contains by 3.6 a subgroup isomorphic to Q and Q contains a subgroup isomorphic to R. Thus we have Q 4 P G R G Q, and yet Q does not contain any proper subgroup isomorphic to itself by 1.E.8. Hence P = R N Q and the lemma is proved. I In particular we see that if P and Q both are Sylow p-subgroups of a locally group G satisfying min-p, then P and Q are isomorphic. However it may not be possible to induce such an isomorphism by an automorphism of the whole group G. In Wehrfritz [8],2.4 gives an example of a countable periodic metabelian group G satisfying min-p that contains Sylow p-subgroups P and Q such that for every automorphism a of G one has P" # Q. Sylow p-subgroups of a locally finite group satisfying min-p are more than just isomorphic, they have a very useful local-conjugacy property. This I S our next objective; but first a preliminary lemma. If G is any group and n any natural number, put Gn = (9E G; g n = 1). 3.9 Lemma. I f A is a radicable abelian subgroup of jinite index n in the group P , then P contains a conjugacy cZassX(P) of subgroups of P that is left invariant under all automorphisms of P and such that f o r every K E S ( P )one has A n K = A,, and A K = P. Proof. The subgroup A is characteristic in P. Let X = { x l , . . ., x,} be any transversal of P over A . If g E P , then x i g = x i O a ifor some a, E A and r~ an element of the symmetric group on {I, . . ., n}. Define the mapping 6, : P + A by g6x = a , . This map is a derivation, for if h E P and x i h = x i z b iwith bi E A , then x i g h = xib.a:bib, and
fli=
n
n
n
P u t K X = ( g E P ; g a X = 1 ) .I f a f A , t h e n a a X = a " ; h e n c e A n K x = A , . Since A is radicable, there exists an element b E A such that gax = b-", for a given g E P . Then (gb)ax = (gax)b(bax) = 1, and hence g E AK,. Therefore P = AK,. Let Y be any other transversal of P over A , and order Y = {yl, . . ., y,} in such a way that y i = x i c i for some c , E A .The radicable c i . Now group A contains an element c such that c" =
nf=
Consequently,
y.g
=
x I. c1. g
=
x. a.c? ' , = yi.ci:'aic;.
CH.
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SYLOW THEORY
89
Therefore one has K i = K y , and the kernels of the derivations corresponding to any two transversals of P over A are conjugate. Since A is a characteristic subgroup of P, an automorphism of P, leaves A invariant and permutes the transversals of P over A ; that is to say it leaves the conjugacy class of the subgroups Kx of P invariant. Thus p u t X ( P ) = {Kx; X transversal of P over A } . I This class of subgroups . X ( P ) that we have just constructed is a most convenient tool and whenever we need it, we shall refer to it as the class X(P). 3.10 Theorem. Let G be a locally jinite group satisfying min-p and P and Q Sylow p-subgroups of G. Let A and B be the minimal subgroups of jinite index in P and Q, respectively. Then for each natural number n the subgroups A, and B, are conjugate in G. I ~ H X(P) E and K EX ( Q )then , the subgrog s HA, and KB, are conjugate in G. In particular, H and K are conjugate in G. Proof. Since P and Q are isomorphic, one has JP: A J = lQ : BI = s, say. There exist transversals Xof P over A determining H a n d Y of Q over B determining K, and there is an integer k divisible by ns2 such that X c HA, and Y c KB, . Since G is locally finite, Sylow's Theorem implies that G contains an element g such that the subgroup S = ( H A k s , (KB,,)') is a p-group. By 3.7 there exists a Sylow p-subgroup R of G containing S. If C is the minimal subgroup of finite index in R, then lR : C /= s. Consequently A, = C, = (Bk)B.Since n divides k,one has A , = (Bn)9. Notice that H has exponent dividing s2, so C n H c C, E C, = A k . But A n H = A, = C, thus C n H = C, and, in the same way, C, = C n K g . Now s = (R:CI = ( H : A n H I = ( K : B n K ( ,
and consequently R = CH = CKg. Thus X and Y g are both transversals of R over C and the subgroups H and Kq both belong to X ( R ) . By 3.9 there is an element c E C such that H = KgC,and hence (KB,)SC = HC,' = HA,. I Remark. The first paragraph of the above proof does not use the existence of the subgroups H and K. Thus if D is any radicable abelian p-subgroup of maximum rank, that is, which is isomorphic to A(or to B), then A, and D,are conjugate. This does not follow directly from the statement of 3.10 since
90
LOCALLY FINITE GROUPS WITH
mh-p
[CH.
3, 8
A
there need not exist any Sylowp-subgroup of G containing D . This is the case, for example, in the group constructed in 3.3. 3.11 Corollary. If G is a locallyfinite group satisfying min-p, then thep-subgroup P of G is a Sylow p-subgroup of G if and only if P contains a conjugate of every Jinite p-subgroup of G. Proof. Suppose P is a Sylow p-subgroup of G, and let R be any finite p-subgroup of G. By 3.7 there exists a Sylow p-subgroup Q of G containing R. Choose K EX(Q),and denote by B the minimal subgroup of finite index in Q. Then there exists an integer n such that R c KB, . By 3.10 some conjugate of KB,, and hence of R, lies in P. On the other hand it follows immediately from 3.8 that the p-subgroup P is a Sylow p-subgroup of G if it contains a conjugate of every finite p-subgroup of G. I
3.12 Corollary. If G is a locally finite group satisfying min-p, then for every natural number n there are only finitely many conjugacy classes of elements of order p" in G. Proof. If x is an element of G of order p" there exists a Sylowp-subgroup P of G containing x by 3.7. Denote by A the minimal subgroup of P of finite index and let m be the least non-negative integer such that xPmE A . There exists a transversal X of P over A of the form
x
=
: i = 1 , 2 , . . ., r; j = 0,I , . . .,pm-1}.
In the notation of 3.9 it is easy to check that xsx = xrP". The group A contains an element a such that arp" = xrpm.Clearly a has order p" and (xa-l)dx = 1. Thus xE
K x A,,
=
F , say.
By 3.10 some conjugate of every element of G of order p" lies in the finite group F and therefore G contains less than IF1 conjugacy classes of elements of orderp". I The Sylow p-subgroups of a locally finite group G satisfying min-p behave quite reasonably with respect to normal subgroups. For any group G denote by Sy1,G the set of Sylowp-subgroups of G (this set may be empty!). Obviously, one has Sy1,G E MaxpG, the set of maximal p-subgroups of G. If N is a normal subgroup of G and S a set of subgroups of G , for example Syl, G or Max,G, we shall write S
A
N = { S n N ; S E S} and SIN = { S N / N S E S}.
CH.
3, 5
91
SYLOW THEORY
A]
With this notation we have the following result. 3.13 Theorem. Let G be a locally Jinite group satisfying min-p and N a normal subgroup of G. Then the groups N and GIN also satisfy min-p and
Sy1,G (Sy1,G)IN
A
N E Sy1,N c Max,N E Max,G A N , Syl,(G/N) E Max, (GIN) -c (Max,G)/N.
=
Each of theseJive inclusions may be strict. Proof. Clearly, the groups N and GIN are locally finite, and N satisfies min-p; further
Sy1,N
c Max,N
C Max,G
A
N, and Syl,(G/N) c Max,(G/N).
Let H / N be any countablep-subgroup of GIN. Then there exists an ascending series of subgroups Hiof G such that m
N=H,GH, L...EH~G...EH, UH,=H, i= 1
and each index [ H i : N ( is finite. Suppose we have found a finitep-subgroup P i - of G such that P i - N = Hi-1. There exists a finite subgroup Fi of G such that P i - , E Fi and FiN = H i . Let P i be any Sylow p-subgroup of Fi containing P i - Clearly one has P i N = H i , In this way one chooses inductively an ascending sequence (Pi)ieNof finite p-subgroups of G. Put P = P i . Then P is ap-subgroup of G and satisfies PN = H. The group Pis a Cernikov group, and so is the factor group H / N N P / ( P n N ) . Thus every countable p-subgroup of GIN satisfies min. By 1.E.3 the factor group GIN satisfies min-p. In particular the p-subgroups of GIN are countable, so if H / N is a maximal p-subgroup of GIN there exists by the preceding paragraph a p-subgroup P of G such that PN = H . Since H / N is a maximal p-subgroup of GIN, for every maximalp-subgroupP , of G containingP one has P , N = H . Therefore Max,(G/N) c (Max,)G/N. If Q is a Sylow p-subgroup of G and if R is any finite p-subgroup of N, then there exists an element g E G such that Rg s Q. But then Rg E Q n N , and Q n N contains an isomorphic copy of every finite p-subgroup of N . By 3.8 the intersection Q n N is a Sylowp-subgroup of N . Hence Syl,G N c Sy1,N. If SIN is any finitep-subgroup of GIN, then there exists a finitep-subgroup T of G such that TN = S. For some element g E G one has T8 Q. But then (S/N)SNE QNIN, and thus by 3.11 the factor group QN/N is a Sylow
,
0,:,
,
92
LOCALLY FINITE GROUPS WITH
mh-p
[CH.
3, 5 A
p-subgroup of GIN. Conversely, let U/N be any Sylow p-subgroup of GIN and let V be any Sylow p-subgroup of U . Then V N / N is a Sylow p-subgroup of UjN, and thus VN = U. If W is any finite p-subgroup of G, then there exists an element x E G such that W" c U. Hence some conjugate of W lies in V, and V is a Sylow p-subgroup of G by 3.11. Since VN = U,we have shown that (Syl,G)/N = Syl,(G/N). It remains to show that the five inequalities can be strict. Let G be the group constructed in 3.3. We shall assume the notation of that construction and make various choices for N. i. If N = H, then C E Sy1,N n Max,G. Thus the subgroup C cannot lie in a Sylow p-subgroup of G and therefore Syl, G A N $ Syl, N. ii. If N = G, then C E Max,N\Syl,N, so Sy1,N $ Max,N. Similarly, iii. If N = (l), then SyI,(G/N) g Max,(G/N). iv. If N = M B then C E Max,G, while C n N = (l), which is not a maximal p-subgroup of N. Thus Max,N
Max,G
A
N,
v. If N = H, then GIN is cyclic of order p, and yet C E N is a maximal p-subgroup of G. Thus Max, (GIN) = {G/N}
g (Max,G)/N. 1
3.14 Corollary. Let G be a 1ocallyJinite group satisfying min-p and let N be a normal subgroup of G. If all the maximal p-subgroups of G are isomorphic, then all the maximal p-subgroups of N (respectively of GIN) are isomorphic. Proof. The data imply that Sy1,G = Max,G. Now 3.13 yields Sy1,N = Max,N and SyI,(G/N) = Max,(G/N). I If P is a Cernikov p-group and if A is the minimal subgroup of finite index in P,then we call the pair (rank A , IP : A ! )the size of P. We well-order the sizes of Cernikov p-groups lexicographically. Thus if Q is another Cernikov p-group and if B is its minimal subgroup of finite index, we say that the size of P is smaller than the size of Q (or, more briefly that P i s smaller than Q ) if either rank A < rank B, or rank A = rank B and IP : A J< J Q :BI. If G is a locally finite group satisfying min-p, then we call the size of the Sylow p-subgroups of G the p-size of G and denote it by JGJ,. Let G be a locally finite group satisfying min-p and N a normal subgroup of G. By 3.13 the Sylow p-subgroups of N are smaller than those of G if the
CH.
3, 5
B]
APPLICATIONS OF SYLOW THEORY
93
factor group GIN is not a p'-group. If the normal subgroup N contains an infinite p-subgroup, then the Sylow p-subgroups of the factor group GIN are smaller then those of G. These observations enable one to apply inductive methods.
Exercise. G is a locally finite group satisfying min-p such that 0 , G is finite. Prove that G contains two Sylow p-subgroups with finite intersection.
3.15 Lemma. If the locallyfinite group G satisfying min-p hasp-size (r, n), then thep-component of G' n ZG has order at most n'". Proof. Suppose that P is finite p-subgroup of G' n ZG. The group G contains a finite subgroup H such that P c H'. Let S be any Sylow p-subgroup of H containing P and put m = IH : SI. Denote by (r the transfer of H into S/S'. Since P is central in S, one has for every element x E P S'
= xu = X m S ' .
Since rn is prime to p, this means P E S'. By 3.7 there exists a Sylow p-subgroup T of G containing S. The minimal subgroup A of finite index in T is abelian of rank r and IT : A1 = n, by assumption. Denote by z the transfer of T into A . Now P c S' c T' c ker z,and P is central in T. Therefore (Pn A)' = (Pn A)" = (1). Thus P has order at most n'+'. I (For the properties of the transfer see Huppert [l], Ch. IV, $0 1 and 2.)
Section B. Applications of Sylow theory to the structure of groups with min-p The main objective of this section is to use the Sylow theory just established to determine the structure of a locally finite group G satisfying min-p, a t least in the case where no infinite simple group is involved in G. This information, our 3.17, was obtained in Kargopolov [4]. In Sunkov [ 5 ] one finds the (at least formally) more general form 3.29 of this result. In fact Sunkov's technique, introducing the notion of the range of an element of a locally finite group, is most useful in showing that proper normal subgroups exist. It is from this point of view that some of the results of this section will serve as rather handy tools for the problems to be tackled in Chapter 5. The first lemma states that, to within a finite factor, one can slide finite or certain infinite p-factors up across p'-factors.
94
LOCALLY FINITE GROUPS WITH
min-p
[CH.
3,
8B
3.16 Lemma. Let G be a locally finite group and N a normal subgroup of G. i r f n is a set ofprimes, i f N isfinite, and if GIN is a n-group, then the index IG : 0, GI is finite. ii r f N is a Cernikov p-group, and i f GIN is a p'-group, then the index IG : ( N x O,,G)I is Jinite.
Proof. i Let C = C, N, Then the index IG : CI is finite. If P = O,,(C n N ) , then P is central in C, and C/P is a z-group. By the Schur-Zassenhaus Theorem every finite subgroup of C is the direct product of a z-group and a n'-group. It follows that C = P x 0, C. Since P is finite, the index JG: 0, C ]is finite. But 0, C E 0,G, and thus the index IG : 0,GI is finite. ii Let A be the minimal subgroup of N of finite index. Now A is normal in G, so it follows from i that H / A = O , ( G / A ) has finite index in G/A. By 1.F.3 the group C = C,A has finite index in G, and A is a central p-subgroup of C such that CIA is ap'-group. Again thdSchur-Zassenhaus Theorem implies that C = A x 0 , X and so ( N , 0, G ) = N x 0,.G has finite index in G. I 3.17 Theorem. In the locally finite group G satisfying min-p for the primep, the index IG : O,,GJ isfinite if and only i f G does not involve any infinite simple group containing elements of order p.
Proof. To see the necessity of the condition, assume that G has subgroups X and Y such that Y is normal in X and such that the factor group X / Y is an infinite simple group containing an element xY of orderp. For every normal subgroup N of G one clearly has X N / Y N N X / Y ( X n N ) . If N = O,, G, then X Y @ Y ( X n O,,,G), and thus the natural projection of the simple group X / Y onto X / Y ( X n 0,.G ) is an isomorphism. Consequently, X n OptG E Y. But then the infinite simple group XIYcontaining an element of order p is involved in the group G/O,. G, which therefore cannot have any p-subgroup of finite index. To see the sufficiency of the condition, suppose that G does not involve any infinite simple group containing elements of order p. Assume by way of contradiction that the theorem is false, and let G be a counter example of minimal p-size. Considering a factor group if necesary, we are free to assume that OptG = (1). By 3.16 we need only consider the case G = OP'G, when G is generated by its elements of p-power order. Let N be any proper normal subgroup of G. The factor group G / N is not a p'-group and thus the p-size of N is smaller than that of G . Consequently the minimality of the p-size of G implies that the index IN : 0," is finite. I f N is infinite, the
CH.
3, 8 B]
APPLICATIONS OF SYLOW THEORY
95
p-size of the factor group G I N is smaller than that of G , and thus the index (GIN : O,,,(G/N)I is also finite. But then 3.16 shows that IG : O,,GI is finite, which cannot be. Hence every proper normal subgroup N of G is finite. In particular, if the proper normal subgroup N of G is not contained in the centre of G, then the centralizer CGN is finite. But this is clearly absurd, since IG : CGNl is finite and G is infinite. Consequently the factor group G/ZG is an infinite simple group. But G is generated by its elements of p power order, so the infinite simple group G/ZG contains elements of order p ; this contradicts our initial assumption. I Remark. It should be observed that by 1.B.4 a simple locally soluble group has prime order, and thus in a periodic, locally soluble group one of the conditions of 3.17 is automatically satisfied. This remark also applies to the next two results.
3.18 Corollary. Let G be a locallyfinite group satisfying min-p f o r every prime p . If no infinite simple group is involved in G , then there is a radicable abelian normal subgroup A in G such that the factor group G / A is residually jinite and every Sylow subgroup of G / A is finite. Proof. For every prime p by 3.17 there exists a normal subgroup H p of finite index in G such that H , contains the subgroup 0,-G and H,IO,.G is a radicable p-group (simply take HJO, G to be the minimal subgroup of finite index in G / O , , G ) . Put A = n,H,. By construction the factor group GIA is residually finite. Also very radicable subgroup of G is contained in each of the subgroups H, and thus in A . By 3.13, the Sylow subgroups of GIA are all finite, Since npO,.G = (l), the subgroup A is residually abelian and hence abelian. Finally for every prime p the p-primary component of the abelian group A is isomorphic to H,/O,, G , and hence A is a radicable group. I
3.19 Corollary. Let G be a locallyfinite group such that no infinite simple group is involved in G. Then the group G is a cernikov group if and only i f all abelian subgroups of G satisfy min. Proof. If all abelian subgroups of G are Cernikov groups, then by 1.6.4 the group G satisfies min-p for every prime p . Hence by 3.18 one has that G has a radicable abelian normal subgroup A such that every Sylow subgroup of G/A is finite. If B / A is any abelian subgroup of GIA, then B is sohbIe and hence by 1.G.2 a Cernikov group. In particular, the group 3IA satisfies min; and since all its Sylow subgroups are finite the abelian subgroup B / A is finite. Now 2.5 implies that G / A is finite. Thus G is a Cernikov group. I
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[CH.
3, 5
B
The above result will be superseded by 5.8, where we only assume that the abelian subgroups of the locally finite group satisfy min. The proof of 5.8 uses the special case of 3.19 in which G is locally soluble. For any group G denote by Z c ( G ) the set of all centralizers of subsets of G. Clearly, the set 2ZC(G)is closed under taking intersections (of arbitrarily many centralizers) and is partially ordered by inclusion. Moreover the restriction to Z c ( G ) of the mapping X H C G X (of the power set of G onto .Yc(G)) is a bijection that inverts the partial order. This Galois correspondence shows that the lattice .Yc(G) satisfies the descending chain condition if and only if it satisfies the ascending chain condition. Denote by %Rc the class of all groups G for which ZJG) satisfies the descending chain condition. One of the principal results of Chapter 5 is that a locally finite %Rcgroup satisfying min-p for every prime p , is abelian-by-finite (see 5.10). The locally solube case of this result, or rather the case of those groups not involving infinite simple groups, is proved separately, and since its proof makes use of 3.17 it is convenient to insert it here; but first we prove two auxiliary results.
3.20 Lemma. I f the group G is abelian-by-finite, then G E nc. Proof. Let A be an abelian normal subgroup of finite index in G, and let { D i }be a properly descending chain of centralizers of subsets of G. Then the chain of centralizers {Ci= C , D , } is (properly) ascending. Since the factor group G / A is finite, the sequence { A C J A } becomes stationary. Hence one may assume that AC, = ACj for all the groups of the sequence {Ci}. Now the sequence {Ci} is ascending, so there is a finitely generated subgroup F common to all the groups C isuch that Ci= F(A n Ci)for all i. Since D i= C G C i , one has Di = CGF n C,(A A Ci).Each of the centralizers C,(A n Ci) contains A , and thus there are only finitely many such subgroups. Consequently, the sequence {Di} is finite. I 3.21 Lemma. I f the locallyfinite %Rc-groupG satisfies min-p for every prime p , thenfor every abelian subgroup A of G the index IN, A : CGA1 isfinite. Proof. The abelian group A is the direct product of its primary components A,. Since the subgroups A , are Cernikov groups, 1.F.3 yields that the index INGAP : CGApI is finite. Clearly we have
N G A E NGA,
and CGA
=
n CGA,. P
By the descending chain condition on the partially ordered set 6pc(G)there
CH.
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APPLICATIONS OF SYLOW THEORY
97
exists a finite set n of primes such that CGA = n p E l r C G AConsequently p. ( N G A CGAI
E
fl INCA,
CGApl,
pen
which is finite. I 3.22 Proposition. The locally f n i t e %RC-groupG satisfying min-p f o r every primep is abelian-byfnite, if it does not involve any infinite simple group. Proof. Assume, if possible, that 3.22 is false. Then among the counter examples G to this statement there must exist one such that every proper centralizer in G is abelian-by-finite. By 3.18 there exists a radicable abelian normal subgroup A in G such that the factor group GIA is residually finite, and every Sylow subgroup of GIA is finite. If Xis any subgroup of G, put X o = C, Y, where the intersection is taken over all subsets Y of G for which the index ( X : C, YI is finite. Clearly for every subgroup X of G the index (X: XoI is finite since G is an %Rc-group. Call the subgroup X connected (in G ) if X = X o . Obviously A , ZG and G are all connected. By 3.21 the index IG : CGAI is finite. Since G is not abelian-by-finite, we have A c ZG. In the infinite, locally finite group G / Z G there exists by 2.5 an infinite abelian subgroup EIZG, its preimage E being nilpotent of class two at most. Let F be any maximal abelian subgroup of E. Then F is normal in E and F = C E F .But then 3.21 implies that the index IE : FI is finite. Consequently, F o is a connected abelian subgroup of G containing Z G such that the index IFo : ZGI is infinite. Choose B maximal among the connected abelian subgroups of G containing Po. Put N = NGB and C = CGB. By 3.21 the factor group NIC is finite, and C is abelian-by-finite, since B $ ZG. Thus B = N o = C o and NIB is finite. Denote by n the set of prime divisors of the order of NIB. By the chosen properties of A there exists a normal subgroup H of G containing A such that the factor group H / A is a d-group and the index IG : HI is finite. Let g be an element of H and suppose
n
a E ( ( H n B ) n ( H n B)g)\ZG.
Then C,a # G , thus B = (CGa)O= Bg and g E H n N . But A G H n B, and so ( H n N ) / ( H n B ) is a n'-group; yet NIB is a n-group. Therefore g E H n N = H n B. Trivially H n Z G 5 ( H n N)" for every element x of G . Also H n B # H and H n B # H n ZG, since the index IG : HI is finite and IB : ZGI is infinite. Therefore the factor group H / ( H n Z G ) is a
98
LOCALLY FINITE GROUPS WITH
mh-p
[CH.
3,
5B
Frobenius group with an abelian Frobenius complement ( H nB)/(H n ZG). By 1.5.2 the group H / ( H n ZG) is soluble. Thus G is soluble-by-finite. Since the group G satisfies min-p for every primep, the infinite, soluble-byfinite group G/ZG contains a non-trivial finite abelian normal subgroup L/ZG, say. Then the centralizer K = CG(L/ZG) has finite index in G. For an element a in L\ZG the map 6 = 6, of K into ZG given by 6 : x w [x,a ] for x E K, is a homomorphism, and its kernel C K ais abelian-by-finite. Since ar = 1 for some positive integer r, we have ( K a y = (1). But G satisfies min-p for every prime p , so the subgroup K' of ZG is finite. Therefore G is abelian-by-finite, contradicting the assumption that G is a counter example to 3.22. I 3.23 Lemma. I f g is a non-trivial element of the locallyjnite group G that does not lie in any proper normal subgroup of G, then there exists a local system Z ofjnite subgroups of G such that for every subgroup M E Z we have g E M and if N is any proper subgroup of M there exists an element x E N, M such that g # N" (thus, in particular, one of the maximal normal subgroups of M does not contain the element g).
Proof. If H is any finite subgroup of G containing g , denote by MH the unique minimal normal subgroup of Hcontaining g. Put Z, = { M H ;H a finite subgroup of G containing g } . If Kis also a finite subgroup of G containing g, then g E H n M ( H , K4 ) H, so MH C M < H , K )and , similarly M K 5 M ( H , K ) . Therefore Z, is a local system for M = MH. Clearly M is a normal subgroup of G containing g, and consequntly M = G. Let N be any proper subgroup of the subgroup MH E Z. By the choice of MHthe element g does not lie in N h . Hence there must exist an element h E H E NGMHsuch that g # N h . I
UH
nheH
3.24 Proposition. Let G be a locallyjinite group satisfying min-p for the prime p. I f for the element g E G there exists a local system Z of G of jinite subgroups H containing g such that there is a normal subgroup NH of H of order divisible by p which does not contain the element g, then there exists a proper normal subgroup N of G not containing the element g. (In particular, the group G is not simple.)
Proof. Suppose that for every finite p-subgroup R of G there exists a subgroup H EZ with R c H and R n NH = (1). There exists a Sylow p-subgroup S of NH that is normalized by R. By hypothesis S # (1). This selection of H and S may be repeated with R replaced by RS. Hence there exists an infinite sequence {Pn}of finite p-subgroups of G such that
CH.
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APPLICATIONS OF SYLOW THEORY
99
for each n E N we have P,P,
. . .P, n P,,,
=
(1)
and P, normalizes P, whenever n < r. Put Q, = (Pr; r > n). Since Q , is ap-subgroup of G , it is a Cernikov group. The sequence {Q,} is a strictly desI), cending sequence of subgroups of Q , ,for P, G Qn+ and if x E (P,nQ,)\( then x = xi,x j , . . . x j v for some xjkE Pjk\(l), where n < j , < . . . < j,. But then xj,, E ( P , P, . . . P j r - n Pjc = (l), a contradiction which proves that P, n Q, = (1). This is an absurd situation, and hence in G there must exist a finite p-group R such that R n NH # (I> for every H E Z containing R . Let D,,. . ., D,be the non-trivial subgroups of R, and put
u
Zi= ( H E C; R c H, R
n NH = D i } .
Since Zi is a local system of G, there exists an integer t such that Zt is a local systemof G . Put D = (D:). Since ( 1 ) Dic D, all we have to prove is that the element g is not contained in D. If g E D, then there exists a finite set X of elements of G such that g E ( O f > .There exists a subgroup H EZt containing the set X . From the definition of the local system I t , it follows that D,c NH 4 H. But then g E (0:)G N H , which cannot be. Hence g $ D.1 The element g of the locally finite group G is said to havejinite range in G , if there is a natural number k such that for every finite subgroup F of G which contains the element g and has a unique maximal normal subgroup M g $ M implies IF : MI
5 k.
The minimal such integer k is called the range of g in G . (If for the element g no such subgroup F with a unique maxima1 normal subgroup M exists with g E F \ M , then define the range of g in G to be 1; thus the unit element always has range 1.) If no bound k exists, the element g is said to have infinite range in G. This definition was proposed by Sunkov [ 5 ] (using the overworked term rank for range). The existence of elements of finite range in a locally finite group G can often be used to show the existence of non-trivial normal subgroups of G, and under suitable further hypotheses these normal subgroups may be located rather precisely. Before doing this for locally finite groups satisfying min-p, we consider some of the more formal properties of the notion of range. The following property is obvious from the definition.
3.25 Lemma.
If the subgrozrp K
of the locally finite group G contains an
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LOCALLY FINITE GROUPS WITH min-p
[CH.3,
5B
element g which is offinite range in G, then g is offinite range in K, and the range of g in K is bounded by the range of g in G.
3.26 Lemma. Let n be a set of primes. If G is a locally finite group, g a nelement of G and K a normal d-subgroup of G, then the range of g K in the factor group GIK is equal to the range of g in G. The restriction that K be a n’-group is necessary. Simply let K = ( a ) be a cyclic group of prime order p , F an infinite, locally finite field, and G = ( a ) x PSL (2, F). If b is any non-trivial element of PSL (2, F ) of order prime to p , then the element ab has range 1 in G, and yet the element bK = abK has infinite range in G/K.
Proof. Let S be a finite subgroup of G , with a unique maximal normal subIf g E NK, then g E S n N K = group N, and suppose that g E S\N. N ( S n K ) . Since N is a maximal normal subgroup of S, one has N ( S nK ) = S and SIN is a d-group. But since g is a n-eleme-ntof S\N, this is a contradiction. Thus g $ NK. Since SK/K N S/(S n K ) it follows that NKIK is the unique maximal normal subgroup of S K / K . Also ISK : N K [ = IS : NI, and thus the range of g in G does not exceed the range of g K in G/K. Now let S and N be subgroups of G containing K such that the factor group SIK is finite, N / K is the unique normal subgroup of SlK and g E S\N. Let T be a finite subgroup of minimal order of S such that g E T and K T = S. If Nl = N n T, then TIN, 1: SIN, and hence Nl is a maximal normal subgroup of T.Let M be another maximal normal subgroup of T. Then T = M N , and S = K M N , . Since N / K is the unique maximal normal subgroup of S / K , we have S = K M . But then TIM = M ( T n K ) / M is a n’-group, and so g E M . This contradicts the choice of T,and hence no maximal normal subgroup of T beside Nl can exist. Consequently the range of the element gK in GIK is bounded by the range of the element g in G. I Exercise. If n is a set of primes, G a locally finite group such that every n-element of G has finite range and K afinite normal subgroup of G , prove that every n-element of G / K has finite range (in G/K). There is an ample supply of locally finite groups in which every element has finite range: 3.27 below and 1.B.4 imply that every element of a periodic locally soluble group has finite range. 3.27 Proposition. r f the locally finite group has a series with jinite factors, then every element of G has finite range.
Proof. Let G be a series of G with finite factors. For every finite subgroup F of G and every subgroup S appearing in one of the pairs of 6the inter-
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APPLICATIONS OF SYLOW THEORY
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section F n S is a subnormal subgroup of F. If (V, A ) is the pair of 6determined by the element g , then the factor group AlVis finite (by assumption), and thus for every finite subgroup F of G which contains the element g one has I(A n F ) / ( V n F)I 5 IA/Vl. Thus the order IA/Vl is a bound to the range of the element g in G. I One wonders whether the opposite implication also holds. Question 111.1. Does every composition series of a locally finite group G haveJinite factors, if every element of G hasfinite range in G? If the answer to this question were affirmative, then the condition that every element of the locally finite group G is of finite range in G would be equivalent to the condition that no infinite, absolutely simple group is involved in the locally finite group G . Under the relatively stringent assumption that G satisfies min-2 we shall provide an affirmative answer in 3.30. A related question, to which 3.28 gives a partial answer, is the following. Question 111.2. Does the existence of an element g # 1 of finite range in G imply that the Iocallyfinite group G is not simple? 3.28 Theorem. Let A and B be normal subgroups of the locally finite group G such that A G By A is locally soluble, IB : A1 is finite, and G / B is infinite and satisjies min-p. If the group G contains a p-element of finite range in G which is not contained in B, then there is a proper normal subgroup of G not contained in B. Proof. Let g be an element of G \ B that has finite range in G and assume, if possible, that every proper normal subgroup of G lies in B. Let C denote a local system of G for the element g of the type guaranteed by 3.23. Put Zl = ( H EC;H contains a maximal normal subgroup QH excluding g and not containing B n H }
Suppose that C, is a local system for G. If H E Zl,then H = ( B n H)QH. Let S, be the set of pairs (U, V ) where U is a normal subgroup of H excluding g and V / U is a soluble normal subgroup of HIU of finite index at most J B: A ( . If H , K E Z, and H E K, then the mapping
A ; * : ( U , V ) H ( H U~ , H v), ~ (U, V)ESK maps the set SK into S,. The collection {SHYA,"; H, K E Cl, H C K) is an inverse system of non-empty finite sets over the directed set Cl, and thus by 1.K.1. its inverse limit is non-empty. Let ((UH, V,)) E ]im SHY where
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LOCALLY FINITE GROUPS WITH
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[CH.
3,
0B
UHEIr (UHEZI
( U H ,V H )E SH. It is easily seen that X = UHis a normal subgroup of G excluding g(so X G B), and that V H ) / Xis a locally soluble normal subgroup of G / X of finite index (see 1.K.2). Since the factor group GjB is infinite and simple, 1.B.4 yields a contradiction. Therefore, the set Zlis not a local system of G, and consequently Zz = Z\Z, is a local system of G, Put Z3= ( H E Zz; : H contains a maximal normal subgroup QHexcluding g such that lQH : B n HI is divisible byp}. By 3.24 applied to i3= {HB/B; H E Z3>,the factor group G/B cannot be simple if is a local system of G/B. Since G/B is simple the set Z3 is not a local system of G, and thus Z4 = Z2\Z3 is a a local system of G. Let H EZ4.If Q is any maximal normal subgroup of H that does not contain the element 9, then B n H E Q and Q / ( B n H ) is a p‘-group. If R is any other maximal normal subgroup of H, then by 3.23 there exists an element x E N,H such that g I$ R”. Hence B n H c R” and R“/(Bn H ) is d p’-group. Since x normalizes B and H, the intersection B n H lies inside R, and R / ( B n H ) is a p’group. Now R # Q , and thus H = RQ. Therefore H / ( B n H ) is a p’-group. But g E H B \ is ap-element. This contradiction shows that no such maximal normal subgroup R can exist in H, and thus Q is the unique maximal normal subgroup of H . By the definition of the range k of g in G, one has IH : Ql 5 k. Since this holds for all subgroups H E Z4,1.K.2 applies and yields that in G there is a proper normal subgroup of index bounded by k , which does not contain the element 9. This again contradicts the assumption that G/B is infinite. This final contradiction shows that our initial assumptions are untenable and thus proves the result. I Since Question III.1 remains open, there is definite interest in Sunkov’s generalization of 3.17, were it only to illustrate further the power of the notion of the range of an element.
z3
3.29 Theorem. If the locally $nite group G satisfies min-p,then every pelement of G hasjinite range in G if and only i f the index IG : Op,p GI is Jinite. Proof. By 3.26 everyp-element of G has finite range in G if and only if every p-element of G/O,G has finite range in G/O,,G. I f the index ( G : O,,, GI is finite, then every composition series of G/O,, G has finite factors, and thus - by 3.27 - every element of C / O ,G has finite range in C/O,, G. In the opposite direction, we assume that the result is false and choose among the hypothetical counter examples a group G with an abelian-byfinite normal subgroup B such that the Sylowp-subgroups of G/Bare as small as possible. By 3.26 and 3.16 we may assume that 0,. G = (1) and OP’G=
CH.
3,
6 B]
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APPLICATIONS OF SYLOW THEORY
G . Therefore B is a finite extension of a p-group and so has min. If PIB is an infinite normalp-subgroup of G/B, then P is abelian-by-finite by 3.16 and the Sylowp-subgroups of GIP are of smaller size than those of GIB. This contradiction of the choice of G and B shows that O,(G/B) is finite. Suppose that GIB is simple. By 3.28 G contains a proper normal subgroup N not contained in B. Then G = BN. Since B satisfies min we can find a supplement M of B in G such that B n M is minimal. M / ( B n M ) N GIB, so by 3.28 again M contains a proper normal subgroup M , not contained in B n M . Then B M , = G and B n M , c B n M , a contradiction to the minimality of B n M . Therefore G / B is not simple. There exists a normal subgroup B, of G such that B c B, c G. Since OP’G= GI the group G/B, is not a p’-group. Therefore the Sylow p-subgroups of B J B are smaller than those of GIB and consequently IB, :O,.,B1l is finite, But 0,.G = (1) and O,(G/B) is finite. Thus BJB is finite and nontrivial. We can repeat the above argument with B, in the place of B. Continuing in this way we can define an ascending series B = Bo c B, c B2 c . . of normal subgroups of G such that B,/B,_ is finite and non-trivial for each i. If all but a finite number of the Bi/Bi- are p‘-groups, then D = Bi contains a non-trivial normal p‘subgroup by 3.16. Therefore DIB contains infinite p-subgroups. Let Q be the subgroup generated by the divisible abelian p-subgroups of DIB. The group Q is infinite and contains no proper subgroup of finite index. But clearly IQ : C,(B,/B)I is finite, so Q lies in the centre of D . Consequently Q is an infinite p-group, contradicting the finiteness of O,(G/B). The theorem is now proved. I In view of the result of Feit and Thompson [l] that every finite group of odd order is soluble, we obtain the following result as an immediate consequence of 3.29.
.
,
uz
3.30 Corollary. The locally finite group G satisfying min-2 is an extension of a local/y soluble group by afinite group if and only if each of its 2-elements has finite range in G . There also is a variation on 3.18. 3.31 Corollary. Let G be a locally finite group satisfying min-p for every prime p. If every 2-element of G hasJinite range in G, then there is a radicable abelian normal subgroup A in G such that there factor group GIA is residually finite, and every Sylow subgroup of GIA isfinite.
Proof. From 3.30 and 1.B.4 it is obvious that no infinite simple group can be involved in G, and thus the conditions of 3.18 are met. I
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5B
Observe that for results like 3.29 or 3.30 to hold some condition with bearing on the normal structure of G is necessary, since the simple groups PSL(n, F ) over infinite, locally finite fields F satisfy min-p for every prime p except the characteristic of F. In view of several of the preceding results we ask Question 111.3. Does the locallyfinite group G have a locally soluble subgroup offinite index if it satisfies min-p for every prime p. The following lemma reduces a number of problems to the study of certain infinite simple groups. It allows numerous variations, for example 1.F.4 could have been incorporated into it.
3.32 Lemma. Let G be a Iocallyfinite group satisfying min-2 such that no locally soluble subgroup of G has finite index in G. Then G contains subgroups H a n d N such that H is perfect, N is normal in H , and the factor group HIN is infinite and simple. Also every subgroup of H of 2-size smaller than H (in particular N ) is (locally soluble)-by-finite. Further, if G satisfies any one of the following four conditions, then so does H / N . a The group satisfies min-p; b Every abelian subgroup of the group satisfies min; c The group is an XRc-group and satisfies min-p for every prime p ; d The group is an mC-group satisfying min-p for every prime p , and the centralizer of each of its non-central subgroups is abelian-by-finite. Proof. G contains a subgroup H that is not (locally soluble)-by-finite and is of minimal 2-size with respect to this property. Clearly we may assume H = 0 2 ' H .If K is any proper normal subgroup of H, then the 2-size of K is smaller than that of H, and by the minimal choice of H in G, the subgroup K is locally soluble)-by-finite. In particular, the group H is perfect. Let N denote the product of all proper normal subgroups of H . By 3.17 (or 3.30, if you prefer) the characteristic subgroup N of H is (locally soluble)by-finite, and hence H / N is an infinite simple group. By 3.13 if G satisfies min-p, then so does H/N. If AIN is any abelian subgroup of H / N and if G satisfies b, then A is (locally soluble)-by-finite and satisfies min by 3.19. Thus every abelian subgroup of H / N is a Cernikov group. Now assume that G E mCand that G satisfies min-p for every prime p . By 3.22 the normal subgroup N of H contains an abelian characteristic subgroup B = N o of finite index. Now both C H B and C,(N/B) have finite
CH.
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APPLICATIONS OF SYLOW THEORY
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index in H (see 3.21) and the stability group ( C H Bn CH(N/B))/CHN is abelian ( l . C . l ) . Hence H = N C H N , H / N N C , N / Z N , and C H N is not (locally soluble)-by-finite. Thus we may assume that H = CHN, that is, N = ZH. Let a E H\N, and consider the map 6 = 6, of K = CH((a,N > N)/N) into N given by 6 : x H [x, a ] . The map is a homomorphism. Moreover, if I(a)l = r, then ( K a y = (l), and since N satisfies min-p for all primes p, the image K S is finite. Clearly we have Ker 6 = C H a ,and thus IK : CHal is finite. In particular, if every proper centralizer in G is abelian-by-finite, then every proper centralizer in H / N is abelian-by-finite. Let S1 c S , c . . . c Si c . . . be an ascending chain of subsets of H , and put K i= C H ( ( S i ,N ) / N ) . In order to prove that H / N E % , , we have to show that Ki = K j for all large i and j . Since G E E,, the collection of centralizers { C H S i ;i = 1, 2, . . .} is a finite set, and we may assume that CI,S, = C H S ifor i = 1,2, . . .. There exists a finite subset F of SI such that C H S 1= CHF, since H E %I?=. By the previous paragraph, the index ICH((a,N ) / N ) : C,al is finite for each element a E F, and thus the index JC,((F, N ) / N ) : CHFI is finite. Trivially CH((F,N ) / N ) 2 K , 3 K i3 CHSl = C H F for each i,
and the result follows. I The next result (Theorem 5 of Sunkov [2]) is of a different nature. It is closely associated with the results of Chapter 2, but we give a proof using 3.29. This theorem may in fact be used to give another proof of the result of Kargapolov [ 6 ] and Hall and Kulatilaka [I] (contained in 2.5) that every infinite, locally finite group G contains infinite, abelian subgroups, since (together with the Feit-Thompson Theorem) it provides a reduction to the 10cally soluble case, which may be dealt with by 2.3. In the proof of this result we shall need a special case of Theorem 2H of Brauer and Fowler [11, namely that if G is a finite group and if the involution i of G satisfies (CGiI = n, then there exists a normal subgroup L # G of G with IG :LI 5 [+n(n+2)]!. 3.33 Theorem. If the involution i of the IocaIlyJinite group G hasfinite centralizer C , i, then there exists a normal subgroup ofJinite index in G which does not contain i. (Hence G cannot be inJinite and simple.)
Proof. Let X be any finite subgroup of G containing i, and denote by SX the set of all normal subgroups N of X such that i # N a n d such that every composition factor of X/N has order at most m = [$lC,il(lC,il+2)]! The
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[CH.
3,
5B
set Sx is not empty, for let M be the smallest normal subgroup of X such that every composition factor of X / M has order at most m, and suppose i E M . Then by the result of Brauer and Fowler [l] quoted above, M has a normal subgroup MI with 1 < IM : NIl 5 m. Hence the intersection M , = nxsXMf is a normal subgroup of X such that every composition factor of X / M 2 has order at most m , and yet M2 < M . This contradicts the choice of M , and consequently i 4 M and M E Sx. If Y is any finite subgroup of G with i E: X c Y, then the mapping A; : N w N n X , N E S ~maps , the set Sy into the set S,. The collection {S,, A:; i e X E Y } is an inverse system of non-empty finite sets over the directed index set of all finite subgroups of G containing i. By l . K . l the inverse limit b.m Sx of this system is not empty. Let (M,) E lim Sx,where M x E S, . Then the properties of the inverse limit yield that M = u x M x is a normal subgroup of G not containing i, M x = M n X , and M X / M =! X / M x . Consequently the composition factors of every finite subgroup of G/M have order at most m, and so every element of GIM has finite range. By 3.2 the group G satisfies min-2, and so by 3.13 the group GIM satisfies min-2. By 3.29 the group G contains normal subgroups M c L c K c G such that G/K is finite, K / L is a radicable, abelian 2-group, and L / M is a 2'-group. Let P be any Sylow 2-subgroup of G containing i and A its minimal subgroup of finite index. If a E A , there exists an element b E A with b2 = a. But then a = (bb')(bb-'), bb' E C,i, and (bb-')' = ( b ! ~ - ~ ) - In ' . particular A = (CAi)[i,A ] . But A is radicable and CGi is finite. Consequently A = [i, A ] , and thus a' = aL1 for every element a E A . Now clearly 3.13 implies that A L = K , and so the involution i inverts every element of K/L. Hence if i E K, then K = L . Now L / M is a 2'-group and so i E M. Since this cannot be, we have i$ K. I 3.34 Lemma. If the element x of prime order p in the locally Jinite group G hasjnite centralizer in G, then every normal subgroup of G containing x has finite index in G.
Proof. Let N = (x'), the normal closure of x in G. By 3.2 the group N satisfies min-p, and by 3.7 and 3.10 there is a finite p-subgroup P of N containing x such that the conjugacy class of P in N is invariant under all the automorphisms of N. Hence the Frattini argument yields G = N . NGP. Since P is finite, INGP : CGPI is finite. Clearly C G P c C G x , and so the index IG : NC,xl is finite. But C,x is finite, thus IG : NI is finite. I An automorphism of a group that leaves fixed only a finite number of elements is called a nearly regular automorphism. Involutary nearly regular auto-
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morphisms arise in several contexts (the involution i of 3.33 induces such an automorphism on G). The reader will have no difficulty in extracting the proof of the following lemma from the last paragraph of the proof of 3.33. 3.35 Lemma. I f A is a radicable abelian group and i f i is an involutary nearly regular automorphism of A , then ai = a - l f o r every a E A . 3.36 Theorem. Let G be a locally Jinite group in which every 2-element has finite range. If G admits an automorphism a of order two such that the centralizer C G a is a C'ernikov group, then the subgroup 0 2 , z G of G has finite index in G and contains an ascending series with abelian factors consisting of subgroups invariant under G and a. If the centralizer CGais finite, then the subgroup O Z pG2is soluble. We do not know whether the condition that all the 2-elements of G have finite range can be weakened or even left out without affecting the conclusion. In fact, we are not aware of any examples of a locally finite group G which is not soluble-by-finiteand which admits an automorphism a of order two such that the centralizer CGais a Cernikov group.
Proof. Denote by K the semi-directproduct K = G X (a). By 3.2 the group K (and hence also G) satisfies min-2. Consequently, by 3.29, the subgroup 02,2 G is of finite index in G. The factor group OZr2 G/02.G is soluble, and by the result of Feit and Thompson [l] the subgroup 02' G = OG is locally soluble. Let N be any K-invariant subgroup of OG with N # OG and put H = O G / N . Observe that C,a = N C o G ~ / NWe . want to show that in the group H # ( 1 ) there exists a K-invariant abelian subgroup A # (1). Let M be maximal among the K-invariant subgroups X of H satisfying X n C,a = (1). Then clearly a acts regularly on M , and since M i s locally finite, it must be abelian. So if M # (l), we are done. Hence assume that every K-invariant subgroup # ( 1 ) of H intersects the centralizer CHm nontrivially. By 3.12 there are but a finite number of conjugacy classes of elements of prime order in C , a. Since every K-invariant subgroup # (1) of H contains at least one of these classes there exists a minimal K-invariant subgroup M # (1) of H. Now 1.B.4 asserts that M is abelian. Thus 02, G has an ascending series of K-invariant subgroups with abelian factors. If COGa is finite, then induction on the order ]COGa1 shows that the derived length of OG is bounded by 2 log 31C~GaI 1. I
+
Exercise. Let G be a locally finite group and a an involutary nearly regular automorphism of G. Prove that for every a-invariant normal subgroup
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mk-p
[CH.
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0B
N of G the automorphism induced on GIN by a is still nearly regular. (See Sunkov [12], Lemma 8.) Exercise. Let G be locally soluble group admitting an automorphism a of prime order. Show that G is hyperabelian if the centralizer C,a is a Cernikov group, and that G is soluble if the centralizer C,a is finite. (Use Thompson’s Theorem - see Huppert [l], V.8.14 - and the results of G. Higman [ l ] - see Huppert [l], V.8.8a.) 3.37 Theorem. If G is an infinite, locally finite group and V is an elementary abelian subgroup of G of order four, then for at least one involution i E V the centralizer C , i is infinite.
Proof. Suppose, if possible, that this result is false, and let G be any counter example. By 3.2 the group G satisfies min-2. Let S be any maximal 2-subgroup of G containing V, and denote by A the smallest subgroup of finite index of S. If i and j are distinct involutions of V then a’ = aj = a-l for every a E A by 3.35. But then A E CGij,which is finite. Therefore the 2-subgroups of G are all finite. We shall now assume that G is chosen to be a counter example with Sylow 2-subgroups of minimal order. By 3.33 - applied to each of the involutions of V - the group G has a normal subgroup N of finite index such that N n Y = (1). Suppose that N is a 2‘-group. For every finite subgroup F of G containing V and for every prime p there exists a Sylow p-subgroup P of F n N normalized by V. By Gorenstein [l], 5.3.16 we have
P c L = (CGi;i e V\(I)). Hence F n N G L . But then N c L , and yet L is a finite group and N is infinite group. Thus N must contain an involution. There exists an involution z in N centralizing V (the centre of a Sylow 2subgroup of G has a non-trivial intersection with N ) . If i is any involution of V, then the Sylow 2-subgroups of ( i , N ) have smaller order than those of G and hence there exists an involution j in the elementary abelian group ( i , z ) of order four such that the centralizer C < i , N )isj infinite. Put C = CGj. The infinite group C contains V,and the image of V in C/(j ) has order four. Further it contains an involution with infinite centralizer in C / (j ) , since the Sylow 2-subgroups of C/(j)are of smaller order than those of G. So let k e V\(l) be such that the centralizer K = C c ( ( j , k ) / ( j ) ) is infinite. But C c k = C&, k ) and so the index IK : C,kl is finite, which implies that the centralizer C,k is infinite. This contradicts the assumption that G is a counter example. Hence no counter example can exist and the result is proved. I
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We conclude this chapter with a further application of the characteristic conjugacy classes of subgroups constructed in 3.9 that allows many properties of the automorphism group of a Cernikov group to be read off from corresponding properties of linear groups. For example both parts of 1.F.3 follow at once from the proof below and the fact that a periodic linear group over the field of p-adic numbers is necessarily finite. Note also that since for any prime p the ring of p-adic integers is an integral domain of characteristic zero, there always exist rings R satisfying the requirements of 3.38 below. 3.38 Proposition. Let G be a C'ernikov group, A its minimal subgroup of finite index and p l , . . ., p t the primes involved in A . I f R is any integral domain that f o r i = 1, 2, . ., t contains a subring isomorphic to the ring of pi-adic integers, then the automorphism group AutG of G contains a subgroup L28 with Aut G = (Inn G)W, that is linear over R . Moreover the outer automorphism group Out G of G is linear over the quotient field of R.
.
Proof. Let d = Aut G, 4 = Inn G, K E Z ( G ) (see 3.9) and put 3 = N,(K) = {$ E d ; K# = K } . Since A is characteristic in G, C,A = {$ ~ d ; u4 = a for all a in A } is normal in d.Also d / C , A is isomorphic to asubgroup of Aut A , which in turn is a direct product over i of linear groups over the pi-adic integers. Hence d / C , A is linear over R. Since A K = G, C,A n C,K = (1). Thus C,K is linear over R. But g / C , K is a finite group as K is finite, so 53'too is linear over R. If $ E d then K@E%(G), By 3.9 there exists g in G such that KO = K g .If Ag denotes the inner automorphism x H gxg-' of G then $Ag E and so d = #a. Now 4 n C,A = {Ag; g E C G d }and so 9/($ n C,A) is finite. Hence I$*C,K: ( 4 n C C , A ) C , K J is finite. But 4 L 2 8 = d and I$: C,KI is finite, so Id : ( 4n CdA)C,Kl is finite. Now
(9n C,A)C, K 3 n C,A
N
c, K = C,K, 4 nC,A nC , K
and therefore d / ( 9n C d A ) is linear over R . Again 4/(9n C,A) is finite and thus Out G = d/9is linear over the quotient field of R (Chevally [I], p. 119). 1
Note. The concept of range (there called rank) and the fundamental theorem 3.29 is taken from Sunkov [ 5 ] , though the treatment given here follows Wehrfritz [8 1, from where the Sylow theory of locally finite groups with min-p is also taken. Of the other less trivial results of this chapter, 3.2 appears in Sunkov [3] and Kegel [3], 3.3 in Wehrfritz [7], 3.4 in Zalesskii [ 1 ]
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[CH.
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and Wehrfritz [l] and [7], 3.5 in Baer [ 5 ] , 3.17 in Kargarpolov [4], 3.22 in Kegel and Wehrfritz [l], 3.24 in Sunkov [ 5 ] , 3.33 in Sunkov [ 2 ] , and 3.37 in Sunkov [3]. A quite different approach to the linearity in 3.28 is given in Merzljakov [l].
CHAPTER 4
Locally finite simple groups
In the earlier chapters of this book we either did not need any assumptions about the simple groups involved in the various groups G under consideration (as, for example, in Chapter 2) or we essentially assumed that the relevant simple sections of G were all finite (see Section B of Chapter 3). Further we saw that the existence of counter examples to certain types of conjectures implies the existence of infinite, simple counter examples (1.F.4, and 3.32). Such counter examples must be studied, of course, in close connection with the particular conjectures they contradict, but there are a few general considerations about infinite, locally finite, simple groups that may be of help in any such closer scrutiny. For example, we show that any infinite simple group has a local system consisting of countable simple subgroups. Moreover a countable, locally finite, simple group is a limit (of sorts) of a sequence of finite simple groups, and such a sequence contains a good deal of information about any locally finite, simple group that may be its limit. Section A below studies such sequences and contains some results that prove quite strong whenever we know infinitely many terms of the sequence rather well. For example, if all the finite simple groups in such a sequence are linear of degree at most n (over possibly different fields) then any limit of the sequence is linear of degree n, and then 4.6 yields that we may assume that the terms of the sequence are subgroups of the limit (and not just sections). It is essentially this situation that we consider in Section B. We give an axiomatic (and hence considerably more general) treatment of the theorem that a group G with a local system of finite subgroups, all isomorphic to PSL(2, q ) for various prime powers q is isomorphic to PSL(2, F ) for some locally finite field F. In this axiomatic approach, the symbol PSL(2, 4) is considered essentially as a functor from tKe category of locally finite fields to that of locally finite (simple) groups. The simplicity of the groups PSL(2, ) is nowhere required in the axioms, indeed we give examples of functors that satisfy our axioms and yet map into the category of soluble groups. 111
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However, we exploit the results of this axiomatic treatment exclusively in order to study certain types of infinite, locally finite, simple groups and the structure of groups involving sections of this kind. In Section C these results are used to extend several rather deep structural theorems about finite groups to locally finite groups. Of course, the principal labour in these results resides in the finite case, which we usually admit without question. With the ever expanding supply of extendable theorems on finite (simple) groups the subject of this section has vast potential for growth. We give only a few salient examples of such extendability, our choice being greatly influenced by the tools that we require for Chapter 5, where some of the results of Section C will assume a critical importance. Section A. Infinite and finite simple groups, an approximation principle
Every infinite, simple group has a local system of countably infinite, simple subgroups. If the simple group G is locally finite, one might ask whether there is already a local system consisting of finite simple subgroups. For linear groups this question will be answered affirmatively. But in general, we can only show that a countably infinite, locally finite, simple group G can be approximated by a sequence of finite simple groups that appear as sections of G, and is linearly ordered by involvement. We study, superficially, the influence that such a sequence has on the structure of G. This study will also show that, should several “natural” conjectures on locally finite, simple groups be wrong, then there would be infinitely many “new” finite, simple groups still to be found - an easy, but intriguing observation. For the study of infinite groups in general the following observation is sometimes quite useful. 4.1 Theorem. The group G is simple if and only if there is a local system Z of G such that for every subgroup S E C and every normal subgroup N of S with (1) # N # S there exists a subgroup T E Z containing S such that every normal subgroup M of Tsatisfies M n S # N . (If G is simple, then every local system C of G has this property.) Proof. Let the local system C of the group G have the given property. Suppose G has a non-trivial normal subgroup X . Then there is subgroup S E Z of G such that (1) # S n X # S. For every subgroup T EZ of G containing S we have
(T n X ) -=I T and ( T n X ) n S
=
S nX,
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contradicting the assumption on the local system C. Thus, such a normal subgroup X of G cannot exist, and the group G is simple. Now, let G be a simple group and C any local system of G. If for some pair S, N of subgroups of G such that S E C, N 4 S, (1) # N # S, there is no subgroup T E C containing S such that M n S # N for every normal subgroup M of T, then put C' = { T EZ; S c T ) and for T EC' put A , = { M a T; M n S = N } . In this case 2' is a local system of G , and for T EZ' the set A , is non-empty. If N7' = M , then N T is thenormal closure of N in T (hence the notation), and it follows that for T, U E C' with T E U one has N T c N U .Put X = N T ; then X is a subgroup of G (in fact, it is the normal closure of N in G) and X # (1). On the other hand we have
nMEdT
UTEf,
X nS =
(u
NT) nS
TEZ'
(NT nS ) = N # S.
= TEZ'
Hence X is a proper normal subgroup of G, contradicting the assumption that G is simple. Thus the simplicity of G forces every local system Z of G to have the desired property. 1
4.2 Corollary. Let G be a simple group with a local system C.Ifthe subgroup S E C of G has only finitely many normal subgroups, then there is a subgroup T E C of G containing S such that every normal subgroup of T meets the subgroup S of G trivially (that is, in (1) or s). Proof. By 4.1 for each non-trivial normal subgroup N of S one may choose a subgroup TNE Z containing S such that M n S # N for every normal subgroup M of T N .There exists a subgroup T E C of G containing each of the finitely many subgroups TN . For every normal subgroup M of T one has M n S = ( M n T N )n S # N for each proper normal subgroup N of S . Hence M n S is either (1) or S. I This corollary shows that such a subgroup S of the simple group G embeds naturally into some chief factor of the subgroup T of G. One would, of course, like to have some control on the position in T and on the structure of such a chief factor. This can be achieved by placing severe restrictions on the local system C.
4.3 Corollary. If the local system Z of the simple group G is such that every subgroup S E Z of G has only finitely many normal subgroups, and i f with S also all the normal subgroups of S belong to the local system Z, then for every subgroup S E Z there is a subgroup T EC containing S, such that f o r some maximal normal subgroup M of T one has M n S = (1).
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In general such a stringent condition on the local system of a simple group will have a strong influence on the structure of this group. But if Z consists of finite groups, one obtains a local system Z' of G satisfying the assumptions of 4.3 by adjoining to Z the subnormal subgroups of members of Z. Thus, in a locally finite group G, the conditions imposed on the local system Z of finite subgroups in 4.3 have no real influence on the structure of the simple group G. Proof of 4.3. Fix the subgroup S E C of G. For each subgroup X of G containing S denote by Sx the normal closure of S in X . For each pair X , Y of subgroups of G with S c X c Y one has Sx G Sy. Consequently, the set C, = (Sx; S c X E Z} of subgroups of G is a local system of the group V = UXEISX, which is clearly a normal subgroup of G. Since we assumed that the group G is simple, the set Zlis a local system of G. By our assumption on the local system Z of G, the local system Z1is a subset of Z. Applying 4.2 to the local system Zlof G, we know that there is a subgroup U E C containing S such that for every normal subgroup N of Su one has
(*)
either S G N or S n N
= (1).
Now Su E Z, so Su contains only finitely many normal subgroups. Let K be a maximal normal subgroup of Su and put L = nxeUK".Since L is normal in U and properly contained in Su we have L n S = (1). Further K has only finitely many conjugates in U and consequently Su/L is a direct product of a
finite number of (isomorphic) simple groups. Let M be a normal subgroup of Su containing L such that M n S = (1) and maximal subject to this. Then if N / M is any minimal normal subgroup of S U / Mwe have that N / M is simple and that S c N , by (*). I Before exploiting 4.3 for the study of locally finite simple groups, we present another general fact about infinite simple groups that in many instances allows us to confine our attention to countable groups. 4.4 Theorem. The injinite group G is simple if and only ifit has a local system C consisting of countably infinite simple subgroups of G.
Proof. By 4.1 the existence of a local system of G consisting of simple subgroups forces the group G to be simple. Thus we need only show that every infinite simple group has such a local system. Let G be an infinite simple group. For every pair c, d of elements of G with c # 1 # d choose a finite set S(c, d ) of elements of G such that cE
(d";x
E S(c, d ) ) ;
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since the simple group G is generated by the set of elements conjugate to the element d, such a choice is always possible. With respect to this choice we define for every countably infinite subgroup C of G another countably infinite subgroup C* by C*
=
If M is any normal subgroup of C* with M n C # (l), pick an element d # 1 in M n C. Then one has c E (d"; x E S(c, d ) ) c M for every element c of C. Thus C C M . Now put C = C1 and choose, inductively, Ci+l = (C,)* for every natural number i. The group C = Ci is countable, since it is the union of an ascending sequence of (countably many) countable subgroups. The set {C,}of subgroups of the group C is a local system of C and 4.1 yields that C is a simple group. Since the set of all countably infinite subgroups of G forms a local system of G, the set Z of the subgroups C, one for each countably infinite subgroup C of G, forms a local system of the simple group G . I
u
Remarks. Obviously 4.4 allows variations with respect to the cardinality of the subgroups which form the local system. Another variation, an elaboration of the preceding argument, gives that every infinite group G can be embedded into some simple group S having the same cardinal as G. (For locally finite groups see 6.5.) It may be of some interest to observe that results like 4.1 and 4.4 characterise simplicity in much more general structures than just groups; they are essentially results in universal algebra. Finding general formulations and proofs for such results is left to the interested reader. Since we intend to direct our attention to locally finite groups, 4.4 suggests a problem, for which we offer a solution in a special case only. Question IV.1. Does every locally finite, simple group have a local system consisting of finite, simple subgroups? Because of 4.4 it suffices to find an answer to this question for countable, locally finite, simple groups.
4.5 Lemma. If G is a countably infinite, locally finite, simple group, then there exists a strictly ascending sequence {Rn}nsN offinite subgroups of G with G = UnENR,,such that for each natural number n there exists a maximal normalsubgroup M,,, of R n f l satigying M,,, n R , = (1). Proof. By 1.A.9 there is an ascending sequence {Si}ieN of finite subgroups Replace the local system {Si}isN of G by the local of G with G = UieNSi.
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A
system Z consisting of all the subnormal subgroups of the Si E {Si)ieN.Now choose the sequence {R,,),,, inductively. Put R , = S, , and assume that for the natural numbers 1 5 k 6 n subgroups Rk have already been chosen according to the second requirement. Then there is a first term Si(,,) E {Si)isN which is not contained in R,,. Put T, = (R,,, Si(,,)), and choose as R n f l any finite subgroup of G belonging to .Z which contains the finite subgroup T,, and such that for some maximal normal subgroup M,,,, of R n f l one has M,,,, n T,, = (1). By 4.3 applied to the local system Z, such a choice is always possible. The sequence {Rn)n,Nof finite subgroups of G so obtained meets the requirements of 4.5. I
Remark. Contrary to what has been asserted in Kegel [3], Hilfssatz 2.2, this property of the countably infinite, locally finite group G does not ensure that the group G is simple. To see this, let {S,} be a sequence of finite perfect simple groups, such that for each n E N there exist two isomorphisms y,,, ,,)I of S,, into S,+ such that the images SF and S p commute element-wise. (Clearly thejoin (Sin,Sp) is isomorphic to the direct square of S,, .) Such sequences exist within the set of all finite alternating groups for example. Put G,, = S, x S, and define a monomorphism 8, of G,, into G,+ by
,
en :(x, y ) I+
(xYny$fn,Y@L-) for x, y
E
s,,.
,
The subgroups
Hn
=
((x , 1); x E S,,) and K,, = ((I, x ) ; x
E S,,)
are the only non-trivial normal subgroups of the group G,. One checks easily that the following two relations hold: H n f l n G?
=
H?
and K,,+l n Gff. = (1).
For the direct system {G,,, On> of groups and maps, let G = l k G,, be the direct limit; since the G,, are (locally) finite, the group G is locally finite, and the images of the G,,in G form an ascending sequence of subgroups the union of which is G . As the second of our relations shows, this local system of the group G satisfies the property described in 4.5. The firsr of the two relations, however, shows that G cannot be simple. In fact, H = l h H,, ~ is a proper normal subgroup of G. (Clearly, this construction has nothing to do with the finiteness of the groups S,,.) In the situation of 4.5 the sequence (S,,} of finite simple groups S,, N R,,/M,, contains a wealth of information about the simple group G. Taking some liberty with established language we shall say in this situation that the group G is a limit of the (approximating) sequence S, of finite simple groups.
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The only method that we have to answer Question IV.1 affirmatively for a countable, locally finite, simple group G, is to choose a sequence {R,J of finite subgroups R, of G according to 4.5 and to show that infinitely many of the R, are in fact simple. In order to make this idea work, we must assume that G is “small” in some sense. One such situation is given in the next result. It must be stressed, however, that the situation described here seems very special indeed. 4.6 Proposition. If the infinite, locally finite, simple group G is the union
of an ascending sequence { G,} offinite subgroups G, satisfying G, c G, + such
that G,, I has a maximal normal subgroup M,, with G, n M,, = (I>, then almost all of the G, are simple (i.e. almost all the M, = (I)), provided there exists afield F and a natural number d such that G is isomorphic to some subgroup of GL(d, F). The countability of G is in fact redundant by 1.L.2.
Proof. Let d be the least natural number so that there is an algebraically closed field F such that G is isomorphic to a subgroup of GL(d, F). Since G is infinite, it follows from a theorem of Jordan [1] (see 1.L.4 or Schur [l]) that the characteristic c of the field F is different from zero. The result of Brauer and Feit [1] (see 1.L.4) yields that there is no bound to the order of the finite, elementary abelian c-subgroups of G . On the other hand, for every prime p # c, a finite p-subgroup of G (and of GL(d, F ) ) has at most b(d)generators, where b is a monotonically increasing function of the degree. By deleting the first few terms, if necessary, of the sequences {G,} and {Sn> we may assume that the simple section S , of G contains an elementary abe+ l .ensures that for every field of characlian subgroup of order ~ ~ ( ~ )This teristic different from c, every non-trivial, linear representation of Sl over that field will have degree at least b(d)+ 1. By Mal’cev’s Theorem (see 1.L.9) at most finitely many of the groups G, and S, can have a faithful linear representation of a degree less than d(otherwise G would have such a representation also, contrary to the minimality of d). Again deleting finitely many of the terms of {G,}, if necessary, and renumbering, we may assume that every one of the groups G, and any subgroup of G which is isomorphic to one of the groups S, is an irreducible subgroup of GL(d, F). Let m be the least natural number so that G has a non-trivial, projective representation over F (see Huppert [l], V. 9 24). Since G is isomorphic to a subgroup of GL(d, F), there is a projective representation of this degree and hence m 5 d. Let
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p :G + GL(m, F)
be such a projective representation of minimal degree of G , and put H = E GL(m, F). Again by Mal’cev’s Theorem (1.L.9) only finitely many of the finite, simple sections S,, of G can have a non-trivial, projective representation over F of a degree less than m.Thus we may, and shall, assume that S , does not have any non-trivial, projective representation over F of degree less than m. In the groups G and H we shall consider “substitutes” for the section S , : the finite group S will be called a substitute for Sly if S/@S N S,. Clearly, in the groups G and H there exist substitutes for S,; every minimal supplement of M , in G , is a substitute for S , in G. We shall prove: Every subgroup S of H which is a substitute for S,, is an irreducible subgroup of GL(m, P ) , and CDS = Z S . The Frattini subgroup @S of the substitute S for S , in H is nilpotent. For the prime p # c, let N denote the Sylow p-subgroup of CDS. Since m 5 d, the group N has at most b(d) generators, and so the group S , N S/@S operates trivially on the vector space N/@N over the prime field GF(p). Since the perfect group S is generated by p’-elements, and since each of these operates trivially on N, the group S operates trivially on N . Thus N is contained in the centre of S. Since every irreducible, linear representation of S over the field F has the largest normal c-subgroup 0,s of S in its kernel, such a representation of S is, in fact, a projective representation of S , , and hence of degree at least m. This shows that S is an irreducible subgroup of GL(m, F). This result has a very convenient consequence in the group G. Every subgroup S of G, that is a substitute for S , ,is isomorphic to S, . If we denote by I;* the group of all multiplications of the vector space V = F”’ by elements of the multiplicative group F* of the field F, that is, the centre of GL(m, F ) , then there is a natural homomorphism
v :H
--f
HF*/F N G.
For the subgroup S of G, which is a substitute for S , ,let T be any finite subgroup of H with T’ = S, and denote by K the inverse image under v of CDS in T. Every minimal supplement L of K in T (that is LK = T, and for every proper subgroup L , of L one has L , K # T)is a substitute for S,. But then CDL = Z L , and by Schur’s Lemma one has Z L E F*, and L”N S , . But from the noncomplementation of the Frattini group it follows that L‘ = s. Now we have enough information to show that M,, must be trivial for
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n > 1. Suppose it is not, and let P # (1) be a Sylowp-subgroup of M, and consider the group C = GI M,,. The normalizer N,P covers C/Mn,and so it contains a substitute S for S , , which is therefore isomorphic to S , . As S is an irreducible subgroup of GL(d, F), so is the group SPYand so P cannot be a c-group. Since P has at most b(d)generators, the group S will centralize P,and SO by Schur’s Lemma P c ZGL(d, F ) . But since G is a simple subgroup of GL(d, F ) , one has
P
E
G n ZGL(d, F ) = (l),
a contradiction. 1
Remark. A different proof of 4.6 can be extracted from the proof of Lemma 2 of Kargapolov [5], for another variation see Kegel [3], Satz 2.6. In a countably infinite, locally finite, simple group G an approximating sequence of finite subgroups according to 4.5 need not contain a subsequence consisting of simple groups. Example. Let {ui;i E N] be a basis of the countably infinite-dimensional vector space V over the finite field F # GF(2). For every natural number n denote by V,, the subspace V, = On= Fui.Every automorphism of V, can be trivially extended to an automorphism of V by letting it act as the identity on the vectors v j , j > n. Let R, be the group of automorphisms of V thus obtained from the group SL( V,) of all automorphisms of determinant 1 of V,. One clearly has R, c R,, and R, N SL( V,,).The centre of R, is the unique maximal normal subgroup of R,; each element of it is the expansion to V of a multiplication of the space V, by an n-th root of 1 in F. Now an application of 4.1 shows that the “stable linear group”
,
G = SLo(V) =
u
R,,
nEN
is simple. To exhibit a local system of the locally finite simple group G that does not have a subsystem consisting exclusively of finite simple subgroups of G choose C = {R,; (n, IF1- 1) # I},
(all the groups R, in C have non-trivial centres). It should also be observed that the necessary condition 4.5 for a countably infinite, locally finite group to be simple can also be used as a non-simplicity criterion: 4.5 says in fact, that if Y is the set of all finite simple groups involved in the infinite, locally finite group G, partially ordered by involvement, then 9cannot have maximal elements and must be directed whenever
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A
G is simple. This yields, in particular, the well-known fact that a locally finite soluble group cannot be non-abelian and simple (see 1.B.5). In order to get some hold on a countably infinite, locally finite simple group G, we shall study now sequences of finite simple groups of which the group G is a limit as in 4.5.
Definition. The non-empty, two-parameter family 9 = (F(n, q ) } of finite simple groups F(n, q ) is said to be classically parameterized by the pairs (n, q ) of natural numbers, if 1 the field parameter q takes prime power values only; 2 there is a function dF = d : N + N such that each group F(n, q ) E 9has a faithful linear representation of degree d(n) over the algebraic closure GF(q) of the field G F @ ) if q > 1, and over every finite field if q = 1; 3 if the subset Fl of 9is linearly ordered by involvement, and if in 9, the rank parameter n is unbounded, then for no prime p is there a bound to the orders of elementary abelian p-subgroups of the groups F(n, q ) E PI. The family 9= {F(n, q ) } will be said to be very classicallyparametrized, if it satisfies the preceding conditions 1 and 2 and furthermore 3‘ if the subset F1 of 9is linearly ordered by involvement, and if in 9, the rank parameter n is unbounded, then for no prime p is there a bound to the exponents or a bound to the derived lengths of the p-subgroups of the groups F(n, q ) E 9,. Observe that in a (resp. very) classically parametrized family of finite simple groups the same abstract group may appear with different parameter values. Clearly, every non-empty, two parameter subset of a (very) classically parametrized family of finite simple groups is itself (very) classically parametrized.
Exercise. Show that every very classically parametrized family of finite simple groups is classically parametrized. Question IV.2. Is every classically parametrized family of jinite simple groups necessarily very classically parametrized? The standard example of a very classically parametrized family of finite simple groups is the family 9 = {PSL(n, 4)). The group PSL(n, q ) acts naturally and faithfully on the space 2 of all n-by-n matrices of trace zero over the field G F ( q ) and hence also on the space~.Z = Z 0 GF(q); these spaces being of dimension n2 - 1 over GF(q) and G F ( q ) respectively. Thus if we put d(n) = d - 1 , then the family 2’clearly satisfies condition 2.
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Furthermore the alternating group of degree n is a subgroup of the group PSL(n, q), and thus the family 9satisfies condition 3’. Denoting the alternating group A, of degree n by PSL(n, 1) one may incorporate the alternating groups into the family 9. The known, “natural” infinite families of finite simple groups as described, for example, in Tits [l] or Carter [l] appear as two-parameter families. They are more or less elaborate variations of the family9, where the ranges of the rank parameter n and the field parameter q may be restricted. If the rank parameter in the “natural” family 9 is bounded, then trivially 9 also satisfies conditions 3 and 3’. The only (known) “natural” families, for which the rank parameter is unbounded are (using the standard codification of the known finite simple groups in Tits [ l ] or Carter [l])
From this point of view it is not surprising (and it is obvious from the descriptions of these families in Tits [ l ] and Carter [l]) that these two-parameter families satisfy conditions 1 and 2 of the definition of a (very) classically parametrized family. In fact, if 9= (F(n, q)) is such a “natural” family, then the group F(n, q ) involves an alternating group A, of degree m = m(n) depending linearly on n (and of course, also on the family 9); hence S satisfies also condition 3 . Thus, the known “natural” infinite families of finite simple groups are very classically parametrized. According to Tits [l] and Carter [ l ] - and this seems still to be the state of knowledge at the moment of writing - every known finite simple group appears in one of the finitely many known, “natural” very classically parametrized families, in the family of groups of prime order, or in a finite set of “sporadic” finite simple groups. So the question arises naturally whether (up to a few more “new” sporadic groups) all finite simple groups are obtained that way. Let A’” be a set of finite simple groups satisfying the maximal condition with respect to involvement. Observe that the union of finitely many such sets still satisfies the maximal condition with respect to involvement, SO the set of all groups of prime order together with finitely many sporadic groups clearIy satisfies the maximal condition with respect to involvement. This observation now leads us to paraphrase the above question on the set of all isomorphism types of finite simple groups in the following considerably weakened, but flexible, form.
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Question IV.3. Are there finitely many families Fv,1 5 v S r, and JV of finite simple groups where each Fvis (very) classically parametrized and Af satisfies the maximal condition with respect to involvement, such that everyfinite simple group is isomorphic to some group belonging to one of these families?
The notion of a (very) classically parametrized family of finite simple groups and the discussion following its introduction provide us with a condition to impose on countably infinite locally finite simple groups that are a limit of some family of finite simple groups. The following discussion, though rather brief, raises several problems which we formulate not in order to make any contribution towards their solution, but rather to point out the close connection that exists between the theory of finite simple groups and some aspects of the theory of locally finite simple groups. For this discussion the following notion will be convenient. Definition. The group G is large if it contains, for every primep, an infinite elementary abelian p-subgroup; it is enormous if, for every prime p , it contains finite p-subgroups of arbitrarily high exponent and arbitrarily long derived length. Exercise. Show that every enormous locally finite group is large. 4.7 Proposition. I f the countably infinite, locally finite simple group G is a limit of the sequence {Si} offinite simple groups Si,and if each of the groups S, is isomorphic to a group appearing in the (very) classicallyparametrized family 3 = {F(n, q)} of finite simple groups, then there either exists a field F of finite characteristic and a natural number n such that the group G is isomorphic to a subgroup of GL(n, F), or the group G is large (resp. enormous).
Proof. To say that G is a limit of the sequence { S , } means that the group G has a local system Z consisting of a sequence { R , } of finite subgroups Ri of G such that Ri c R i + l and such that there exists a maximal normal subgroup M i + , of R i + l with R , n Mi+l = (1) and Ri+lllkfi+l N S i + t . Since each of the groups S , is isomorphic to some F(n, q ) E 3, we choose one such group F(ni, q,) N S , for each i E N and denote by F1the subset of 9 so chosen. If the rank parameter n, is bounded on the family Fland if d = maxiGN d(n,), then for each i E N we have the embeddings
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C+F(ni+l, q i + l ) 4 -_GL(d(ni+11, GF(qi+1)) 4 * . . 4 GL(4 GF(qi+I)), where GF(q) denotes the algebraic closure of the finite field GF(q). Thus G has a local system of subgroups R iwhich all have faithful linear representations of degree d over (possibly) varying fields. But by 1.L.9 there then exists a field F such that the group G has a faithful linear representation of degr& d over F. The field F must have finite characteristic, for if F had characteristic zero then every locally finite subgroup of GL(d, F ) would have an abelian normal subgroup of finite index bounded by a function of d, compare 1.L.4; and thus G could not be simple. Now assume that the rank parameter ni is unbounded on the family PI. If for some prime p the group G belonged to the class Min-p, then the elementary abelianp-sections of G would have bounded order by 3.13, and so in particular elementary abelian p-subgroups of the groups Si N F(ni, qi) would have bounded order, contradicting the assumption that Fl is a classically parametrised family. If F1is a very classically parametrized family, then clearly the group G is enormous. I Observe that whenever the locally finite group G is isomorphic to a subgroup of some GL(n, F ) the maximal primary subgroups of G are soluble and have derived lengths bounded by a function of n. If the prime p is the characteristic of F, then the maximal p-subgroups of G are nilpotent of class 5 n and of exponent S p"; if the prime p is not the characteristic of F, then G E Min-p, and the p-size ]GI, of G is bounded by a pair of numbers depending only on n and not on p . Thus the two cases encountered in 4.7 are at opposite ends of the spectrum. What happens between these two extremes?
Ri
C-+Si+l
Question N.4. Is there a non-linear, locally jinite simple group G which is not large (enormous)? Question IV.5. Is there a large, locallyjinite simple group G which is not enormous? An affirmative answer to these questions would be of considerable interest for the theory of finite simple groups because of the following result; but unfortunately, one has no idea how to show the existence of such groups. 4.8 Theorem. Let Pv , 1 S v 5 r, be$nitely many (very) classically parametrized families of jinite simpIe groups, dV a set of jinite simple groups satisfying the maximal condition with respect to involvement, and %' the set of all isomorphism types of jinite simple groups not appearing in A' or in one of
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the families F, ., Then the existence of a non-linear, locally finite simple group G which is not large (enormous) implies that the set V is infinite. Remarks. If we choose for the Fvthe known, natural infinite families of simple groups and for A’” the set of all groups of prime order together with the finitely many known sporadic groups, then every finite simple group known to date appears in one of these families. Thus the existence of a nonlinear, locally finite simple group that is not large would show that there are still infinitely many new finite simple groups to be found (V would denote the set of unknown finite simple groups), and that they cannot be arranged in finitely many classically parametrized families. Thus an affirmative answer to Question IV.4 would answer Question IV.3 negatively. From the point of view of infinite groups it seems rather unlikely that every locally finite simple group either is linear or enormous; since it seems most implausible that such a vague finiteness condition on a single prime as: the localty jinite simple group G is not enormous should, in fact, force the strong and homogeneous finiteness conditions on all primes implied by: the locally Jinite simple group G is linear. The results in Chapter 5, however, seem to point in the opposite direction. Since all the locally finite, simple groups which are not enormous, that we know, are linear, and so by 1.L.2 countable, we want to point out a variation to Question IV.4.
Question IV.4’. Is there an uncountable, locally finite, simple group which is not large (enormous)? A negative answer to this would provide a rather interesting nonsimplicity criterion. Proof of 4.8. Let 9 be the set of isomorphism types of finite subgroups of prime power order of the infinite, locally finite group G. The set 9 is countable. For each isomorphism type in 9 choose a single subgroup of this type in G, and let Go be the subgroup generated by these countably many finite subgroups of G . If all the finite subgroups of the group G had a faithful linear representation of bounded degree, then by 1.L.9 the group G would be linear, contrary to our assumption on G. Thus in G there is a sequence (Di} of finite subgroups D iof G such that the sequence of the minimal degrees of faithful linear representations of D iis unbounded. The subgroup D = (Di; iE N ) is a countable, non-linear subgroup of G . Consequently, the join (D,G o ) is a countable, non-linear subgroup of G . By the proof of 4.4 there is a countable simple subgroup H of the simple group G containing (0, Go), and so H i s also non-linear.
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The countably infinite, locally finite simple group H has by 4.5 a local system consisting of a sequence of finite subgroups R i , iE N,with R , c R i + l such that each R i + l has a maximal normal subgroup M i + l with Rin Mi+ = (1). Put Si+ = R i +JMi+ Then the group H is a limit of finite simple groups. We shall now show that alof the sequence (Si> most all of the finite simple groups Sibelong to 9. The pigeon-hole principle asserts that infinitely many of the finite simple groups Si belong to A’”, one of the families S,,, or to %?. Since Siis involved in S i + l ,it is impossible that infinitely many of the Sibelong to the set M which satisfies the maximal condition with respect to involvement. So assume infinitely many of the Sibelong to the (very) classically parametrised family S,,. The group H i s a limit of this subsequence {Sit>. Since the group H is not linear, 4.7 yields that H is large (enormous). But the assumption that G is not large (enormous) is a condition on the set 9 of isomorphism types of finite groups of prime power order involved in G. Since every isomorphism type of 9 is represented as a subgroup in H , the assumption that G is not large (enormous) entails that the group H cannot be large (enormous). This contradiction is derived from the assumption that there are infinitely many of the finite simple groups Si belonging to one of the families S,,. Thus almost all of the Si belong to %?, and hence %? is infinite. I A class 23 of groups is called a variety if every subgroup and every homomorphic image of a @-group belongs to @ and if the Cartesian product of any set of @-groups also belongs to @. The variety 8 is locallyfnite if every group G E @ is locally finite. Clearly, the groups of a locally finite variety must be of bounded exponent. It is an open question as to “how finite” the groups of a locally finite variety are. Let us specify one interpretation of this vague question as Question IV.6. Is there a locally finite variety @ containing an infnite simple group? A positive answer to Question IV.3, would answer Question IV.6 negatively, since such a simple group S would be linear of finite degree over a suitable field F, and a theorem of Platonov [2] asserts that a linear group S belonging to a variety different from the variety of all groups must be soluble by-finite. Thus the simple group S is finite. Let us give another twist to the question: “how big” (or how intricate) is an infinite, locally finite, simple group? Question IV.7. If G is an infinite, locally$nite, simple group, is the variety of all groups the only variety of groups containing G?
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An introduction to the theory of varieties of groups can be found Hanna Neumann [I]. The importance of locally finite varieties for this theory and questions related to this section are discussed in KovBcs [3]. If some locally finite, simple group G is a limit of a (very) classically parametrized family of finite simple groups, by 4.7 there are two possibilities, either G is large (enormous) - and apparently, in this case, we have no means to make any further structural statement - or G is linear, and then by 4.6 almost all the simple groups of such an approximating sequence appear indeed as subgroups of G. These subgroups have, of course, bounded rank parameter. It is essentially this situation that we shall study in the next section, although the axiomatic approach that we choose will look quite different.
Section B. An embedding theorem and concrete examples In this section we study groups G that have a local system consisting of finite subgroups all of the same type. The desired conclusion is that then the group G itself is also of this type. For example, a group having a local system consisting of subgroups isomorphic to PSL(2, q), for varying q, is necessarily isomorphic to PSL(2, F ) for some locally finite field F. In the following account locally finite fields and finite simple groups will appear only at the very end. Instead, we give an axiomatic treatment of our main results, the embedding theorem 4.10 and the identification theorem 4.13, and only then indicate how certain families of (possibly twisted) Chevalley groups fit into this pattern. However, it may be helpful for the reader to bear in mind the particular example X = PSL(2, ) and Y = PGL(2, ) in all that follows and to check the progress made in terms of this example. Simply to point out that our axiomatic set-up has a much wider range than will be exploited here, we give two examples where the groups involved are soluble (instead of simple).
Definition. An index category on the set M is a category U whose objects are subsets of M such that 1 M = u A s e A (abusing notation, we denote the set of objects of V also by U), and 2 if A , B E V, then the set %(A, B ) of morphisms from A to B is empty if A $ Band it consists only of the canonical embedding eAB:A --f B otherwise. The objects of an index category V will be called indices.
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Definition. Ajinite (group) functor on the index category $? is a functor F from V to the category of groups such that 1 the group F(A) is finite, if A is a finite object of V, and 2 the map F(e,,) is a monomorphism, if A and B are finite objects of V with A c 3. The preceding two conditions are satisfied by any group-valued functor on the index category V, if V does not have any finite objects. It is part of the usual definition of functor that if A , B and C are indices of the index category V with A s B c C, then F(e,B)F(eBc) = F(e,c).
We shall not need this coherence property in this section. However, since in our applications F really will be a functor in the usual sence, it does not seem worthwhile introducing a new term. The reader may include or exclude this axiom according to his taste. We shall say that the index category Vl on the set M , is wider than the index category V on the set M , if M G MI and V E V, . We shall say that the group-valued functor Y on V, is wider than the functor X on V, if the index category V, is wider than the index category V, if for every finite index B E V the group Y ( B ) contains a subgroup isomorphic to X(B), and if for each finite index A E V with A s B every subgroup S N X ( A ) of Y ( B ) is contained in a subgroup T N Y ( A ) of Y ( B ) . Definition. A pair of functors X , Y on the index categories V, %,, respectively, into the category of groups is coupled if Y is wider than X and 1 whenever A E V and B E V, are finite indices, then the number of conjugacy classes of subgroups of Y ( B )that are isomorphic to Y ( A ) is bounded by an integer r,, independent of B; 2 whenever A€%' and B E V ~ are finite indices, then any subgroup U of Y ( B ) that is isomorphic to X ( A ) satisfies IN,(,,U : UI 5 ,S for some integer s, , independent of B. If the functor X is coupled to X,then we shall call X a self-coupled functor. Observe that if X and Y are coupled functors on the index categories V, Vl , respectively, then the functor Ylo is self-coupled. One has only to show that for every finite index A E V there exists an integer tA such that for every finite index B E V and for every subgroup U N Y ( A ) of Y ( B ) one has INY(,)U : Ul S t A . To see this, let kA denote the number of subgroups V of U that are isomorphic to X(A). With N = N I - ( ~ ) Uthe , Frattini argument yields
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IN : UI 2 max IN, V : Vlk, V
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5 max INY(B)V : Vlk, 5 s A k A . V
It seems unlikely that in general the functor X will also be self-coupled on %,' but it clearly will be in the important special cases where for all finite indices A , 3 E 9 the group Y(B) contains a unique subgroup isomorphic to X(B) and the index of this subgroup in Y(B) is bounded over all B for which Y(B) contains a subgroup isomorphic to Y(A). Notice that the functors X = PSL(n, ) and Y = PGL(n, ) meet these conditions for the index family of all locally finite fields. 4.9 Lemma. Let X and Y be coupled functors on the index categories 9
and V l , respectively, and suppose that the group H has a local system A of Jinite subgroups such that for each subgroup L E A there exists a j n i t e index F = F L E Vl with L N Y(F). If A E V and B E V are Jinite objects and if S is afixed subgroup of H isomorphic to X ( A ) then the number of subgroups of H which contain S and are isomorphic to Y(B) is bounded by a function of A and B only. Proof. Choose (if possible) subgroups S and T of H satisfying S C T, S N X ( A ) and T N Y(B). In the local system A of H there are arbitrarily large subgroups L containing T such that L N Y(F) for some finite index FE . We first prove 4.9 for L (and the pair A , B) instead of for H. Let a = u ( T ) denote the number of subgroups of L that are conjugate to T i n L and contain S , dA,Bthe number o f subgroups of T conjugate to S in L, and eA,B the number of subgroups of T isomorphic to S. Then the number of conjugates of S in L is Thus
IL : NLSI
= dA,BIL
: NL T1a-l.
U l L NLSI = d , , ~ l L : NL TI 5 eA,BlL : N, TI. Now in T the set of the eA,Bsubgroups of Twhich are isomorphic to S splits into n conjugacy classes, say, each containing at most IT : SI elements. Hence eA,B nlT : SI. Also since the functors X and Y are coupled, one has
IL NLSI = IL : SI/INLS : SI 2 sA'IL SI. Putting these inequalities together, we obtain a * sA1lL : SI
that is:
alL : NLSl 5 e,,,IL
: NLTI
- n * I T :Sl IL :NLTI I
a = a(T) 5 n
S nlL :Sly
.sA.
Again, since the functors X and Y are coupled, the number of conjugacy
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classes of subgroups of L that are isomorphic to T N Y ( B ) is bounded by rB. Hence the number of subgroups of L that contain S and are isomorphic to T is at most n'rB'sA. Note that this bound is independent of the subgroup L E A of H. Thus the number of subgroups of H that contain S and are isomorphic to T is also at most n.r,.s,. I The following embedding theorem is technically the central result of the whole section.
4.10 Theorem. Let X and Y be finite coupled functors defined on the index categories $? and Vl , respectively, and suppose that G and H a r e groups having local systems .Z and A , respectively, such that for each S E C there exists afinite index E, E %' with S ?: X(Es) and for each L E A there exists afinite index FL E %, with L N Y(F,). rffor every S in C there exists an L in A such that Es E FL, then the group G is isomorphic to some subgroup of the group H. Note that, since X and Y are finite functors, each S and each L is finite; thus the groups G and H in 4.10 are assumed to be locally finite. Further the condition Es E FL implies that some subgroup of L is isomorphic to S ; for since Y is wider than X, the group Y(Es) contains a subgroup isomorphic to X(Es) N S, and Y(eEsFr,) embeds Y ( E s ) into Y(F,) 1: L . In general, there may be many embeddings of S into H the problem is to choose such embeddings in a coherent way as S ranges over Z. Proof. Let S be any element of C. By assumption S N X(E,) for some finite index E, E V, and there exists a subgroup L E A of H such that L N Y(F,) for some finite index FL E V, , Es c F,, and some subgroup of L is isomorphic to S. Let ( T , , . . ., T i , . . .} be a maximal set of pairwise non-conjugate subgroups of H that are isomorphic to Y(E,). The cardinality t of this set is finite, since the group H has the local system A of subgroups L l N Y(FL,) and the number of conjugacy classes in L , of subgroups isomorphic to Y (Es)is at most Y E S . For each of the groups T i let ( S i , , . . ., Sim} be a maximal set of pairwise non-conjugate subgroups of Ti that are isomorphic to S (m is finite since the group Tiis finite). Denote by lijthe set of all isomorphisms of S onto S i j . By 4.9 for each S , E C with S c S , , the set of all subgroups of H containing S , j and isomorphic to y(&,) is finite. For every element a E l i j let Zija(Sl)denote the set of all monomorphisms of S , into subgroups of H isomorphic to Y(E,,) that extend a. Clearly, for any such choice of
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i, j , u and S , E Z the set Zija(Sl)is finite. Conceivably it may be empty, but fortunately the following holds. (*) There exist i', j ' and u E ZiPj. such that for every S , E C with S E S1 the set Zii.j.a(S,) is non-empty. Suppose that S , ,S , are elements of .Z with S c S , c S, . Then the restriction of S , to S , maps the set Zija(S2)into the set Zija(Sl).For if /3 E Zija(S2) then S! E. T c H for some subgroup N Y(Es,) of H . Since Y is wider than X, there exists a subgroup T of Twhich contains S , and is isomorphic to Y(E,y,).Thus BlS, E Zija(S1). If (*) is false, then for every pair i, j and every u E Zij there exists some subgroup WijaE .Z of G such that S c Wijaand the set Zija( Wija)is empty. The local system Z contains a subgroup Z of G which contains all these (finitely many) subgroups Wijaof G. Then for every choice of i, j and cz the set Zija(Z)is empty (as resgiJm: Iija(Z)-+ Zija(Wija) = 0). But by assumption there exists some embedding q of Z into a subgroup isomorphic to Y ( E J of H . Again, since Y is wider than X there exists a subgroup of H which contains s'' and is isomorphic to Y(Es). Consequently, S" is conjugate in H to some S i j and so the set Zija(Z)is non-empty for some u E Zij(S). This completes the proof of (*). Pick one such triple (i', j ' , u ) for which (*) holds and write Z for Zi,,.a. The system {Z(S,), re,: : S , , S , E Z, S E S , E S,} is readily checked to be an inverse system of finite sets and any element of the non-empty inverse limit of this system defines an embedding of G into H . I We are interested in embedding groups of type X ( A ) into groups of type Y ( B ) for infinite indices. Since virtually all our definitions up to now refer only to finite elements of V and V, , we will have to make a definition that ensures that such groups X(A) and Y(A) have local systems of finite subgroups of the same type. Then one can apply 4.10.
Definition. Let V be an index category on the set M. We shall say that $7 is a normal index category if 1 denoting { E E V; E finite and contained in C} by &c for every C E V, then a C = U E E I , Efor every C E V, and b if C E V and E l , E, E b, then there exists a finite index E E &, satisfying El L: E, c E; 2 for any ascending chain 7 of finite elements of %? the subset UTE,Tof A4 belongs to V. A functor F from the index category V on the set M into the category of groups is normal if V is a normal index category, F is a finite functor, and if
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1 whenever A , 3 are finite elements of V such that F ( A ) is isomorphic to some subgroup of F(B) then A E B; 2 for every C E V the group F(C) has a local system C of subgroups such that S E C is isomorphic to some group F(Es) where Es E gC,and the partially ordered sets C and €, are order-isomorphic via the map S H &.
4.11 Lemma. Let F be a normal functor on the index category V on the set M . Then the set M is countable, and f o r every C in V the group F(C) is a countable, locally finite group. Proof. M is the union of all the finite elements of V by the definition of normality for .'%5 Since F is also normal, the finite elements of %' are in one-to-one correspondence with non-isomorphic finite groups. There exist but countably many isomorphism types of finite groups and hence M is countable. The group F(C) has a local system Z of finite subgroups such that C is order isomorphic to gC.Since €, is countable, F(C) is a countable, locally finite group. I We can now prove the rollowing basic corollary of 4.10.
4.1 2 Corollary. Let X and Y be normal coupled functors defined, respectively, on the index categories V and V, and suppose that the group G has a local system Z such that f o r each S in Z there exists a finite index Es E V with S N X(Es). Then there exists some N in %? such that G is isomorphic to a subgroup of Y ( N ) ,and in particular G is countable. Proof. Put N = usssEs; then N is a subset of the set M upon which %? is defined. By 4.11 the subset N is countable. The set {Es; S E C} of finite subsets of N is filtered above by inclusion since C is a local system for G and X is normal. Hence {Es; SEC} contains an ascending sequence the union of which is N. Since V is a normal index category, one has N E V and since the functor Y is wider than X, the group Y ( N ) is defined. Further Y ( N )has a local system A of subgroups isomorphic to Y ( E s)as S ranges over C. Therefore by 4.10 the group G is isomorphic to some subgroup of Y ( N ) . The latter group is countable by 4.11. I In the situation of 4.12, if E is any finite subset of N then E c Ui=1 Esi for some Siin Z. There exists S i n C satisfying Ur= Sic S. By the definition of normality Esi E Es. Hence S contains a subgroup isomorphic to X(E). Thus we may assume that the map S I+ Es maps C onto 8,.Further, the map is order preserving. (But clearly it need not be one-to-one, PSL(2,q') contains many subgroups isomorphic to PSL(2, q).) Call a group G an X-group of index N E V, if G has a local system C for
u;=
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which there exists a map S H Es of Z onto Q, satisfying S N X(Es) for all S E Z. Clearly X(N) is an X-group of index N. Are all the X-groups of index N isomorphic? If not, are all those X-groups of index N for which there exists a local system C of the above type with the map S H Es a bijection isomorphic (to X(N))? If X is self-coupled the answer is yes. 4.13 Theorem. If X is a normal self-coupled functor and if G is an X-group of index N then G is isomorphic to X(N). Further every injection of X(N) into itself is an automorphism.
Proof. Let H = X(N). By 4.10 there exists an embedding 4 of G into H. By the same result, with the roles of G and H interchanged, there exists an embedding $ of H into G. Hence for both parts of the theorem it suffices to prove that any embedding 8 of an X group K of index N into itself is onto. Let y denote the first uncountable ordinal. For every ordinal a < y we define inductively X-groups K, of index Nand embeddings A,+ ;K, -+ K,, such that K, = l&,<,K, for limit ordinals fi. Take Kl = K. Suppose K, is defined for all a < fi and that I, is defined for all non limit ordinals a < p. If is a limit ordinal define K,= b m < , K a The . group K, is the union of its X-subgroups of index Nand so it is itself an X-group of index N. Suppose now that p- 1 exists. By 4.10 there exist embeddings 'p, : Kg-l+Kand$,:K-+K,-l.DefineKg = K , - l a n d l e t I , = qfith,hp.We have now defined K, and I,, for all a < y. Put Ky = 1ha K,.
,
Definition. Let X and Y be normal coupled functors defined respectively on V and Vl . Then we call X, Y a 1-class pair if 1 Whenever A E %?and B E C, are finite indices the, group Y ( B ) contains at most one conjugacy class of subgroups isomorphic to Y(A) (that is r A 5 l); 2 Whenever A E %' and B E Vl are finite indices with A E B, then any two subgroups of Y ( B ) isomorphic to X(A) are conjugate in Y(B);
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3 If A E $2 is a finite index, then Y ( A ) contains a unique subgroup isomorphic to X(A). Notice that if X, Y is a 1-class pair, then in the above notation YIce,YI, is also a 1-class pair. 4.14 Lemma. Let {Li}isN be an ascending sequence of subgroups or the group G such that the normalizer NGL , is jinite and whenever i < j any subgroup of L j conjugate to L i in G is already conkgate to L in L j . Let { be a second ascending sequence of subgroups of G such that for each i g N the subgroups L, and M iare conjugate in G. Then there exists an element g of G such that Lig = M ifor every i E N.
Proof. Put S j = { g E G : Ljg = Mi)- and T j = ( 9 E G :Lig = M i for every i with 1 5 i s j } . By assumption the sets Sj are not empty, and one has Tj E S j for all j E N. Clearly for each j E N
Tj E Tj-, E
. . . E: Tl = S , .
Now S , is a coset in X of the normalizer N,L,, and so is finite. Hence each T j ,j E N , is a finite set and the intersection T = n j G N Tisj non-empty if and only if each of the sets Ti is non-empty. In order to prove for a fixed natural number that the set Tj is not empty we show by induction on k, 0 5 k 5 j - 1 , that there exists an element s of G with L: = M ifor j - k 5 i 5 j . This is clearly true for k = 0. Assume that this is true for all natural numbers k 5 r( < j - l), so that there is an element s E S j with L; = M i for all i with j - r 5 i 5 j . Then the groups L j - r - l and My-',-, are subgroups of L j - , and are conjugate in G by hypothesis. But then by assumption there is an element t of L j - , such that MS-1 L>-r-l = j - r - l . Thus we have Lfi"_,-l= A t j - , - , , and if j - r 5 i S j also LfiT = L; = M i . Hence the element ts E S j is such that Ly = M ifor j - (r+ 1) 5 i 5 j . Therefore Ti is not empty. I 4.15 Theorem. Let X, Y be a 1-class pair of functors dejined on the index categories $2, respectively. If N E %? and C E V, satisfy N E C, then all the X-subgroups of Y(C) of index N are conjugate in Y(C). In particular all the X-groups of index N are isomorphic (to X ( N ) ) .
Proof. Let G be any X-subgroup of index N of Y(C). The group Y(C) contains a subgroup H isomorphic to X ( N ) . We prove that G and H are conjugate. By 4.12 the group G is countable. Thus G contains an ascending sequence {A,}nsNof finite subgroups such that G = U,A, and for each n in N, A, N X(E,) for some En E gN.By the normality of X the set {En},sN
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is an ascending sequence in gN and UnEn= N. Now 2 of the definition of normality ensures that H contains an ascending sequence {B,},ENof finite subgroups such that H = U n B nand satisfying B, N X(E,) for every n. For each natural number n define a chainfor A , to be finite ascending chain { U j ] : , of subgroups of Y(C) such that U j N Y ( E j ) and A j c Ui for 1 6 j S n. Denote by B, the set of all chains for A,. By 2 of the definition of 1-class pair there exists a subgroup U, of Y(C) isomorphic to Y(E,) and containing A,. In the same way U, contains a subgroup U,-l isomorphic to Y(E,-l) and containing A,-l. A simple inductive argument shows that (5, is not empty for any n E N. By 4.9 the set B, is also finite. If m > n there is an obvious map of B, into &, simply obtained by deleting from any chain {Ui},, €6, the terms U,,, ,. . ., Urn.It is equally clear that this collection of sets B, and maps is an inverse system. The inverse limit (5 of this system is not empty. Any element of 0; is (or rather, may be interpreted as) an ascending sequence {L,JnEN of finite subgroups of Y(C) such that A , E L , N Y(E,) for every natural number n. Put L = UnENLn. For m > n every subgroup of L, isomorphic to L , is conjugate in L, to L,. Further, by the definition of coupled functors JNycoA1I5 lAIIsA,,which is finite. Since A , is characteristic in L we have that Ny(,,L, is finite, and the sequence {L,} satisfies the conditions required for 4.14. In the same way there exists a sequence (M,,} of subgroups of Y(C) such that for every n we have B, -c M , N Y (En).Further, by the definition of I-class pair the subgroups L , and M , are conjugate in Y(C). Hence by 4.14 there exists an element y of Y(C) satisfyingLi = M, for every n E N. But A , (resp. B,) is the only subgroup of L , (resp. M,,) isomorphic to X(E,). Hence A: = B, for every n and therefore Gy = H . I Although the notion of coupled functors arose from the study of locally finite infinite simple groups and all our applications will be made within that context, the notion itself is much more general and has nothing to do with simple groups.
Exampies. 1. Let V = Vl be the set of all (isomorphism types of) locally finite fields and z any fixed natural number. For each CEV denote by X(C) the Frobenius group obtained by taking the split extension of the additive group Cf of the field C by the multiplicative group C*. The map X clearly extends to a finite functor on V. For any finite field F let P' denote the (unique) extension of F of degree z and for C E V let C' denote the composite of all the F' as F ranges over all the finite subfields of C. Define Y by Y(C) = X(C'). Then X and Y are coupled normal functors for which
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1 for all finite A in V. In particular, the case z = 1 shows that
X is self-coupled.
2. Let (pi}ieNbe an enumeration of the odd primes and for each i~ N let Pibe a finite abelian pi-group admitting the cyclic group (f) of order four as a group of fixed-point-free automorphisms. Put V = V, equal to the set of all non-empty subsets of N. For each C E V let X(C)denote the split extension of the direct product of the P i for i E C by the group (f ') acting componentwise, and let Y(C) denote the split extension of the direct product of the Pifor i E C by the group ( f ) , also acting componentwise. Then X and Y can be extended to form a 1-class pair of functors. We leave it to the reader to check the details of these rather trivial examples (both of which clearly allow many variations) and return to our main theme. It is an easy consequence of standard representation theory that PSL(n, ) and PGL(n, ) are coupled normal functors on the set of all locally finite fields of order exceeding max (5, n!]. The point of restrictingourselves to fields of order greater than max (5, n ! ] is that if PSL(n,p') is isomorphic to a subgroup G, of PGL(n, q s )wherep and q are distinct primes then pr S max (5, n!}. To see this let P,,be a Sylow p-subgroup of G,. Since p # q, Curtis and Reiner [I], 52.1 implies that P, contains an abelian normal subgroup of index dividing n! It is very easy to check that every abelian subgroup of P, has index at least pr and clearly P3 is a homomorphic image of P, for all n 2 3. Hence either p' S n! or n = 2. But now a glance at the list of subgroups of PSL(2, @),given in Huppert [I], 11.8.27 shows that if n = 2 then p' S 5. Thus we have taken our fields large enough to avoid mixing the characteristics. (One can reduce the bound somewhat, but we are always free to dispose of a finite number of finite indices, so there is no advantage for us to do this.) For rA we simply take the number of projective representations of PGL(n, A ) of degree n and characteristic p = char A (see Curtis and Reiner [l], 9 53), such a representation being necessarily absolutely irreducible if JAl = pr > 3, since PGL(n, A ) contains an elementary abelian t-group of rank n - 1 for some prime t dividing pr- 1. The centralizer in PGL(n, B ) of any such representation of PSL(n, A ) has order at most n2,since a set of cOset representatives of this centralizer in GL(n, B ) is linearly independent over By and hence s, may be taken to be n2 times the order of the outer automorphism group of PSL(n, A ) . It remains only to point out that PGL(n, ) is wider than PSL(n, ). All that is in doubt, given that PSL(n, p') is isomorphic to a subgroup x
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of PGL(n,p'), is whether there exists a subgroup Y of PGL(n, p') containing X and isomorphic to PGL(n, p'). This follows from the facts that a Sylow psubgroupP of X lies in a unique Sylow p-subgroup Q of PGL(n, p'), that a complement H of P in its normalizer N,P lies in a unique complement of Q in its normalizer N P ~ ~ps) ( "Q,, and that H has exactly n fixed points in the projective space on GF(ps)on which PGL(n,p s )operates. In view of the comments after the definition of coupled functors we have: 4.16 Proposition. The finite normal functors PSL(n, ) and PGL(n, ) are self-coupled on the index category of all (isomorphism types o f ) locally jinite3elds of order exceeding max (5, n!}. Let C denote the Chevalley functor attached to a given finite dimensiona1 simple Lie algebra over the complex numbers, which associates to every perfect field (with sufficiently many elements) the simple group C(F). The functor C is certainly normal on the index category of sufficiently large locally finite fields.
Question IV.8. Is every Chevally functor self-coupled on the index category of all (isomorphism types o f ) suficiently large locally finite fields? I f the answer to this question is positive then 4.13 implies that for no locally finite field F does the group C ( F ) contain a proper subgroup isomorphic to itself. Does this statement hold for the group C ( F ) if F is an absolutely algebraic field of characteristic zero? This does not seem to be known even for the groups PSL(n, F). Let n denote the smallest integer independent of F for which C ( F ) has a non-trivial projective representation of degree n over the algebraic closure of F. Question IV.9. Are the functors C and PGL(n, ) coupled on the index category of all (isomorphism types o f ) suficiently large locally jinite fields? If one knew that PGL(n, ) was wider than C this would indeed be the case (see Steinberg [l], 7.5). It seems harder to provide examples of 1-class pairs. We present the following pick from the literature. 4.17 Proposition. The following pairs of coupled normal functors are 1class pairs on suitable index categories of locally finite3elds. a PSL(2, ), PGL(2, ) a' PGL(2, ), PGL(2, ) b PSL(3, ), PGL(3, ) b' PGL(3, ), PGL(3, ) c PSU(3, ), PU(3, ) c' PU(3, ), PU(3, ) d PSU(3, ), PGL(3, ) e Sz( 1, Sz( ).
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Proof. Case a is contained in Dickson [ l ] Chapter XII, especially 0 260, see also Huppert [ I ] Chapter 11, 0 8. For fields of odd characteric cases b, c and d are contained in the remarkable paper Mitchell [I]; for fields of characteristic two these three cases are contained in R. W. Hartley [l]. Case e is contained in Suzuki [l]. For x = a, b and c the case x’ follows from the case x by the remark after the definition of 1-class pair. 1 Little seems to be known about 1-class pairs beyond what has been said above. Apparently it is an open question whether PSL(n, ) and PGL (n, ) form a I-class pair on an index category of (sufficiently large) locally finite fields. Does there exist for every Chevalley functor C a functor C, such that C and C, form a 1-class pair on an index category of sufficiently large locally finite fields? It is convenient for later reference to combine into a single statement all the results from this section that we shall use again later and the reader need remember only this. The following theorem is a direct consequence of 4.16, 4.11, 4.13 and 4.15. 4.18 Theorem. Zf the locally Jinite group G has a local system Z consisting of finite groups S that consistently are isomorphic to groups of type PSL(n, Fs),PGL(n, Fs),PSU(3, Fs),or Sz(Fs)f o r suitable jinite jields Fs, then there exists a locally Jinite field F (essentially the composite of the finite fields Fsfor S E Z) such that, according to the type of the groups S E C, the group G is isomorphic to PSL(n, F), PGL(n, F), PSU(3, F) or Sz(F). Section C. Immediate applications to a few structure theorems Many of the theorems characterizing certain classes of finite simple groups still make sense when re-worded for locally finite groups. Can we prove these new statements? In the following we shall give a few instances where this can be done. The proofs will consist essentially in applying the identification result 4.18 appertaining to these particular simple groups. This programme is very much an open-ended operation since with every new result on finite simple groups the corresponding question is posed for locally finite groups. The results given here - most of them are needed in applications later - are just examples of how one type of such a theorem can be extended to locally finite groups. Statements on permutation groups involving multiple transitivity still pose serious problems. 4.19 Theorem. Let G be a locally finite group such that the finite 2-subgroups of G are cyclic or dihedral, and denote by O G the greatest normal sub-
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group of G consisting of elements of odd order. Then one of the following statements holds: 1 Thefactor group G/OG is a iocally cyclic or a locally dihedral 2-group. 2 Thefactor group G/OG is isomorphic to the alternating group of degree seven. 3 There exists a iocaiiy JiniteBeid F of odd characteristic such that the factoi group G/OG is isomorphic to a subgroup of the group PrL(2, F ) containing PSL(2, F). This result extends (and its proof uses) the main result of Gorenstein and Walter [2].
Proof. If H/OG is a finite 2-subgroup of G/OG, then there is a finite 2-subgroup H , of G such that H = H,OG. Thus H , N H/OG and every finite 2-subgroup of G/OG is either cyclic or dihedral. If the group G/OG is finite, then the main result of Gorenstein and Walter [2] applies. If the group G/OG is locally soluble, then it is either a 2-group, and thus locally cyclic or locallydihedral, or it is isomorphic to PSL(2, 3) or to PGL(2, 3), and thus is finite. Hence we may assume that the factor group G/OG is infinite and that it has a finite, non-soluble subgroup. By 1.B.9. for every finite subgroup R there is a finite subgroup S of G/OG containing R such that for every finite subgroup T of G/OG containing S one has R n OT = (l}. This means in particular that there is no bound on the index IT : OTI as T ranges over the finite subgroups containing S of the infinite group G/OG. Now let R be any finite, non-soluble subgroup of G/OG with IRI 2 7!. Then by the isomorphism theorems, R
N
R / ( R n O T ) ‘v ROTIOT
and
R‘
2:
c T/OT
R‘O(T’)/O(T’)E T’/O(T’)
The main result of Gorenstein and Walter [2] gives that each such factor group T ’ / O ( T ’ )is isomorphic to some group PSL(2,q). Since for every nonsoluble subgroup of PSL(2, q ) the subgroup ( l } is the only normal subgroup of odd order (cf. Huppert [l], II.8.27), we get that O ( R ‘ ) =
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G/OG is trivial, and G/OG embeds into PrL(2, F ) since this group is the full automorphism group of PSL(2, F). I There are other such results, inferring information on the structure of the group G from information on the structure of the Sylow 2-subgroups of G (and not from their embedding in G). We give one more example of such a result; for finite groups it is the main result of Alperin, Brauer, Gorenstein
PI-
4.20 Theorem. If the locally jinite simple group G has jinite Sylow 2-subgroups that are quasi-dihedral, then either G is isomorphic to theMathieu group MI of degree eleven, or there is a locally jinite jield F of odd characteristic, which is not quadratically closed, such that G is isomorphic either to PSL(3, F ) or to PSU(3, F). For the locally finite field F of odd characteristic the group PSL(3, F ) has (finite) quasi-dihedral Sylow 2-subgroups exactly if the characteristic p of F satisfies p E - 1 (mod 4), and if F does not contain the field GF(pZ). In the other cases the Sylow 2-subgroups of PSL(3, F ) are isomorphic to the wreath product C 2 Cz where C is a locally cyclic 2-group. In order to define the unitary group PSU(3, F ) one needs a quadratic extension of the field F, so that the field F is not quadratically closed. For the locally finite field F of odd characteristic the assumption that F is not quadratically closed implies that the Sylow 2-subgroups of the groups GL(n, I;) are finite. Thus, for F any such locally finite field of odd characteristic, the Sylow 2-subgroups of PSU(3, F ) are finite. They are quasi-dihedral exactly if either thecharacteristic p of F satisfies p = 1 (mod 4) or if F contains the field GF(p2).From Alperin we learn that the work on the characterisation of those finite simple groups which have their Sylow 2-subgroups isomorphic to a wreath product C l C, with C some finite cyclic 2-group has now been completed. These groups are of type PSL(3, q ) and PSU(3, q ) only. Thus we get the following result as a companion to 4.20. 4.21 Theorem. If the Iocallyjinite simple group G hasjinite Sylow 2-subgroups which are isomorphic to a wreath product C C , with C a cyclic 2group, then there is a locallyjinite field, which is not quadratically closed, such that G is isomorphic to PSL(3, F ) or PSU(3, F). It should be observed that in view of the results of Alperin, Brauer, Gorenstein [l] and [2] the only finite sections of the wreath product C l C,, where C is the infinite, locally cyclic 2-group, which are isomorphic to Sylow 2subgroups of finite simple groups are the finite dihedral groups, the quasidihedral groups, and the wreath products of finite cyclic groups with the
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group of order two. Thus we shall combine 4.20 and 4.21 with 4.19 (and it will be in this form that we prove 4.20 and 4.21). 4.22 Theorem. Let C denote a Priifer 2-group. If in the locallyfinite, simple group G all thefinite 2-subgroups of G are isomorphic to sections of the wreath product C t C , , then either the group G is isomorphic to the alternating group of degree seven or to the Mathieu group M i l of degree eleven, or there is a IocallyJinitefield F odd characteristic such that G is isomorphic to one of the groups PSL(2, F), PSU(3, F), or PSL(3, F).
Proof. We may assume that G is infinite. First assume that the simple group G is countable, and according to 4.5 let {Gn}nsM be a sequence of finite subgroups of G with G, c G,, and G = UnsN G, such that there is a maximal normal subgroup M,,, of G,,, with G,, n M,,,, = (1). Since the Sylow 2-subgroups of the finite simple groups GJM, are dihedral, quasidihedral or wreath products of finite cyclic groups with the group of order two, the results of Gorenstein and Walter [2] and Alperin, Brauer, Gorenstein [l] and [2]yield that 4.22 holds for the finite simple groups G,/M,. As each of the simple groups G,/M, has a faithful, linear representation of a degree independent of n over some field, there is a field F by 1.L.9 such that G has a faithful linear representation over F. But then 4.6 implies that almost all of the subgroups G,, of G are already simple. If infinitely many of the groups G are of the form PSL(2, ), then by 4.18 the group G itself is isomorphic to PSL(2, F ) for a suitable locally finite field F of odd characteristic. If infinitely many of the groups G, are of the form PSU(3, ) then by 4.18 the group G itself is isomorphic to PSU(3, F ) for some locally finite field F of odd characteristic. And if infinitely many of the groups G, are of the form PSL(3, ) then by 4.18 the group G itself is isomorphic to PSL(3, F ) for some locally finite field F of odd characteristic. NOW,in the general case, by the preceding argument and 4.4 the infinite, locally finite, simple group G has a local system of simple subgroups of type PSL(2, ), PSU(3, ), or PSL(3, ) over infinite, locally finite fields of odd characteristic. Now for such fields PSL(2, ) does not involve groups of type PSU(3, ) and PSL(3, ) nor does PSU(3, ) involve PSL(3, ). Thus we get that there is a local system of such subgroups all of the same type. This in turn implies that there is a local system of finite simple subgroups of G all of the same type. Application of 4.11 yields that G is countable anyway. (Alternatively, the countability of G in the general case may be proved by using the fact that the groups of type PSL(2, ) PSU(3, ), and PSL(3, ) of this local system admit faithful, linear repre-
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sentations of bounded degrees, and thus by 1.L.9 the group G admits a faithful linear representation. Now 1.L.2 yields that G is countable). 1 The next result deals with groups of small type over locally finite fields of characteristic two. 4.23 Theorem. Let G be a locally Jinite, simple group in which every 2subgroup is abelian. If G contains elementary abelian 2-subgroups of order 16 then there is a locallyjnite$eld F of characteristic two (containing at least 16 elements) such that G is isomorphic to PSL(2, F). For finite groups, this is a special case of the main result of Walter [l]. We leave the extension to locally finite groups as an Exercise. If in the countably infinite, locally finite simple group G every 2-subgroup is abelian, and if some maximal elementary abelian 2-subgroup of G has order eight, then - by Walter [l] and 4.5 - G is a limit of an ascending sequence of finite simple R-groups (which are suspected to coincide with the simple groups 'G,( ) of Ree [l] over finite fields of characteristic three which do not contain the field GF(9)). For every involution i of G the centralizer CGican be determined from 4.18: it is isomorphic to ( i ) x PSL (2, F ) where F is a locally finite field of characteric three which does not contain the field GF(9). The Sylow 2-subgroups of G are elementary abelian of order eight. F orp # 2 , 3 the Sylowp-subgroups are locally cyclic. The Sylow 3-subgroups are nilpotent of class three and of exponent nine. If some maximal elementary abelian 2-subgroup of a locally finite simple group G in which every 2-subgroup is abelian, has order four, then this is a maximal 2-subgroup of G, and by 4.19 the group G is isomorphic to PSL(2, F ) for a suitable locally finite field F of odd characteristic. The following results all deal with conditions restricting the way certain subgroups are embedded. Definition. The proper subgroup H of the locally finite group G is strongly embedded in G if H contains an involution, and if for every element g E G H \ every element of the intersection H n H 9 has odd order. If G is a locally finite Frobenius group, then the Frobenius complement C of G is strongly embedded in G if C contains an involution (which then is unique). Thus, in a sense, the notion of a strongly embedded subgroup is obtained from that of a Frobenius complement by relaxing the condition that the intersection C n Cg be trivial for each g E GC \. That one can still prove an incisive result in this more rarified situation was discovered by Bender [I J who proved the result below for finite groups. In his proof Ben-
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C
der reduces the situation to one where results of Suzuki characterizing certain doubly transitive groups become applicable. In a way, Bender has freed Suzuki's results from the assumption of double transitivity, making possible the following extension of his result (and also of Suzuki's results) to locally finite groups. 4.24 Theorem. If the locally finite group G has a strongly embedded subgroup H , then one of the following properties holds: 1 Finite 2-subgroups of G are cyclic or (generalised) quaternion; 2 OG = H g ,and the factor group GjOG has a unique minimal normal subgroup M/OG which is isomorphic to one of the groups PSL(2, F), PSU(3, F), or Sz(F) f o r a suitable locallyfinite field F of characterlstic two and G / M is an abelian group the elements of which all have odd order.
OgEG
Observe that the first case corresponds to the Frobenius situati on (compare 1.5.2). Proof. If in the group G there is no elementary abelian subgroup of order four, then clearly 1 pertains. Assume that in G there is an elementary abelian 2-subgroup V or order four. Let S be any finite subgroup of G satisfying VGSandlfHnSfS. Then clearly the intersection H n S is strongly embedded in S. By Bender [ I ] one has that 0s = n s e S ( Hn S)" and that the commutator group Ms = (S/OS)' is the unique minimal normal subgroup of S/OS, is simple and is isomorphic to some PSL(2, 2"), PSU(3, 2"), or Sz(2"). Let Z be the local system of G consisting of all such subgroups S. If there is a subgroup S E C of G such that Ms is isomorphic to PSU(3, 2") (respectively to Sz(2")), let Z, = {XEZ; S E X } . As neither PSL(2, 2@)nor S~(2~)(respectively, neither PSL(2, 2@)nor PSU(3,2*)) involves the group PSU(3, 2")(respectively Sz(2")), it follows that the simple group M x for each X E Clis of the same type as M s . If for every S E C the group Ms is of type PSL(2,2"), then put Z1 = C. Thus we have achieved that in any case, for X , Y EZl the groups M x and M y are of the same type. Observe that for every simple group M of type PSL(2, 27, PSU(3, 2"), or Sz(2") every subgroup T of M , such that T/OT is a simple group of the same type as M , satisfies OT = (1). For every subgroup S E Zl of G denote by S the subgroup of S generated by 05'together with all the elements Sof order a power of two. Then the set 2= S E Z,} forms a local system for' the subgroup G, of G which is generated by OG together with all the elements of G of order a power of two.
{s;
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APPLICATIONS TO STRUCTURE THEOREMS
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For X , Y E with X 5 Y the result of Bender [I] implies that the group X / ( X n O Y ) is isomorphic to a subgroup of the simple group Y/OY, and by the above observation this is possible only if X n O Y = O X . Thus the set {OX; X E is a local system of OG. Again appealing to Bender [l 1, we have obtained: a O G , = ngEG,(H"G,)9 = n g o G H g , and b The group G,/OG has a local system of simple groups isomorphic to M , where S ranges over Z,. Now, for each of the three possible types 4.18 is applicable and yields the asserted structure for GJOG; clearly G,/OG is the only minimal normal sub-group of G/OG. I In analysing the structure of quite general locally finite groups, one often encounters the existence of strongly embedded subgroups as a special case that one can dispose of thanks to the powerful structure theorem 4.24. We shall give only one rather special example for this occurrence which is well known but will be useful to us later.
z}
4.25 Proposition. If the subgroup H of the locallyfinite group G contains an involution and i f f o r every involution i of H every involution of G which centralizes i is contained in H, then either there exists a strongly embedded subgroup of G or the subgroup H , generated b y all the involutions of H is a normal subgroup of G. It should be noted that the following proof only uses that the group G is periodic (and the obvious extension of the notion of a strongly embedded subgroup to periodic groups). Unfortunately it seems unlikely that the 4.24 extends to periodic groups.
Proof. For any involution i E H, let %,(i) denote the set of all involutions of G commuting with i. Inductively, define and put q i )=
u
Un(i).
nsrm
Clearly, for any involution j E U ( i ) one has U ( j ) = U ( i ) ; and if the involution j~ G is not conjugate to i, then the group (0) has even order and thus contains an involution centralizing both i andj; hence j E U , ( i ) . By the assumption on H one has U ( i ) E H for every involution i of H. Denote by C(i)the subgroup generated by the set U ( i )of involutions. Consider any involution j~ C ( i ) : Either %(j) = V(i), or the element 0 has odd order, and hence there exists an element c E (g) E C(i) such that i' = j .
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C
But then one has (%?(i))’ = %(i“) = %(j),
and hence
C ( j ) = ”)= (V(i)) = C(i)
Put N ( i ) = N,C(i); we show that N ( i ) = NGN(i): Assume that there is an element nENGN(i)\N(i); then one has C ( i) # (C(i))”. Put D = C( i ) n (C(i)>”.Since C(i)is generated by involutions, by the preceding remark, there are involutions u E C(i)\D
and v E (C(i))”\D.
In the product C(i)(C(i)>”of normal subgroups of N ( i ) the involutions u and u cannot be conjugate, and thus v
E%?(u)
c
C(u) = C ( i ) and U E % ( V= ) (C(i))”
contrary to the choice of the involutions u and v. The subgroup C ( i ) contains all the involutions of the subgroup N(i); for i f j were an involution in N(i)\C(i) then the subgroup ( i j ) would contain an involution centralising i andj, and j would belong to U,(i) s C(i), a contradiction. Thus, if N ( i ) = G, then one has C ( i) = H,, the subgroup of H generated by all the involutions of G and hence also of H . If, on the other hand, G # N ( i ) ,then the subgroup N ( i ) is strongly embedded into G. Let g E G\N(i) and consider the intersection N ( i ) n (N(i))9. Every involution j of this intersection must belong to the intersection C ( i ) n (C(i))B. But then C ( i ) = C ( j ) = (C(i)>g,and g E N,C(i) = N(i), contrary to the choice of the element g. Thus there cannot be any involution in the intersection N ( i ) n (N(i))g. I Under the name of 2-injinite isolation we shall study in Chapter 5 a slightly different embedding condition for subgroups which is related to strong embed ability . Another sort of embedding condition for 2-subgroups is to prescribe the structure of centralizers of involutions. In a series of papers Suzuki has studied finite groups in which the centralizer of every involution is first a 2-group, then nilpotent, and finally 2-closed. This study culminates in the following result. 4.26 Theorem. If for every involution i of the locally jinite group G the
centralizer C,i is 2-closed, then for the structure of G one has the following alternatives: Either 1 The group G is locally soluble; then it is either 2-closed or its Sylow 2-sub-
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groups are locally cyclic or locaIly (generalized) guaternion and have trivial intersection; or 2 The group G has two normal subgroups GI and G, with 1 ) E G, c G, c G such that Gz is 2-closed, such that the factor group GIG, does not contain any involutions, and such that the factor group G,/G2 is either simple or isomorphic to the non-split extension of PSL(2, 9) by the cyclic group of order two. If GI/G2 is simple, then it is isomorphic to one of the following groups: a. PSL(2, p ) where p is a Fermat or Mersenne prime (> 3); b. PSL(2, 9); c. PSL(2, F), PSU(3, F), PSL(3, F), S z ( F ) , where F is a suitable locally finite field of characteristic two.
<
This result extends Theorem 2 of Suzuki [2] to locally finite groups. Proof. From Theorem 2 of Suzuki [2] it is clear that the structure of a locally soluble group in which the centralizers of all involutions are 2-closed i s given in 1. Thus let us assume that the group G is not locally soluble, and let C be a local system of G consisting of finite, non-soluble subgroups. Observe that by Theorem 2 of Suzuki [2] for each of the groups S E C statement 2 applies. Denote by G , the subgroup generated by all the elements of G of order a power of two. For each S E C, let T, denote a Sylow 2-subgroup of S. Then the set C, = ((T,)'; S E C) will be a local system of G1 . Put G, = O,, 2 , G, . If the quotient group G,/G2 is finite, then again the result follows from Theorem 2 of Suzuki [2]. Thus we may assume that G,/G2 is an infinite group. For every finite subgroup S E C ~ there , is by 1.B.9 a subgroup T E Z ~ with S s T such that S n G, = S n O,, ,'T, and S / ( S n G,) is isomorphic to a subgroup of T/O,, ,,T, which is a finite simple group in the above list. The order of TIO,, ,, T is not bounded, since G,/G2 is infinite. If p andp, are distinct primes exceeding five, then the group PSL(2, p ) is not involved in the group PSL(2, p , ) . Therefore the simple group T/O,,,,T will, in general, be one of type c, that is, of characteristic two. If there is a subgroup S E Z, with SjO,, ,.S N PSL(3, 2") (or Sz(2")), for some a, then put ,X2 = { T E Z , ; S E T ) . Since the group PSL(3, 2") (respectively Sz(2")) is not involved in the groups PSL(2, 2B),PSU(3, 2'), and S~(2~)(respectively PSL(2, 2 9 , PSU(3, 2@)and PSL(3, 2')), one has for every L E C3an isomorphism L/o,, ,,L
N
PSL(~, 2") (respectively L/o,, ,,L
N
sz(2"')) for some a'.
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If there is no such subgroup X E Zlwith X/O,, ,, X N PSL(3,2")(or Sz(2")), for some c(, but if there is a subgroup S E C, with SjO,, ,, N PSU(3,2")for some c(, then put C, = ( L E C ~S; c L } . Since no group PSU(3, 2") is involved in a group PSL(2, 28), we get for every L E C, an isomorphism L / O , , , , L N- PSU(3, 2"') for some a'. If no such subgroup S exists either, then put Z, = E l . Since in a simple group X of type PSL(3, ), Sz( ), PSU(3, ) or PSL(2, ) over finite fields of characteristic two there is no subgroup U such that O,, 2 r U # (1) and such that U/O,, 2 , U is of the same type as X , we get for every subgroup S E C, of G , the relation O,, ,, S = S n G , . Thus the group G J G , has a local system 2, consisting of the finite simple groups = S/O,, , S, S E Z, all of the same type. Now 4.18 is applicable, and hence the group G,/G2 is isomorphic to PSL(3, F), Sz(F),PSU(3, F), or PSL(2, F) for some locally finite field F of characteristic two. I There is another, equally obvious, weakening of the condition that the centralizer of every involution be a 2-group7 namely the condition that the centralizer of every involution be 2'-closed. It is this condition that is studied in Gorenstein [2]. Observe that in all the groups PGL(2, F ) , where F is a locally finite field of odd characteristic, the centralizer of every involution is 2'-closed. For convenience, we introduce some notation first: let F be a locally finite field of odd characteristic, which is not quadratically closed, but which contains the quadratic extension of the prime field. The group PrL(2, F) is the split extension of the group PGL(2, F) by the group A of all automorphisms of F. By the choice of F the group A contains an involution a, so that the abelian factor group PTL(2, F)/PSL(2, F) contains an elementary abelian subgroup of order four. To each of the three involutions of this group there corresponds an extension of PSL(2, F) by the group of order two, one being PGL(2, F),another being the split extension PSL(2, F)(c(), and the third we denote by PGL*(2, F).The finite Sylow 2-subgroups of PGL*(2, F) are quasi-dihedral.
s
4.27 Theorem. Let G be a locallyjnite group such that f o r every involution i of G the centralizer C G i is 2'-closed. Then the group G is either locally soluble or one of the following structural properties holds f o r G : 1. The group GjOG has a normal subgroup NjOG such that the factor group GIN does not contain any involution and either N / O G is isomorphic to one of the groups PSL(2, F), PGL(2, F), or PGL*(2, F)f o r some locallyfinite JieId F of odd characteristic, or N / O G is isomorphic to the alternating group
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of degree seven; 2. The group G/OG is isomorphic to PSL(2, F ) or to the Suzuki group S z ( F ) for some locally finite field F of characteristic two, or it is isomorphic to
PSL(3, 4); 3. The subgroup O G = O,, G is properly contained in O,,,, G, and G/O,,,, G is isomorphic to PSL(2, F ) or to Sz(F) for some locally finite field of characteristic two. Proof. Consider the factor group G/OG- it clearly satisfies the assumptions of 4.27. In view of Gorenstein [2], we may assume that the group G/OG is infiniteand not locally soluble. Let N/OG be the subgroup of G/OG generated by the elements of order a power of two in G/OG and let C be the local system of N/OG consisting of all those finite non-soluble subgroups of N/OG which are generated by their elements of order a power of two. The statement of 4.27 applies to SEC. For each X E C there exists by 1.B.9 a subgroup Y E Z such that X c Y and X n OY = 1 ; hence X is isomorphic to a subgroup of Y / OY.But for a non-solublesubgroup X of PSL(2, q), PGL(2, q), PGL*(2, q), or Sz(q), q a power of some prime, one has that the largest nilpotent normal subgroup, the Fitting subgroup nX = (1); and for an extension of a 2-group by PSL(2, q ) or by Sz(q), q a power of two, every non-soluble subgroup X satisfies O X = (1). NOW,by 4.18, the group N/O,.,,G is isomorphic to PSL(2, F), PGL(2, F), PGL*(2, F), or Sz(F) for some locally finite field F, and if the characteristic of F is odd, then O G = O,,, G. (In the case of PGL"(2, F ) one considers derived groups as in the proof of 4.19.) If the characteristic of the field F is two, then we have to show that G = N. For this purpose, consider the local system Zlof G/OG consisting of all finite subgroups of G/OG containing a non-soluble subgroup S. By the preceding remarks it is clear that every X E Z1has only one non-abelian chief factor, and this is isomorphic to PSL(2, q ) or to S z ( q ) for q a power of two. As the statement of 4.27 applies to finite groups X E C, (by Gorenstein [2]), one sees that each group X E Zlis generated by its elements of order a power of two. Thus we indeed have G = N in this case. I In the recent paper Sitnikov f l ] the two preceding results are generalized, at least as far as simple groups are concerned. This is a further step in the direction of investigating those groups in which the centralizer of every involution is locally soluble of 2-length one. For convenience, let us call cgroups those locally finite groups in which the centralizer of every involution is 2-closed. Then the following extends the results of Sitnikov [I] to locally finite groups.
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4.28 Theorem. Let G be a locallyfinite group such that f o r euery involution i of G the centralizer CGi has a normal 2-subgroup N ( i ) in its hypercentre such that the factor group C G i / N ( i )is a (possibly trivial) Frobenius group such that either the Frobenius kernel or the complement is a 2-group. Then one of the following holds: 1. G/QG is a C-group; 2. G/QG is isomorphic to the alternating group of degree seven; 3. There is a locally finite field F of odd characteristic such that G/QG is isomorphic to PSL(2, F), PGL(2, F ) , or PGL*(2, F). It should be pointed out that the simple groups in these three classes of groups, viz. the groups PSL(2, F ) , F any locally finite field with more than three elements, PSL(3, F ) , PSU(3, F ) , or S z ( F ) for suitable locally finite fields of characteristic two, and the alternating group of seven elements all satisfy the above condition on the centralizers of the involutions, in fact the centralizers of involutions in these groups are all either 2-closed or 2'closed. Accepting the proof of 4.28 in Sitnikov [4] for finite groups, the reader is confronted with the Exercise to prove the extension to locally finite groups. Probably the characterizations of certain (infinite families of) finite simple groups by rather precise structurai information on the centralizers of certain involutions (in the centre of Sylow 2-subgroups) like the cases described in Wong [l],Fong and Wong [I], and Suzuki [4], just to mention a few, also extend to the locally finite case. This type of question has not yet received much attention (except for the preceding easy results), probably because the strong information needed on the approximating finite subgroups is not yet available. In all the preceding results of this section the method of proof consisted in first transferring global information on the locally finite group G at hand to the members of some local system for G consisting of finite subgroups, then applying to these subgroups some structure theorem for finite groups, and finally transforming the local information so obtained into some global statement on G (by applying 4.18). In the cases considered, the globally-given conditions on G localize trivially, but in general this is not always so. Consider, for example, a locally finite, doubly transitive permutation group on the set SZ. One would like to find a local system of finite subgroups of G which each have an orbit in Q on which they induce a non-trivial, doubly transitive action. It is not clear at all (to us), that and why this should be SO. Consequently, the assumption of multiple transitivity for locally finite permutation groups seems very hard to use. There are very few instances where this dif-
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1 49
ficulty has been overcome: for Frobenius groups, see 1.5.2, for Zassenhaus groups and in particular sharply triply transitive groups see Kegel [4]. Then there is Bender’s characterization, essentially contained in 4.24 of the groups PSL(2, F), PSU(3, F ) and S z ( F ) over suitable locally finite fields of characteristic two in their “natural” doubly transitive permutation representations. Recently Hering [ l ] has given another characterization of this class of simple groups. In all these cases, the condition of two-fold transitivity on the permutation group has been replaced by simple transitivity together with some other (very restrictive, of course) property of the natural (doubly transitive) permutation representation of these groups. From our point of view, the main advantage of the weakening of the conditions does not lie so much in the principle of wider applicability of the results, but rather in the fact that the new (formally weaker) conditions carry over to suitable local systems of finite subgroups. If one accepts it in the finite case, it is now a routine Exercise to extend Hering’s Theorem to locally finite groups.
4.29 Theorem. Let G be a locallyfinite, transitive permutation group on the set 52 such that, for a E Q, the stabilizer G, = ( g E G; ag = ct> has a normal subgroup N, containing an involution, such that every element n # 1 of N, leaves exactly one point a of Q fixed. Then, if H denotes the subgroup of G generated by all those elements which have exactly one fixed point in Q, one has the following alternatives: either there is a locally finite field F of characteristic two such that H is isomorphic as a permutation group to PSL(2, F), PSU(3, F ) , or Sz(F) in their natural (doubly transitive) permutation representations or O H is transitive on Q, and IH ; O H ] = 2. The final result of this chapter is a paraphrase of an important result of Brauer [l]; it plays a crucial role in Chapter 5 .
4.30 Theorem. Let G be an infinite, locally finite, simple group with afinite Sylow 2-subgroup S. If for every involution i E S the factor group CGilOCGi of the centralizer C, i is finite, then for every pair s, t of involutions of S and every element x E S there exist elements a, b, c E G such that sa,
tb,
xc E
s,
satb = xc,
and the order lC,xcl is maximal among the orders of centralizers in S of elements of S which are conjugate to x in G. The centre of S is elementary abelian. Proof. Because of 4.4 we need only consider the case when the group G is countably infinite. Then by 4.5 there is a sequence {G,} of the finite sub-
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groups G, of G with G, c G,,, and G = UneN G, such that each of the groups G,+l has a maximal normal subgroup M,+, with G, n M , + , = (1) (and hence G, embeds naturally into the finite simple group G,+JM,,+ For almost all n E N one has: S c G,, any two elements of S which are conjugate in G are already conjugate in G,, and the normal subgroup M, of G, has odd order. If i is any involution in S, then one clearly has for almost all n E N that
I C ~ , , ~ , , iM, : O C ~ , , ,~~M, , I5 lcG,i : O C ~i~, 5
I C , i : OC, i~ I p = Max IC,j : OC,ji i
where j ranges over all the involutions of S. If in the group S there is a pair s, t of involutions and an element x such that the conclusion of 4.30 is false, then for the images of the elements s, t, x in the group G,/M, the statement of 4.30 is false for almost all n E N . But Proposition 7 of Brauer [l] then asserts that the simple groups G J M , have a faithful, irreducible linear representation of a degree d, bounded by a function depending on IS1 and on 11, over a field of characteristic zero. However, Jordan’s Theorem, see Curtis and Reiner [I], p. 258, gives a bound to the order of the simple linear group G J M , of degree d. But the orders of the simple groups GJM, are unbounded. This contradiction proves the point. No central element of S of order 4 can be the product of two involutions of S, so the centre of S is elementary abelian. I For every infinite, locally finite field F of odd characteristic that is not quadratically closed, the group PSL(2, F ) satisfies the assumptions of 4.30, and these are the only examples that we know. Question IV.10. if G is an infinite, locally finite simple group with a finite Sylow 2-subgroup such that for every involution i of G the index lCGi : OCGil is finite, can one obtain the structure of G?
Slightly more generally, one may ask for the structure of those infinite, locally finite simple groups G such that for every involution i of G the centraliser C G i does not involve any infinite simple group. The only examples that we know have in fact CGisoluble. Note. Except for the larger part of Section C, the material of this chapter is essentially taken from Kegel [3]. In the concrete cases of the functors PSL(2, ), PGL(2, ), and Sz( ) the results of Section B - and in principle also the methods - appear in Kegel [l]. The results 4.1,4.5 and the remark on non-simplicity on p. 119 appear essentially in Kargapolov [4], see also Kovhcs [ 2 ] .4.4 was pointed out to us by P. M. Neumann. The results in Sec-
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tion C represent a pick from the current literature on finite (simple) groups of examples of results that may be extended to locally finite groups by the methods of Section B. In the same spirit as the examples given in Section C are the results on locally finite groups presented in Starostin [4]and [ 5 ] ; the first studies groups in which the centralizer of every element admits a nontrivial partition (i.e. is covered by a set of more than one nontrivial subgroups such that every non-identity element belongs to precisely one of these subgroups), the second studies groups in which any two maximal locally nilpotent, non-primary subgroups intersect trivially. Of course, there are many other results on finite simple groups which may be extended to locally finite groups that have not been mentioned here. Perhaps the most important of these is the result announced in Thompson [2] which purports to give a complete survey of those non-soluble finite groups in which the normalizer of a nontrivial soluble subgroup is again soluble. Unfortunately, at least one such group has been omitted from the list (rumours are that this is the only omission). If it can be proved that there are at most finitely many omissions to the list, then one still has a complete survey of all infinite locally finite groups with the property that every finite non-trivial soluble subgroup has a locally soluble normalizer; their commutator subgroups are simple and isomorphic to PSL(2, F ) or S z ( F ) for suitable locally finite fields F.
CHAPTER 5
Characterizations of the groups PSL(2, F) and of certain locally-soluble-by-finitegroups
In this chapter we present a few rather particular results. Despite their special nature, the proofs of these results draw on many of the subjects treated earlier. As a matter of fact, these results bring one recurrent theme of our discussion to fruition. Locallyjnite groups satisfying the minimal conditionfor (abeliaiz)subgroups are Cernikov groups. Although it was the quest for this result that directed and spurred work in many quarters, it seems to us in hindsight that the methods and tools used to obtain this result, and other closely related results, are of more interest and importance than the result itself. In fact, we have the impression that here is a fresh chapter of group theory which has scarcely opened. Questions abound. Which are the “right” ones that will give rise to new ideas, new results and new questions? Now at last, some inroads have been made, a few methods are available. No doubt, new ideas, new methods and new tools will be needed; they will be developed, and much more far reaching results are waiting to be discovered. The results that we treat here are two kinds. In Section A we obtain two characterizations of the groups PSL(2, F ) for infinite, locally finite fields of odd characteristic. The first of these characterizations concerns embedding properties of a single subgroup and characterizes PSL(2, F ) for quadratically closed, locally finite fields of odd characteristic; in these groups there are infinite 2-subgroups. If no infinite 2-subgroups exist in PSL(2, F), then we characterize this group by embedding properties of a set of subgroups strongly linked to centralizers of involutions. These results assume a key role in Section B where an affirmative answer is given, under additional structural assumptions, to Question m.3 (whether a locally finite group G has a locally soluble subgroup of finite index if all its primary subgroups are Cernikov groups). In an attempt to prove such a result one argues by the absurd, and studies the structure of a hypothetical counter example G of minimal 2-size. One quickly obtains embedding properties of certain subgroups connected with the centralizers of involutions. Suppose one can strengthen these 152
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embedding properties so much that one can show that either in G there exists a strongly embedded subgroup, when G is by 4.24 isomorphic to PSL(2, F), PSU(3, F ) or to Sz(F) for some infinite, locally finite field F of characteristic two, or the situation characterizing PSL(2, F ) for locally finite fields F of odd characteristic pertains. In either situation, the maximal c-subgroups, where cis the characteristic of F, do not satisfy the minimal condition, and so such a counter example cannot exist. Section A. Characterizations of the groups PSL(2, F) for locally finite fields of odd characteristic
The basic embedding property discussed in this section is that of 2-infinite isolation. Definition. The infinite, proper subgroup H of the group G is 2-infinitely isolated in G if it contains at least one involution and if for every involution i of H with infinite centralizer C , i one has C, i = CGi.
If the infinite subgroup H i s strongly embedded in the locally finite group G, then G is 2-infinitely isolated in G, since it contains the centralizer in G of each of its involutions. If the 2-infinitely isolated subgroup H of the locally finite group G contains the centralizer in G of each of its involutions, then by 4.25 and 4.24 one obtains a firm grip on the structure of G. If, on the other hand, for some involution i to H the centralizer CGiis not contained in H, then the centralizer C H iis finite, and by 3.2 and 3.33 one has some hold on the structure of H. If furthermore, the 2-infinitely isolated subgroup H of G contains an infinite 2-subgroup7then 5.1 will yield very strong information indeed, on the structure of G: either there is a proper normal subgroup of finite index in G or G has a unique non-abelian composition factor that is isomorphic to PSL(2, F ) for some quadratically closed, locally finite field F of odd characteristic. If in the group G every 2-subgroup is finite, it does not seem possible to pull the argument of the proof of 5.1 through. In fact, in order to characterize PSL(2, F ) in this case, we impose very restrictive embedding conditions on a certain family of infinite abelian subgroups related to the centralizers of involutions and we require that the locally finite group G, which we wish to show to be of type PSL(2, F), is already simple to start with. Out of these conditions we shall be able to draw conclusions on the structure of the finite Sylow 2-subgroups of G; eventually these will be shown to be dihedral. Thus we have G 1~ PSL(2, F). Now one realizes - unfor-
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[CH.
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5A
tunately only afterwards - that the normalizers of these singled-out, infinite abelian subgroups are 2-infinitely isolated in G . With respect to these embedding conditions it is natural to consider variations in order to gain flexibility. We shall only give one such variation, useful for Section B, in which we shall not require the 2-subgroups of G to be finite. Here many questions wait to be formulated and to be answered. 5.1. Theorem. Let G be a locallyfinite group containing a 2-injinitely isolated subgroup H such that H contains an infinite 2-subgroup. Suppose further that H contains involutions i such that the centralizer C H i is finite and that every such involution of H is contained in every normal subgroup offinite index in G. Then there exists a quadratically closed, IocailyJinitejield F of odd characteristic such that the factor group G/OG has a normal subgroup ofjinite odd index isomorphic to PSL(2, F); and H = CGifor some involution i. Proof. The proof of this result is broken up into a series of steps the first few of which give us rather precise information about the structure of the 2subgroups of G , so that, eventually, 4.19 becomes applicable. a. If E is an elementary abelian 2-subgroup of H and if i is an involution of E such that the centralizer C H iisfinite, then E has order two or four.
Proof. Suppose that IE! = 8. Then E = ( i ) x V for some subgroup V of E or order four. Obviously E s CGi,and so by 3.32 and 3.37 there is an involution j~ V with infinite centralizer in CGi. If k = j or ij and C,k is infinite then CGk E H. But clearly C G k A CGi = CGjn C G i is infinite so C, i is infinite. This contradiction of the finiteness of C, i shows that every involution of ( i , j ) must have finite centralizer in H. But this is in direct contradiction to 3.37, and consequently E has order at most four. 1 Let now i be a fixed involution of H with finite centralizer C, i. By 3.2 one has H E Min-2. Let S be a Sylow 2-subgroup of H (as in Chapter 3 ) such that i E S, and let A = JS be the smallest subgroup of finite index in S. Then A is an infinite, radicable abelian group.
b. The maximal 2-subgroups of H are extensions of Prufer 2-groups by finite groups. The involution i incerts the elements of A. Proof. Since by definition every maximal 2-subgroup of H is isomorphic to some subgroup of S, it is sufficient to prove property b for the Sylow 2subgroup S of H. (By 1.D.12 maximal 2-subgroups of H are necessarily infinite, since S is infinite.) If a E A , there exists an element b E A such that b2 = a. Then a = (bb')(bb-'), bb' E C,i, and (bb")' = (bb-')-*.
CH.
5,
5 A]
In particular,
CHARACTERIZATIONS OF
A
=
PsL(2, F )
155
( C A i ) [ i ,A ] .
As A is radicable and infinite and C,i is finite, this implies that A = [i, A ] . Since every commutator [i,a ] is inverted under conjugation by i, the involution i operates by inversion on A . Thus the subgroup 9,A generated by all the involutions of A is centralised by i. So the subgroup ( i , 9 ,A ) of H is elementary abelian, and thus of order four because of a and the fact that i# A . Therefore A is locally cyclic. 1
c. G contains a subgroup Q conjugate to A such thut H n Q = (1). Proof. If this is false, then for z, the unique involution in A , one has N = c H. Since H E Min-2, by 3.10, there is a finite 2-subgroup T of N containing z such that the conjugacy class of T in N is characteristic in N. Therefore, by the Frattini argument, H = N * NHT and G = N - NGT. Clearly, the factor group NGT/C,T is finite. Suppose that the index IH : NI is infinite. Then CHT must be infinite, and so C,T G H , since H is 2-infinitely isolated. Hence the index IG : HI is finite. But then the centraliser C,i is finite, and by 3.33 there is a normal subgroup X of finite index in G which does not contain the involution i, contrary to our assumption on G. Thus the index IH : N1 must be finite, and since A does not have any proper subgroup of finite index, one has A E N. N n S is a Sylow 2-subgroup of N by 3.13 and A is clearly its minimal subgroup of finite index. Hence 3.10 implies that we may choose T = ( z ) . Since the centralizer C,z is infinite, one has NGT = CGT 5 H for this choice of T. But this means G = H, contrary to the definition of 2-infinite isolation. I (2')
d. If t is any incolution of H such that the centralizer CHt is infinite, then t is contained in some Priifer 2-subgroup of H.
Proof. By c the group G contains a Priifer 2-subgroup Q conjugate to A such that H n Q = (1). Denote by k the unique involution of Q and put x = tk. If x has odd order, then t and k are conjugate in the dihedral group ( t , k ) , say t = kg.Now C,t is infinite, and since H is 2-infinitely isolated one has C,t E H, t E Qg E C,t E H, as contended. Suppose now that x = tk has even order. Then there exists an involution j E (x) centralizing ( t , k ) . Since the centralizer C,t is contained in the 2infinitely isolated subgroup H of G, one has j E H . If the centralizer C , j were infinite, then we would have k E C,j E H , which cannot be. Therefore the centralizer C,j is finite. We may assume i = j , and we may choose the Sylow 2-subgroup S of H to contain the subgroup (i, t ) . It then suffices to show that the involution t equals z, the unique involution of A .
156
PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITEGROUPS
[CH.5, 5 A
Suppose, if possible, that t # z. Clearly z is central in S, and so the subgroup ( i , t , z ) of N is elementary abelian. By a the subgroup ( i , t, z> has order four, and thus t = iz. In the radicable group A there exists an element y such that y z = z. As i inverts the elements of A , one has iy = i i y - l i y = i y 2 = t . But then C,t = CHi y is finite, contradicting the assumption on the involution t. I e. The Sylow 2-subgroup S of H is an infinite locally dihedral group.
Proof. If t is any involution of C,A, then by d there exists a Priifer 2-subgroup Q of H containing t . If we put IS : A1 = n, then there exist elements s E Q and y E A with s" = t and y" = z. By Sylow's Theorem there exists an element h E C, t such that the subgroup (s", y ) is a 2-group. By 3.11 some conjugate of (sh,y ) in H lies in S; hence one has t h = z, since A is locally cyclic. Then the choice of the element h E C H tentails t = z ; thus there are no involutions in the set C,A\A. For every element x E C s A the abelian subgroup ( x , A ) of S has the radicable subgroup A as a direct factor. The absence of involutions in CsA\A forces the equality A = C s A . The Prufer 2-group A has but two automorphisms of finite order. Since S contains the involution i $ A and i inverts the elements of A , the group S is an infinite locally dihedral group. I f. The Sylow 2-subgroup S of the subgroup H of G is a Sylow 2-subgroup of
the group G.
Proof. Let z be the central involution of S ; since H i s 2-infinitely isolated and since the centralizer C,z is infinite, one has C,Z E H. By b one has that H and hence also C,Z satisfies min-2 and so 3.2 yields G E Min-2. Let T be some Sylow 2-subgroup of G, and put IT: JTI = n. Let y be an element of S such that y" = z. By 3.11 we may assume that T contains the subgroup ( i , y ) of S. Then clearly z E JT, and thus JT C H , forcing JT to be a Prufer 2-group, since JT is isomorphic to a subgroup of S, and thus to A . For the same reason, one has C T z = JT; and so ( y , i ) E T forces T to be an infinite locally dihedral 2-group. Hence S and T a re isomorphic and so f follows from 3.8. I
g. The group GjOG is isomorphic t o some subgroup of PrL(2, F ) containing PSL(2, F ) f o r some infinite, locallyfinitefield F of odd characteristic. Proof. Because of e and f one can invoke 4.19 in order to obtain information on the structure of G. This yields that, should the statement of g
CH.
5,
5 A]
CHARACTERIZATIONS OF
PsL(2, F )
157
not apply, the group G/OG is either isomorphic to the alternating group of degree seven or it is a 2-group. The first cannot be the case, since the group G/OG is infinite - it contains an infinite 2-subgroup. If G/OG is a 2-groupY then by 3.13 one has G/OG N S and G = S - O G . But then the subgroup AOG has index two in G and does not contain the involution i, contrary to our assumption on G. Thus the property stated in g is, by 4.19, the only remaining possiblity. I Using some of the well-known properties of the group PTL(2, F ) we can now finish the proof of 5.1. The group PTL(2, F)/PGL(2, F ) is essentially the full group of automorphisms of the locally finite field F, and so it is residually finite. But since the subgroup A of G is a Prufer 2-group and so does not have any proper subgroup of finite index, one has AOGIOG c PGL(2, F ) . Further since the index lPGL(2, F ) : PSL(2, F)I S 2, one even has AOGIOG c PSL(2, F). Thus the Sylow 2-subgroups of PSL(2, F ) are infinite locally dihedral 2-groupsYas they are in PGL(2, F). In infinite, locally dihedral 2-groups the non-central involutions form a single class of conjugate elements (1.1.2), and so one gets PSL(2, F )
=
PGL(2, F).
Since the field F is not of characteristic two, this means that the field F is quadratically closed. A quadratically closed, locally finite field does not have any automorphism of order two. Now PTL(2, F ) splits over PGL(2, F), and so by Dedekind’s modular law the locally finite group G/OG splits over PGL(2, F ) and the elements of a complement C of PGL(2, F ) in G / O Ghave odd order. Consider C as a group of automorphisms of the field F. Then for each element c E C, the field F, of all elements of F left fixed by the automorphism c is such that F has finite dimension as a vector space over F,. In particular, the quadratic closure F of the prime field of F is contained in F,. Consequently, the centralizer C p ~ L (F z) C , contains a subgroup isomorphic to PGL(2, F ) , and so C centralizes some elementary abelian subgroup of PGL(2, F ) of order four. In view of f and 3.10 we may assume, for some elementary abelian subgroup V of S of order four, that C centralizes VOG/OG. If k is any involution of G, then the Frattini argument applied to ( k , O G ) shows that CGpG(k. O G )
=
CGk. OG/OG.
Clearly the central involution z of S lies in V, and
SO
C E C G z .OG/OG c H . OG/OG.
158
PsL(2, F ) AND
LOCALLY-SOLUBLE-BY-FINITE GROUPS
[CH.
5,
5
A
But by 1.1.2 and e some involution of V has finite centralizer in H. Hence C is finite. It follows from Huppert [I], 11. 8.27, for any quadratically closed locally finite field F of odd characteristic, that the centralizer of any involution is a maximal subgroup of the simple group PGL(2, F), and HOG cannot have finite index in G by 3.33 and the basic assumptions on i. Consequently H . OG = OG C,z. Now the involutions i and iz are conjugate in S, so the centralizer CH iz is also finite. By a result of Brauer and Wielandt (see Gorenstein and Walter [l], Lemma 4) we have
.
H n OG = (OG n C, z)(OG n C, i)(OG n C, iz), and thus CGzhas finite index in H. If Kdenotes the largest normal subgroup of H contained in C H z , then the quotient group ( H n OG)/(Kn O G ) is finite and therefore is centralized by every Prufer subgroup of H. But the involution z lies in the Priifer 2-subgroup A of G and so C,z covers this factor. This means that H = CGz,and the result is proved. I It seems appropriate to state a special case of 5.1 separately. 5.2 Corollary. Let G be a locallyjinite simple group containing a 2-injinit ely isolated subgroup H with an infinite 2-subgroup. If H contains an involution i such that the centralizer C, i isfinite, then there exists a quadratically closed, IocalIyjinitejieId F of odd characteristic such that G N PSL(2, F), and H is the centralizer of an involution in G.
Kemarks. One may wonder whether, under the assumption of 5.1, the group G actually has a normal subgroup isomorphic to PSL(2, F). That this need not be so can be readily seen by taking as G the split extension of the three-dimensional vector space V over F by PSL(2, F), where PSL(2, F ) acts on V as it acts naturally and faithfully on the space of all 2-by-2 matrices over F of trace zero. As subgroup H choose the centralizer of an involution of G. Since for this action of PSL(2, F ) on V there also exists a non-split extension" G which also satisfies the assumptions of 5.1, again choosing H to be the centralizer in G of some involution, we see that in the situation of 5.1 the group G need not even have a subgroup isomorphic to PSL(2, F). This situation can be drastically improved by strengthening the embedding properties of the subgroups H into G apparently only slightly.
* Such an extension can be obtained in the following way. Let F1 be a commutative local ring with Jacobson radical J and residue class field F,/J '2 F, such that the additive group of FI is a direct sum of cyclic groups of order p z where p is the characteristic of F. The group PSL(2, Fl) E SL(2, F1)/Z(SL(2, F I ) has just that structure.
CH. 5, 5 A]
CHARACTERIZATIONS OF
PsL(2, F )
159
5.3 Corollary. Let G be a locallyfinite group with a subgroup H containing an infinite 2-subgroup and assume that f o r every element h # 1 of H such that the centralizer C,h is infinite and contains an involution one has c, h = C,h. If in H there is an involution i with finite centralizer C,i and if every such involution of H is contained in every normal subgroup of finite index in G, then there exists a quadratically closed, locally Jinite field F of odd characteristic such that G N PSL(2, F), and H is the centralizer of an involution in G. This result first appears in Sunkov [3]; there the definition of a 2-infinitely isolated subgroup is more restricted than ours, and this is the reason why in 5.3 we have to formulate more conditions. The paper Sunkov [3] is significant in that it gives the first example of a theorem characterizing the locally finite group PSL(2, F ) by embedding properties of a single subgroup.
Proof. Clearly 5.1 applies to the group G, and so there is an involution j i n G such that H = C,j. Furthermore, G'OG/OG N- PSL(2, F ) for some quadratically closed, locally finite field F of odd characteristic, and thus there is a finite subgroup A of G containing the elementary abelian 2-subgroup V = ( i , j ) o f order four of H such that the involutions of V are conjugate in A . Let t; be the set of all finite A-invariant subgroups S of the locally soluble group OG. Clearly C is a local system of OG. For every subgroup S E Z, denote by nS the largest nilpotent normal subgroup of S, the Fitting subgroup o f S. Then V operates on ttS, and so by a result of Brauer and Wielandt (see Gorenstein and Walter [l], Lemma 4) one has
nS
=
(nS n C,i)(nS n C,j)(nS
n C,ij).
Since A permutes the involutions i,j and ij transitively, one has for S # (1).
(*I
(nSn CGil = InSn C,jl
=
InSn C,ijl # 1;
and thus H n nS # (l), and similarly H n ZnS # 1. Let g be an element of prime order p of H n ZnS # 1. If CHg is finite, then H satisfies min-p by 3.2, and by 3.10 there is a finite p-subgroup K of H n OG containing g such that for every automorphism a of H n OG there is an element h E H n OG with K" = Kh. Therefore, the Frattini argument yields H = ( H n OG)N, K, and hence a Priifer 2-subgroup of H centralizes K. This contradicts the assumption that CHg is finite. Since the centralizer C,g is infinite and containsj (since H = C,j), one has by the general assumption on H that C , g = CGg. Consequently, nS c H for every subgroup S E x. But the centralizer C,i is finite, and so by (*) one has lnSl ICHil3, and this gives a bound to the order of S, namely IS1 IC,iI3. (lCHi13)!.Since
s
s
160
PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITE GROUPS
[CH.5, 5
A
the elements of the local system Z, are of bounded order, this entails that OG is finite. If the finite normal subgroup O G is non-trivial, then the centralizer of OG has finite index in G , and every Prufer 2-subgroup of G centralizes OG. Consequently, OG c C , j = H. The general assumption on the subgroup H yields that CGOG G H , and hence that H has finite index in G. By 3.37 this contradicts the assumptions of 5.3. Hence O G = <1). Now G is a split extension of the normal subgroup G' N PSL(2, F ) = PGL(2, F ) by the finite cyclic group C (of odd order). Put C = (c) and assume C # (1). Then the element c may be made to correspond to some non-trivial automorphism y of the field F. The field F is a finite algebraic extension of the field F, consisting of all the elements of F left fixed under the action of y. Hence the centralizer C,.c contains (in fact, is) a subgroup isomorphic to PSL(2, F,). Thus c centralizes some conjugate i o f j and hence also the infinite centralizer of i in C,,c. We may take i = j , and then also c E H a n d the centralizer C, c is infinite. By assumption, this means that C G c E H. Sincein the simple group S = PSL(2, F ) one has S =
5.1 Theorem. Let G be a locally finite group containing an infinite maximal radicable abelian 2-subgroup A with normalizer H = NGA, such that f o r every element h # 1 of H with infinite centralizer C , h one has CGh = C Hh. Then one has one of the following possibilities: 1. H = G ; 2. G is a Frobenius group with H a Frobenius complement; 3. there exists an involution i in H with finite centralizer C , i and a normal subgroup N of jinite index in G such that i 6 N ; 4. there exists a quadratically closed, locally Jinitefield F of odd characteristic such that G N PSL(2, F). In the preceding results on the structure of a locally finite group G with a 2-infinitely isolated subgroup H essential use was made of the assumption that H contains an infinite 2-subgroup. Without that assumption, one still has a wealth of information but apparently not quite enough to obtain any definitive structural result. In our next result 2-infinitely isolated subgroups will appear, but we shall not be able to exploit this fact in any decisive way. Instead, we shall impose very restrictive embedding conditions on certain
CH.
5,
5
A]
CHARACTERIZATIONS OF
PsL(2, F )
161
subgroups closely linked to the centralizers of the involutions, and we shall consider this situation only for simple groups. These results will give characterizations of the groups PSL(2, F ) for infinite, locally finite fields F of odd characteristic. The first one may be viewed as an affirmative, if conditional, answer to Question IV.10. 5.5 Theorem. Let G be a locally finite, simple group with finite Sylow 2subgroups such that for every involution i of G the index IC,i : OC,il isfinite. Then the group G is isomorphic to PSL(2, F ) for some infinite, locallyjinite field F of odd characteristic, which is not quadraticalle closed, if and only if there exists in G a non-empty family B of iizJnite abelian subgroups such that 1. for every X E 3 and every g E G, also X g E Z; 2. if for X E 3 and the involution i E Nc X one has C , i $ N, X,then the centralizer C xi is finite. 3. for every involution i of G there is a unique X E 8 such that C, i ENG X
Proof. It is rather immediate to see that in the group PSL(2, F ) with F an infinite, locally finite field of odd characteristic, which is not quadratically closed, such a family Z of infinite abelian subgroup actually exists. The centralizer of any involution in PSL(2, F ) is an infinite locally dihedral group. Any two involutions of PSL(2, F ) are conjugate. Now choose E = (OC,i; i involution in G } . This set will, in fact, satisfy condition 2 in the stronger form that C x i is trivial, that is ( i ) = Cxi. Thus it is the converse that one has to prove. Here, clearly, we may assume that for each X E E the normalizer N,X contains an involution, discarding all the other elements of 3, if necessary. With this assumption in force we prove a. For each X E E there is an ~ i i v o ~ u t ~i o E nN , X such that Cc i 4 NG X. Proof. If for some X E B every involution i E N , X satisfies C, i c NG X , then the simplicity of G and 4.25 yield that the normalizer NGXis a strongly embedded subgroup of G, and thus by 4.24 there is an infinite, locally finite field of characteristic two such that G is isomorphic to one of PSL(2, F), PSU(3, F), or Sz(F). In each case, G would contain an infinite 2-subgroup, contradicting the assumption that G has finite Sylow 2-subgroups. 1 For X E E let T be a Sylow 2-subgroup of L = NGX.If i is any involution of T and p is an odd prime, then the p-primary component X , of X is a direct product X , = [X,,i] x C x i , and xi = x-l for every x E [X,, i ] , for if x E X , there exists an element y E X,with y z = x, and x = [i,ylyy'. Thus, if C , i is finite, the commutator group [OX, i ] is a direct factor of finite index in X . Put
162
PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITE GROUPS
x* = x*(T) = n [ O X , i ] ,
[CH.
5,
A
I
where i ranges over all the involutions of T that have finite centralizer in X. For any involution i E T either C , i c N,X or i inverts all the elements of X*.The subgroup X* is normalized by T. From now on we shall assume that 5.5 is false and that the group G is a counter example to the assertion. Let S be some fixed Sylow 2-subgroup of G and z a fixed involution in the centre of S ; by 3 there is a unique subgroup A E B such that S 5 C , z s N,A = N. Let j be an involution of S such that C , j $ N (by a there are such involutions). We shall now focus attention on the structure of the Sylow 2-subgroup S of G. b. Iff o r X E E the inuolution i E N, X is such that CGi $ NG X,then an elementary abelian 2-subgroup of NGX containing i has order at most four; ( j , z ) is the unique maximal elementary abelian subgroup of S containing j , and the centre of S is cyclic of order two. Proof. Assume, if possible, that there is an elementary abelian subgroup E of order eight containing i in N, X . Let T be a Sylow 2-subgroup of NG X containing E. Since E is not cyclic, there is an involution k E E such that the centralizer C x * k = C is infinite by 3.37. Now E = (k) x K for some subgroup K of E. Applying 3.37 again, we see that there is an involution I E K such that the centralizer C c l is infinite. Thus E has a subgroup U of order four centralizing an infinite subgroup of C. By 3 there is a unique Y EB such that C , i E NG Y. Let TI be a Sylow 2-subgroup of NGY containing E, and Y * = Y *(T ,). As before there is a subgroup V of E of order four centralizing an infinite subgroup of Y * . Since E has order eight, U n V # (1). For any involution u E U n V one has that the centralizers C x u and C,u are infinite, so by 2 one gets C , u 5 NGXn NGY, contradicting 3. Thus such an elementary abelian subgroup of N,Xcan have order at most four. If now E is an elementary abelian subgroup of S containing j , then E 0 , Z S is also elementary abelian. By the first part IEf2,ZSI 5 4. Since z # j , the uniqueness of E = ( j , z ) Follows. By 4.30 the centre ZS is elementary abelian. If Z S is not of order two, then it must contain all the involutions of S. The powerful result 4.30 now forces S to be ZS. But then 4.19 yields that G is isomorphic to PSL(2, F ) for some infinite, locally finite field F of odd characteristic, contradicting the assumption that we are dealing with a counterexample to 5.5. Hence Z S = ( z ) . I
c. The Sylow 2-subgroup S of G cannot be quaternion, dihedral or quasi-dihedral. S has a maximal, elementary abelian, normal subgroup of order four.
CH.
5,
5 A]
CHARACTERIZATIONS OF
PsL(2, F )
163
Proof. If every abelian normal subgroup of S were cyclic, then the nonabelian Sylow subgroup S of G would be quaternion, dihedral or quasidihedral (see Gorenstein [l], 5.4.10). Since j # z, the group S cannot be quaternion. So by 4.22 the group G would be isomorphic to PSL(2, F), PSU(3, P ) , or PSL(3, F ) for someinfinite, locally finite field F o f odd characteristic. The first possibility is excluded by the assumption that G is a counter example to 5.5; the other two possibilities cannot occur either, since in these two groups the centralizer of an involution involves PSL(2, F ) whereas the condition that the index ICGi : OC,ij is finite for every involution i of G entails that no such centralizer can involve an infinite simple group. Thus S has an elementary abelian normal subgroup E of order at least four. If (El > 4, then ICEjl 2 4. In this case b shows t h a t j q! E, and so ( j , C E j )is an elementary abelian group of order at least eight, which contradicts b. I Some more notation: There is a unique subgroup B E 8 such that C G j c NG B = M . We shall assume that S contains a Sylow 2-subgroup D of M n N and that this D is maximal (with respect to inclusion) in S with respect to all possible choices o f j (and hence of B). Let T be a Sylow 2-subgroup of N , B containing D , and put B* = B*(T). Then, clearly, every involution of D inverts all the elements of at least one of A* and B*, and hence the elementary abelian subgroups of D have order at most four. We shall now gather information on the structure of the group D and on the way in which it is embedded in S. d. The centre Z D is not cyclic, G , Z D
=
a,D is of order four, and D # S.
Proof. Suppose that Z D is cyclic with involution d and that CGd N. Since C G z E N a n d z is the only involution of ZS, we have S # D. There is a unique X E E with C G d c NGX. But then D $ NSD E NGD E CGd E N G X ,
and this contradicts the maximality of D . Therefore, C G d E N. Now the centralizer CDB* is normal in D and does not intersect Z D non-trivially, since d inverts all elements of B*. Thus no non-trivial element of D centralizes B*. If k # d is an involution of D inverting all elements of B*, then dk centralizes B*, which is impossible. Therefore every invo1ut:on ke D , k # d, has infinite centralizer in B, and such involutions exist, for example j . Consequently 2 yields that the infinite group C = ( C , k ; k involution of D , k # d ) is contained in M , and clearly C is normalized by each element normalizing D. If g E N G D ,then j g # d and so C,jg E M n M g . By 3 we have B = Bg. Also NGD E Cdd E N; consequently N , D 5 M n N, and hence S = D . By 4.30 the involution d cannot be the only involution of D conjugate
164
PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITEGROUPS
[CH.5, 5 A
to d in G. Hence there is an element g E G such that d # dg E D. But then C,dg c M . By 3 one obtains M = N g , and so D and Dg are both Sylow 2subgroups of A4 = N g . By Sylow's Theorem there is an element n E N such that (Dg)""= D, and thus ng E N,D c CGd c N . Hence N = N g = M , a contradiction. Thus Z D is not cyclic. By b now Z D contains all the involutions of D, 1.e. n , Z D = D ; since Z S is cyclic, one has D # S. I
e. C,JZ, D E M n N and C,n, D = D. Proof. By d and b one has Q 1 D = (j, z) and by construction C,j c M and C,z c N . Thus C,j n C,z E M n N . Since D is a Sylow 2-subgroup of M n N , d yields D = Cs52, D . I f. The involutions j and jz are conjugate in N, D ; the element z centralizes A* and the element j centralizes B*. Proof. By e we have D = C,O,D, and so the non-trivial factor group N,D/D acts faithfully on 52, D. Clearly, the central element z of S is fixed by every element of N,D; thus there is some element in N, D that interchanges j and jz. If now x E N, D and C,jz E M , then either or
j Z = j"
and thus CGjZ
=
(CGj)x G M n M"
j = j" and thus C, j E M n M".
In either case by 3 we conclude that x E M and D 2 N,D c M n N , a contradiction to the choice of D . Thus we must have C,jz M. Sincej inverts the elements of A* and j z is conjugate t o j in S, the involutionjz also inverts the elements of A*; thus the element z = j ( j z ) centralizes A*. Since C,jz Q M , the involution.jz inverts the elements of B* by 2, and so does z; thus j = (jz)z centralizes B*. I g . One may choose j (and hence B and M ) s x h that M = N g f o r some g E G.
Proof. Suppose that no G-conjugate of z in S inverts the elements of A*. If the involution i = zgsatisfies C, i c N, we have C,i = (CGz)B-c N n N g . By 3 this means A = Ag and g E N,A = N . Thus i centralizes a subgroup of finite index of A , since by f the involution z does. By 4.30 there are elements a, b, c E G such that z', zb,jcE S and z'zb = j'. By what we have just shown, j' centralizes a subgroup of finite index in A. Since the involution j centralizes B*, we get M' = N. On the other hand, suppose that the involution i = zg of S inverts the elements of A*. Then g N and i E N n N g , and we may choose D to contain i. But then by d we have
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A]
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c,z n CGi E N n N a , and so we may assume that M = N g . I D
E
h. The involutions of D are conjugate in N, D, and D is either the direct product of two cyclic 2-groups of the same order or it is isomorphic to the SyIow 2-subgroup of PSU(3, 4).
Proof. If we choose j according to g, the property d yields that D is not a Sylow 2-subgroup of M , that is there is an element x E NMD D \ with x 2 E D . Since j is the only involution of 0 ,D that centralizes B", we have N,wD E CGj.By e the element x cannot centralize Q,D, hence zx = jz. So the group of automorphisms induced in f2, D by NGD is the symmetric group of degree three. In particular, there is a cyclic group of automorphisms (of order three) of D which permutes the involutions of D transitively. If D is abelian, it is the direct product of two isomorphic cyclic subgroups. If D is non-abelian, then the classification in Higman [3] of those finite 2-groups which admit a cyclic group of automorphisms permuting all the involutions transitively, yields that D is of order 64, and unique up to isomorphism. As the Sylow 2-subgroup of PSU(3, 4) has this structure, we have proved the assertion. I i. 51, D is the only elementary abelian normal subgroup of orderfour of S, and IS : DI = 2. Proof. Since by b the centre of S is of order two, one has for every elementary abelian normal subgroup V of order four of S that the index IS: CsVI = 2. If F'z D, theneyields V = f2,D,D = C,V,and IS: DI = 2. Now assume, if possible, that V $ D. Then j induces an automorphism of order two in V. As the automorphism group of Vis isomorphic to the symmetric group of degree three, there cannot be any element of D having j as proper power. But then, because of h, the group D cannot have elements of order four; hence C , j = CsDl D = D is elementary abelian of order four. By Huppert [I], 111. 14.23, the 2-group S then is of maximal class, that is, S is quaternion, dihedral or quasi-dihedral (see Gorenstein [l1, 5.4.5.). By c this is impossible, and so the assumption V $ D has to be discarded. 1 j. The subgroup D of S is non-abelian. Proof. Assume, if possible, that D is abelian. If D is elementary, then the non-abelian group S is dihedral of order eight, contradicting property C. The exponent of D must thus be at least four. Let s E S \ D , and let d be any element of order four of D with d2 = j . Then s centralizes the element dd" # 1, and dd" is an element of order four in the centre of S. This contradicts b. I k. The SyIow 2-subgroup S of G has exponent four.
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PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITEGROUPS
[CH.5, 5 A
Proof. Assume, if possible, that there is an element t of order eight in S. By 4.30 there exist elements a, b, c E G such that za, zb, t' E S and zazb = t'. Clearly, t' $ D, since by h and j the group D has exponent four. Thus one of za,zb,say za $ D, and the other zb = j or j z . But then j z a = jz, and zazbhas order four, a contradiction. I 1. There is an involution i in S \D in G.
and any two involutions of S are conjugate
Proof. If all the involutions of S lay in D, then no element of order four could be the product of two involutions of S, contradicting the assertion By 4.30 there exist elements a, b, of 4.30. Thus there is an involution i E S\D. c E G such that za, zb, i' E S and zazb = i". The elementary abelian subgroups of S have order at most four, and so z E ( z a ,z b ) . If i' 6 D, then say, z = zb and za = i'z. But icj
=
ic(i9i')j = i'(jz)j
=
i"z = zp,
D, then h yields that i' is conjugate to z. I
and i and z are conjugate. If i'
E
m. The involutions in S\(z)
invert the eIements of A*.
Proof. Suppose if possible that the involution i E D S\ centralizes A* and put C = CsA*.Since S = ( i ) D , Dedekind's modular law yields C = ( i ) ( C n 0).Clearly, C n D is cyclic of order at most four or C n D is quaternion. Also the centralizer of i in D has order at most eight. Hence i has at least eight conjugates under D lying in S \ D and these conjugates all lie in C.Thus C has order 16 and C n D is quaternion. But it also follows that every element of C\D is an involution and so i acts by inversion on C n D . Thus C n D is abelian, a contradiction. We have now shown that i does not centralize A*. By 1 there exists an element g of G such that i = zg.Since i $ CGA* we have g 6 N . But then 2 implies that C,i is finite and so i inverts the elements of A*. 1
The final contradiction. By 1 there exists an involution i E S\D and i has at least eight conjugates under D in SD \. Together withj andjz we therefore have (by m) at least ten involutions of S inverting the elements of A*. Clearly these elements lie in a single coset of C = CsA*, and so [Cl2 10. But, by m,the subgroup C has a single involution, and by k it has exponent four. Thus ICI 5 8, a contradiction. This final contradiction shows that there cannot be any counter example to the assertion of 5.5. I The following variation of 5.5 does not require the finiteness of the 2-sub-
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167
groups of G , but we have to compensate for this by further restricting the normalizers of the subgroups in the set 9. This will allow us to use 5.2 to handle the situation when G has infinite 2-subgroups. 5.6 Theorem. The Iocallyjinite, simple group G is isomorphic to the group PSL(2, F ) f o r some injinite locally finite Jeld F of odd characteristic if and only if there exists in G a non-empty family B of inJnite abelian subgroups such that 1. for every X € 3and every g E G, also X 9 € 2 ; 2. i f X , Y E0” and X # Y, then NGX n N , Y is finite; 3. for every involution i of G there exists a subgroup X E B such that C,i c N,X 4. the centralizer C , i of every involution i of G is (locally soluble) -by-jinite.
Proof. By discarding any superfluous ones we may clearly assume that for each X in 3 there exist involutions i satisfying C , i c N , X. The proof is accomplished in a series of steps. a. If i is an involution and X , Y EB are such that C , i E N, X n N, Y then = Y.
x
Proof. For by 3.33 the centralizer CGi is infinite and thus a is a concequence of2. I
b. If’X E S then the normalizer N , X is 2-injinitely isolated. Proof. Since G is simple N = NGXis a proper subgroup of G and it does contain involutions i. Suppose that C N iis infinite. There exists by 3 some Y in such that C,i 5 N,Y. Thus C N ic N n N,Y and so X = Y by 2. Therefore N is 2-infinitely isolated in G. I
An immediate consequence of b is c. For X E E and i an involution of N , X , whenever C,i CN,+ is finite.
4 N, X we have that
d. If X E S the normalizer N, X contains an involution i with CGi $ N,X.
Proof. For if otherwise by 4.25 there exists a strongly embedded subgroup of G, and so by 4.24 the group G is isomorphic to one of the groups PSL(2, F ) , PSU(3, F ) or S z ( F ) for some infinite, locally finite field F of characteristic two. A maximal 2-subgroup S of G in any of these cases is i n b i t e and has nontrivial centre. Thus S s N,X for some X in 3. If g E N,S, then S E
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PsL(2, F ) A N D LOCALLY-SOLUBLE-BY-FINITE GROUPS
[CH. 5, 5 A
N, X n N, X', and so by 2 we have g E N,X. Hence N,S E N, X. In these cases, the subgroup N,S is a maximal subgroup of G for every maximal 2-subgroup S of G. Since in these cases also for any two maximal 2-subgroups S, T of G the intersection N,S n N,T is infinite, it follows that (N,S, NGT) s NGX for every maximal 2-subgroup T of G. But then N, X = G, contradicting the simplicity of G. This proves d. I
e. G is isomorphic to PSL(2, F ) for some infinite locallyJirlite$eld F of odd characteristic. Proof. If G contains infinite 2-subgroups then by 1.G.4 or 2.5 the group G contains an infinite abelian 2-subgroup. Thus G contains an involution i whose centralizer in G contains infinite 2-subgroups. But then if X is the element of E satisfying C , i E N, X , the subgroup N, X of G satisfies by b and d the conditions of 5.2; consequently the group Gis isomorphic to PSL(2, F ) for some quadratically closed field of odd characteristic. If all the 2-subgroups of G are finite then the set 8 satisfies the hypotheses of 5.5 (by 3.17,1, 3, 4, a and c). Thus again we obtain that G is isomorphic to PSL(2, F ) for some infinite locally finite field F of odd characteristic. !
Exercise. Prove that the locally finite, simple group G is isomorphic to the group PSL(2, F ) for some infinite, locally finite field F of odd characteristic if and only if there exists in G a non-empty family E of infinite abelian subgroups such that 1. for every X E 3" and every g E G, also X g E ti; 2. if X , Y E E and X # Y then the intersection X n Y is finite; 3. for every involution i of G therz is a subgroup X E E such that C G i E NGX, 4. for every X E E the index (N,X :XI is finite. Remark. If we allow infinite 2-subgroups in the more general situation of
5.5 and replace the condition that the index (C, i : OC, il be finite by the con-
dition (equivalent if the 2-subgroups of G are finite) that C,i has a locally soluble subgroup of finite index as in 5.6 or the Exercise above, then there are indeed further examples, e.g. PSL(2, F ) where F is any infinite locally finite field of characteristic 2. It would be of interest to have a complete survey of these groups.
Exercise. Show that a simple minimal counter example to the contention that locally finite groups satisfying min are Cernikov groups satisfies the conditions of 5.6 and so cannot exist. (See 5.8).
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Section B. Structure theorems for groups satisfying certain minimality conditions
In this section we present a few structure theorems that may be considered as partial answers to Question 111.3. They all assert that a locally finite group G in which every primary subgroup is a Cernikov group has a locally soluble normal subgroup of finite index if certain additional conditions are met. In fact, these additonal conditions allow us to make much more precise assertions about the structure of the locally soluble subgroups of finite index. In the proofs of these results we consider counter examples to the theorems which are in some way minimal. By the general reduction lemma 3.32 we may assume that these hypothetical counter examples are simple. Observing that the groups PSL(2, F ) for infinite, locally finite fields F a r e not counter examples (they do not satisfy min-c if c is the characteristic of F), we shall steer the discussion of the properties of the minimal counter examples towards the characterizations of PSL(2, F ) given in Section A . Using these characterizations together with Bender’s result 4.24 we arrive at our final contradictions. 5.7 Theorem. Let G be a Iocallyfinite group satisfying min-pfor every prime p. If the centralizer in G of every involution of G is a Cernikov group, then G contains a locally soluble normal subgroup of finite index. If the group G satisfying the conditions of 5.7 actually contains involutions then, in view of 3.36 it contains a hyperabelian normal subgroup of finite index. 5.7 can also be viewed as a further conditional answer to Question IV.10. We do not know of any infinite, locally finite simple group that contains at least one involution whose centralizer is a Cernikov group. Question V.1. Let G be a locally finite group and i an involution of G such that the centralizer C G i is a Cernikov group. Does G necessarily contain a locally soluble subgroup of finite index?
Proof. If G is any locally finite group satisfying min-2 write E(G) = {primesp; there exists an involution i such that CGi contains a Priifer p-subgroup}. In view of 3.12 the set E(G) is always finite, if the centalizers of all involutions of G are Cernikov groups. If the theorem is false, there exists a counter example G , of minimal 2-size. Put n = E(G,) and E = (2) ( J n;the set n is finite, say n = {pl, . . .,P,}. Amongst the counter examples G2 of minimal 2-size and E(G,) C 7~ there exists one of minimal p,-size, amongst these one of minimal p,-size,
170
PsL(2, F ) AND LOCALLY-SOLUBLE-BY-FINITE GROUPS
[CH.5,
5
B
and so on up to p,. If G3 is one such counter example, then by 3.32 there exists a subgroup G, of G , that is infinite and simple modulo a locally-soluble-by-finite normal subgroup Z. We prove that G = G,/Z is also a minimal counter example of the above type. To do this we have only to show that the centralizer in G of any involution of G is a Cernikov group and that E(G) _c n. If N is any normal subgroup of G , containing no involutions and i is an involution of G,, then C,,,,(iN) = Cc4i.N/N. Hence we may assume that 02 = (1). But then 3.17 implies that Z is a Cernikov group and so G4/CG4Zis a Cernikov group by I.F.3. Therefore G4 = 2 . CG4Z and we may assume that 2 is the centre of G4. Clearly G, contains involutions and is not a Cernikov group, so Z contains no involutions. But then 2 = O Z = (1) and our claim is demonstrated. I We collect some properties of this infinite simple group G. a. If P is a non-trivial radicable abelian %subgroup of G then N,P contains
a locally soluble subgroup of finite index.
Proof. By 1.F.3 the index IN,P : C G P ( is finite, so we may work with C = C,P. If P contains involutions then C is by assumption a Cernikov group, thus suppose that P is a 2’-group. If i is any involution of C, it follows that C,(iP/P) as a homomorphic image of C c i is a Cernikov group and that E(C/P) E n. But clearly the p-size of C/Pis smaller than that of G for at least onep in n. Thus C contains a locally soluble subgroup of finite index. I Let 6 denote the set of maximal radicable abelian subgroups of G. Since G is simple it contains involutions, and the centralizers of these involutions are infinite Cernikov groups by 3.33. Hence 6 is not empty. b. If D is any infinite subgroup of the subgroup X tains a locally soluble subgroup of finite index.
E3
then N = NGD con-
Proof. Suppose otherwise. Then N contains a Sylow 2-subgroup S of G. By 1.1.5 the group S contains an elementary abelian subgroup V of order four. At least one of the involutions of V has infinite centralizer in D by 3.37 and thus for some p in n the Sylow p-subgroup D, of D is infinite. Since D, is normal in N it follows from a that N has a locally soluble subgroup of finite index. I
c. Z f X , Y E = and X # Y, then X n Y isfinite. Proof. If X n Y is infinite, then N,(X n Y ) has a locally soluble subgroup of finite index by b and hence by 3.18 contains a unique maximal radicable
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abelian subgroup, which clearly contains both X and Y.By the maximality of X and Y we have X = Y and c follows. I d. If i is any involution of G then there exists a unique X E 3 such that CGi c NG X .
Proof. The centralizer C , i is infinite by 3.33, and so it contains an infinite radicable abelian normal subgroup A . Clearly there exists X E S with A c X . Now X v CGi E NGA, which by b and 3.18 contains a unique maximal radicable abelian subgroup. Thus X is this subgroup and C, i c N GA NG X . If Y is a second member of 3 with C , i G NGY, since N, Y contains a unique maximal radicable abelian subgroup Y, we have A 5 Y. Thus X n Y is infinite and X = Y as required. I e. If X E 3 and i is an involution of NG X such that CGi $ NG X, then the centralizer CNGXi is finite.
Proof. There exists Y in 3 with CGi c NGY. If CNoXiis infinite, then it contains an infinite radicable abelian subgroup. This subgroup will be in the unique maximal radicable abelian subgroup of NGX , namely X , and similarly it lies in Y. But then X n Y is infinite, so X = Y by c, a contradiction that proves the point. I f. The Sylow 2-subgroups of G are infinite. Proof. If not, by 5.5 the group G is isomorphic to PSL(2, F ) for some locally finite field of odd characteristic c. But then G does not satisfy min-c and thus G must contain infinite 2-subgroups. I If S is a Sylow 2-subgroup of G then the centre of S is non-trivial If z is a central involution of S then there exists an element X of 3 such that S
c C,z
E
NGX = H, say.
g. The subgroup H is 2-infinitely isolated.
Proof. If i is an involution of H such that C,i is infinite then the maximal radicable abelian subgroup R of C H iis contained in X , the maximal radicable abelian subgroup of H . If Y is the member of 9 such that CGi 5: NG y then R E Y in the same way and X n Y is infinite. Therefore X = Y and CGi e H . I h. There exists an involution i of H such that C H iisjnite.
For if not, by e one has CGi c H for every involution i of H . Thus, by
4.25 H is strongly embedded in G and G is isomorphic to PSL(2, F),
172
PsL(2, F ) A N D
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[CH.
5,
5B
PSU(3, F ) or S z ( F ) for some infinite locally finite field F of characteristic 2 by 4.24. Since none of these groups satisfy min-2 we have proved h. I We have now shown that G and H satisfy the hypothyses of 5.2 and therefore the counter example G of 5.7 is isomorphic to PSL(2,F) for some infinite locally finite field of odd characteristic. Since this is again impossible, no such counter example can exist. This proves the theorem. I Obviously, the characterization of the class of all Cernikov groups as the class of those locally finite groups which satisfy min is a corollary of 5.7. A more comprehensive description of this class of groups is given in our next result. 5.8 Theorem. For the locally Jinite group G the following properties are equivalent : a. G is a C'ernikov group; b. G satisfies min; c. The centralizer of every non-identity element of G satisfies min; d. Every abelian subgroup of G satisfies min. The special case of this result where the 2-subgroups of G are all finite is contained in Kegel and Wehrfritz [I], the full result has been announced in Sunkov [12].
Proof. Obviously, b is just a weakened form of a, c is a weakened form of b, and d is a weakened form of c. Thus it only remains to prove that the property d implies the property a. Suppose this is not so. Then among the locally finite groups of minimal 2-size which are not Cernikov groups, but in which every abelian subgroup satisfies min, there are by 3.32 simple groups. Denote by YJl the class of all these simple locally finite groups. By assumption this class is not empty. Observe that for G E 2.R by 1.G.4 each primary subgroup of G satisfies min. We shall obtain by 5.7 a contradiction to the assumption that 2.R is non-empty, if we can show (*) There exists a group G E 2.R such that f o r every involution i of G the centralizer CGi is a Cernikov group. Assume this is not so. Then for every G E ~ R we shall show that there exists an infinite sequence (Mn}nEN of perfect subgroups M, of G such that 1. M , + , $ M , , M,IZM, 1: G,EYJ~; 2. The 2-part T, of the centre Z M , of M , is finite, and T, g T,,, (thus ITnI 2 2.). To start the inductive choice of such a sequence in the group G E 2.R put M o = Go = G . . . . Suppose the subgroup M , has already been chosen subject to these requirements. Since G, E m, there is an involution t E G,
CH.
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GROUPS WITH MINIMALITY CONDITIONS
such that the centralizer CGntis not a Cernikov group. By 3.32 there are two subgroups K and R in CGntsuch that R u K and K/R = G,,, is an infinite simple group. Denote by H a n d K the preimages in M,, of R and R, respectively. Since K is not a Cernikov group, the 2-size of K must be the same as that of G. Since the simple group K/H = G,,, contains involutions, the normal subgroup H of K has by 3.13 smaller 2-size than G and thus H i s a Cernikov group. By l.G.l every abelian subgroup of G,,,, satisfies min. Hence one has G,,, E )JJz, by the definition of this class. Let M,,, be the subgroup of K generated by all the elements of order a power of two. M,, is a perfect group, for otherwise the proper subgroup (M,,,,)' of M,,,, would have smaller 2-size than M,,,, and hence be a Cernikov group; but then also M,,,, would be a Cernikov group, contradicting the properties of M,,, established so far. For the same reason, the group M,, + does not have any proper subgroup of finite index. Since by 1.F.3 the factor group M,,, ,/CMM,+ I(Hn M,,, ,) is a Cernikov group, we get
,
,
,
,
,
,
H n Mn+,= ZMn+1 and Mn+,IZMn+,
,
N
Gn+1.
,,
,
Since the factor group G,,, has the same 2-size as M,,, the 2-part T,,, of ZM,,, must by 3.13 be finite; in fact, T,,, must be contained in the radicable part of every Sylow 2-subgroup of M,,, . By construction, the subgroup M,,,, centralizes the finite group T,,. If T,, $ T,,, then for any Sylow 2-subgroup S of M,,, the 2-subgroup T,, S would have larger 2-size than S, which is impossible. Thus T,, E T,,, ,. This inclusion is proper for T,,, contains a representative for the involution t to G,. By 3.15 the 2-component of ZM,, = (M,,)' n ZM,, has order bounded by a function of the 2-size of G. This contradicts the above construction. I The group G satisfies the weak minimal (respectively, maximal) condition, if every descending (respectively, ascending) sequence { Si}of subgroups of G , such that each of the indices (St : Si,,l (respectively, ISi,, : Sil) is infinite, can only have finitely many terms. In the class of abelian torsion groups the weak minimal condition and the weak maximal condition characterize the same subclass, namely, the class of abelian Cernikov groups. In ZaItcev [ l ] a n elementary proof shows that a locally finite group which satisfies the weak minimal condition in fact satisfies the minimal condition. For US this is clear from 5.8 (for which our proof is rather involved). We formulate this as
,
,,
,
,
,
5.9 Corollary. For the locally fjnite group G the following properties are equivalent:
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PsL(2, F ) AND LOCALLY-SOLUBLE-BY-FINITEGROUPS
[CH. 5, 0 B
a. G is a Cernikov group;
b. Every abelian subgroup of G satisfies the weak minimal condition;
c. Every abelian subgroup of G satisfies the weak maximal condition.
In the next result a considerably more general class of abelian-by-finite groups will be characterized. For this we remind the reader of some notation. The group G is an %Vc-groupif the set OWc(G)of all centralizers of subsets of G, partially ordered by inclusion, satisfies the descending chain condition (or equivalently, the ascending chain condition). With this notation one has the following resalt. 5.10 Theorem. If the locallyjnite group G satisfies min-p f o r everyprimep, then the following properties are equivalent: a. the group G is abelian-by-finite, b. the group G is an %Vc-group. Proof. By 3.20 we know that every abelian-by-finite group is an %V,-group, so we have only to establish the converse under the assumptions of 5.10. If there exists a counter example to this, then there exists a counter example G, in which every centralizer # G is abelian-by-finite. By 3.22 and 3.32 we may even assume that G is an infinite simple group. As in the proof of 3.22 if Xis any subgroup of G we put X o = C x Y, where Y ranges over all the subgroups of G such that ( X :C , Y ( is finite, and we call X connected if X = XO. Let Z denote the set of all maximal connected abelian subgroups of G. By the maximal principle of set theory 3 is not empty. Further, since G contains infinite abelian subgroups (and the centre of G is trivial) the members of 8 are all infinite. a. I f X , Y ~ 8 a n d X #Y , t h e n X n Y = ( l ) .
Proof. For if X n Y # ( 1 ) then CG(X n Y ) is abelian-by-finite. But then C G ( Xn Y)' is abelian and connected and SO X = C,(X n Y)' = Y by the maximality of X and Y. I b. If X E 3, then the index ING X : XI isjnite.
Proof. By 3.21 the factor group NGX/CGXis finite. Also C G X is abelianby-finite SO (CGX)' is abelian. Hence X = (C,X)' and lCGX : XI is finite. Therefore INGX: XI is also finite. I c. If i is any involution of G there exists some X in Z satisfying C, i c NG X .
CH.
5, 5
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175
Proof. G is simple, so by 3.33 the centralizer C G i is infinite. It is also abelian-by-finite, so C = (C,i)O is abelian and connected. Thus, there exists X in 3 with C c X . Clearly X C N, C, and the latter group is abelian-byfinite (3.21). Consequently, X = (NGC)' and C,i c N,C c N,X. I It is now clear that the family 3 satisfies the conditions of 5.6, and so G is isomorphic to PSL(2, F ) for some locally finite field F of odd characteristic c. But then G does not satisfy min-c, a contradiction that completes the proof. I
Remark. In the preceding proof the minimal condition for primary subgroups was used only in two instances, for the prime 2, in order to obtain a counter example of minimal 2-size, and for the prime c, the characteristic of the field F i n the final passage. One might think that the theorem remains true if the minimal condition for all primary subgroups of G is replaced by the condition that the centralizers of involutions b.: Cernikov groups. This is not so, even if G contains involutions (just consider the example 1.H.3 with q = 2); this shows the importance of the part of the result already treated in 3.22. It seems that one should only be able to prove that a locally finite group G, in which the centralizers of its involutions are Cernikov groups, has a locally soluble subgroup of finite index. However, we have not been able to establish this partial answer to Question V.I. Note. It seems that the notion of a 2-infinitely isolated subgroup of an infinite, locally finite group G first appears in Sunkov [3], where it takes a slightly stronger form than in the present text. In Sunkov [3] our result 5.3 is stated and an outline of the proof is given. Analysing this outline, we were led to the stronger result 5.1. The connection of this notion with Frobenius groups and with strongly embedded subgroups (although Bender's definitive result 4.24 was not yet discovered) is cleary realized in Sunkov [4], as is witnessed by 5.4, the main result of that paper. In that same paper Sunkov also points out that a simple minimal counter example to the assertion, that a locally finite group satisfying min is a Cernikov group, can have only finite 2-subgroups; this argument appears in the proof of our 5.6. In Sunkov [9] as Theorem 1, a powerful reduction result is announced for the structure of such a simple minimal counter example, and in Sunkov [12] this reduction is carried out in detail and even strengthened by applying Bender's result 4.24. Then, as an afterthought, it is realized in this paper that an application of the results of Alperin, Brauer and Gorenstein [ I ] (that is essentially 4.20) allows one to show that such a counter example cannot exist. In the meantime, that is, before Sunkov [I21 appeared, the authors had submitted Kegel and Wehrfritz [l]. In this paper we presented 5.1 and then turned the re-
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PsL(2, F ) AND LOCALLY-SOLUBLE-BY-FINITEGROUPS
[CH. 5, 5 B
duction of the properties of such a minimal non-Cernikov group into the characterization of PSL(2, F') given in 5.6. The arguments proving 5.6 are largely parallel to those of the reduction in h n k o v [12]. From 5.6 we deduced 5.10 and hence also answered Question 1.5 for locally finite groups. In this paper we only managed to prove 5.8 in the case where the 2-subgroups of G are finite. In Sunkov [12] the result 5.8 is announced. Complete details appear in h n k o v [14].
CHAPTER 6
Universal Groups and Direct Limits of Symmetric Groups
In Section A of Chapter 4 we called a locally finite group enormous if for every prime p it contains finite p-subgroups of arbitrarily high exponent and of arbitrarily long derived length. The object of defining this class of locally finite groups was, essentially, to set these groups aside, for in a sense they are far too big to handle. Thus, in the preceding three chapters we have dealt only with really very small groups. This final chapter is all that we devote to their larger brethren. In the main, we shall simply construct a certain supply of enormous groups with (or without) various properties. These constructions will provide a pretext for many questions. On the whole the realm of enormous locally finite groups still seems to be in a state of perfect chaos, and there seems to be little hope of subjecting it to any principle of law and order. The first and larger part of this short chapter is an account of the construction and properties of universal groups. These are locally finite groups containing an isomorphic copy of every finite group such that any two of their isomorphic finite subgroups are conjugate. For any given cardinal there exists a universal group of that cardinal, and any two countable universal groups are isomorphic. Every universal group is simple and contains an isomorphic copy of every countable, locally finite group. Trivially universal groups are enormous. It follows readily from the definition of universal groups that they are direct limits of finite symmetric groups. In the second part of this chapter we present a number of other enormous groups with various restrictions on their subgroups. These groups are also obtained as direct limits of direct systems of finite symmetric groups. Their subgroups that are the limit of the corresponding direct system of finite alternating groups are simple and not without interest. In particular, this will give us 2 * O pairwise non-isomorphic, countable, locally finite, simple groups which are enormous. Definition. The locally finite group U is universal if 177
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6
a. every finite group can be embedded into U and b. any two isomorphic finite subgroups of U are conjugate in U.
The term universal is perhaps not a very happy choice, but we are at a loss for a better term. Its justification is that a universal group provides a sort of universe for doing finite group theory. We shall apply the term to locally finite groups only (although an analogous theory can be developed for the class of all groups). It follows directly from property a of this definition that every universal group is enormous. On the other hand, not every enormous group is universal, for in the restricted direct product of countably many finite groups, one for each isomorphism type, property b of the definition is obviously not satisfied. 6.1 Theorem. If U is a universal group, then a. For any two isomorphic finite subgroups A and B of U every isomorphism of A onto B is induced by an inner automorphism of U , b. If A is a subgroup of the finite group B, then every embedding of A into U
can be extended to an embedding of B into U; U contains an isomorphic copy of every countable, locally finite group; d. If C,,,denotes the set of all elements of order m > 1 of U, then C,,,is a single class of conjugate elements and U = C, C,; in particular U is a simple group. c. The group
Proof. a. Let a be any isomorphism of A onto B. Since the holomorph of A is a finite group, it can be embedded into the universal group U. Thus U contains finite subgroups C and G where Cis isomorphic to A, G normalizes C , and G acts on C as its full group of automorphisms. Since the group U is universal there exist elements a and b in U such that A" = Bb = C . Clearly, the mapping x I+x a - l a bdetermines an automorphism of the finite group C . Thus, there exists an element g in G with xa-Iab= xg for every element x E C. Hence, if y E A , we have yu = ( y a y - l a = y 3 b - ' ,
and so transformation by the element agb-' induces the isomorphism A onto B. I
CI
of
b. There exist embeddings q : A + U and $ : B -+ U. Clearly, $-'q induces an isomorphism of A* onto A". By a) there exists an element g E U inducing this isomorphism, that is, a*g = a' for every a E A . But then the map b H b*g is an embedding of B into U, and its restriction to A is just q. I
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c. Let G be any countable, locally finite group. Then G contains a local system of finite subgroups Gi, linearly ordered with respect to inclusion. Let n be any natural number such that for all natural numbers i 5 n embeddings ' p i : Gi 4 U have been determined such that, if i+ 1 S n, the embedding cpi is the restriction to Gi of the embedding cpi+ By b there is an embedding cpn+ of G,, into U extending pn.So, inductively, one may choose a sequence , and this sequence determines an embedding of G into U. I d. Clearly, the group U is simple if it is generated by every non-trivial conjugacy class of elements. If u and v are any two elements of order m of U, then a applied to the subgroups ( u ) and ( v ) shows that u and v are conjugate in U. Now let x be any element of U of order n say. Suppose that there exists a finite 2-generator group (a, 6 ) where a and b both have order m,and the element ab has order n. Then there exists an embedding cp of (a, b ) into U. Since aqbq and x both have order n, there exists an element g in the universal group U such that (aqb")g = aagbag= x. But this exhibits the arbitrary element x of U as a product of two elements of order m. Thus the following lemma completes the proof of 6.1. 1
6.2 Lemma. For any integers m > 1 and n 1 1 there exists a ,finite 2generator group (a, b ) such that a and b both have order m and the product ab has order n. Proof. Let ( a ) be a cyclic group of order m, (c) a cyclic group of order n, and denote by G the standard wreath product (c) 2 ( a ) of (c) by (a). The base group B of G is the set of all mappings of ( a ) into (c) with pointwise multiplication and regarded as an (a)-module via
pa : x H (4, x E ( a ) , p E B. Let cp denote the mapping of ( a ) into (c) given by
(4 cp = c, (a2)(p= c-',
(ai)cp = I, for 3 S i 5 m. (Note that m 2 2). Then for any element x (x)@
. cpa2 -
. . .-
E
(a), one has
. ..- (a"x)cp = fl(ai)cp = 1, m
?arn
= (ax)cp
and thus cpa.
*
(a2x)cp '
. . . . - cpa"
i= 1
= 1.
180
Put b
DIRECT LIMITS OF SYMMETRIC GROUPS
=
[CH.
6
a - l q . Then ab = q, which has order n since (a)qpi= ci.Also b i = q a -q a 2 * .. . . q a i .
since a - i !$ B, and
-
a-i
-
#1 forlSi<m,
b" = qa qa2 . . .
f
q)am =
1.
Thus the element b has order m. Therefore, the subgroup ( a , b ) of G is a finite group such that a and b have order m and the product ab has order n. I (See P. Hall [3], Lemma 5, for a quite different proof.)
',
Clearly C, = C,; and so by 6.1 d one has U = CC-' for every conjugacy class C of non-trivial elements of U. One would think that this is a rare phenomenon amongst locally finite (simple) groups. We have yet to show that universal groups exist, this is part of the content of 6.4 and 6.5. If universal groups exist, it is clear from 6.1 b that they have local systems consisting of finite symmetric groups, since every finite group may be embedded into some finite symmetric group. Let G be any locally finite group and S the full symmetric group on the set G. If p denotes the regular representation of G in S notice that ( x (y))"" = x(y)y = x(y) for all x and y in G. Let S = {c E S; there exists a finite subgroup F, of G satisfying (xFa)"= XF, for all x E G). This set S is in fact a locally finite group for if T = (al,c2,. . ., a,) where the ciE S, then clearly for F = (Fa, , . . ., Far) we have (xF)"= xF for every a E T and x E G. This embeds T into a Cartesian product of copies of the symmetric group on the finite set F and therefore T is finite (e.g. H. Neumann [I], 15.71). We call S the constricted symmetric group on G. Notice that if G is finite then S = S. Observe also that S depends upon the group structure of G and so is not a canonical subgroup of S. 6.3 Lemma. Let G be a locally jinite group and denote by p the regular representation of G in the constricted symmetric group S on G. Then any two jinite isomorphic subgroups of GP are conjugate in S.
Proof. Let K and K* be finite isomorphic subgroups of G and put H = ( K , K*). Denote by x w x* an isomorphism of K onto K*. Let { x i E i E I } be a complete set of left coset representatives of H in G, { y l , . . ., y , } a complete set of left coset representatives of K i n H and ( y y , . . ., y,*} a complete set of left coset represenatives of K* in H. Define the element c of S by
* *
(xiyjx)- = x i y j x
for i E I, 1 5 i 5 r and x G K .
Clearly a E S, we may take H = Fa. For any k E K we have that
cn. 61
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using that * is a homomorphism. Hence a-'kPa = k*p for every k E K, so that K p and K*p are conjugate in S. I 6.4 Theorem. There exist countable universal groups and any two such groups are isomorphic.
Proof. Define inductively a direct system of finite groups (and embeddings) as follows. Let U , be any finite group of order at least 3. If n 3 1 and the group U,, is already chosen, let U,,+ be the symmetric group on the set U,, and embed U,, into U,,, via its right regular representation. This family of groups and enibeddings clearly forms a direct system. Put U = lim U,,. Obviously U is a countable, locally finite group. For convenience of notation we shall identify the group U,, with its image in U. The order I U,,l tends to infinity with n. Hence, if G is any finite group, then there exists an integer n such that IGI IU,,l. But in this case, the group G is isomorphic to a subgroup of U,,, and hence of U. If G and H are any two isomorphic finite subgroups of U, then there exists an integerj such that
,
,
1 = r,
< r2 c . . . and 0 < s1 c s2 < . . .
and two sequences of proper embeddings 'pi
: U,, 4 V,, and
$i
: V,,
-,U,,,,
ieN
such that is the identity on U,, and $icpi+ is the identity on VSi.It fob lows that for each index i the embeddings cpi+ and ~ i + are, l respectively, extensions of (pi and $i. Thus, the sequences {qijisN and {$i)isN determine monomorphisms q : U + V and $ : V -+ U such that cp$ is the
,
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6
identity on U and $cp is the identity on V . Therefore each of these maps is an isomorphism. I
Remarks. Although the countable universal group U of 6.4 is constructed as a direct limit of finite symmetric groups, it can be constructed in many apparently different ways. For example, it is also a direct limit of groups GL(n, q ) for various n and fixed q, or of groups SL(n, q), or of PSL(n, q), etc., since each of these groups contains a copy of the symmetric group of order (n- I)!. Thus for each integer i there exists an integer ia and a subgroup Giof U isomorphic to GL(IU,l, q ) , say, such that Ui s Gi c Ui,. Then for any integer r we have
U, -C G,
-C
U,, C G,,
E Ur,2 -C
. . .,
and U is the union (direct limit) of the sequence (Gr,n}noN. We now show that the restriction to the countable case is unnecessary for the first part of 6.4; we have not been able to prove the second part in general. 6.5 Theorem. Every infinite locally finite group G can be embedded into a universal group of cardinal IGI. In particular there exist universal groups of arbitrary infinite cardinal.
Proof. If S denotes the restricted symmetric group on some countable set, then S is countable, locally finite, and contains an isomorphic copy of every finite group. Let U o = G x S . For i = 0, 1,2, . . . define Ui+l and pi : U i--* Ui+ 1, inductively by: p i is the regular representation of Ui into the symmetric group on Ui, and Ui+l is the subgroup of the constricted symmetric group on U igenerated by Uip and, for each pair of isomorphic finite subgroups of Ufi, an element of this constricted symmetric group conjugating one onto the other. So every pair of finite isomorphic subgroups of U? are conjugate in Ui+ I. Further U iand Ui+l have the same cardinal, which by induction will therefore be IGI. Clearly then U = lim Ui is a universal group of cardinality (GI that contains a subgroup isomorphic to G. I In view of the second part of 6.4 it seems natural to ask the following question.
Question VI.l. Are any two universal groups of the same cardinality isomorphic? If this is not so we suggest a rather drastic alternative. Question VI.2. If for the cardinal N >
N,
there is more than one isomor-
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phism type of universal groups, do there exist 2"' pairwise non-isomorphic universal groups of cardinal H ? In any case 6.1 c suggests the following question. Question VI.3. Does every universal group U contain an isomorphic copy of every IocaIIy Jinite group G satisfying ]GI 5 I UI?
6.6 Lemma. Let M be a subgroup of index d > 1 in the Jinite group K and let H be a semiregular subgroup of order c in the symmetric group L of order ( c d ) ! Suppose that 8 is an embedding of M into H such that the index IH : MeI > 2. Then 8 extends to embeddings 8, and 8, of K into L such that H n ISe1= M e
=
H n KO2 and
Kel # Ke2.
Proof. Let { l} u S and { l} 'J T be left transversals respectively of M in K and M e in H , where 1 4 S u T. Since H occurs as a subgroup of L in its regular representation repeated d times, we may identify the permutand of L (i.e. the set of cd elements permuted by L ) with the set P = H u SH, where S H denotes the set of formal products sh with s E S and h E H, and H operates on P via right multiplication. Since H = M e u TMe we may write P as the disjoint union of the four sets Me, TM', SMe and STM', the products being written formally. Similarly let Q denote the set K u TK of cd formal products and let the elements of k permute Q by right multiplication. In the same way Q is the disjoint union of the four sets M , T M , SM and TSM. Let $ be any bijection from the set TS onto the set ST, and denote by 8 the bijection of Q onto P given by x B = xe, (tx)' = txe, (sx) B
= sxe
and (tsx)'
= (ts)@xe,
where x E M , s E S and t E T. We may use the mapping 8 to transfer the representation of K on Q to a representation p = p ( $ ) of K on P by defining
pkP = (pe-'k)',
where p E P and k E K .
It is easy to check that p is an extension of 8; we check it for the last of the four sets and leave the other three verifications to the reader. Suppose that S E S ,t E T a n d x , EM. Then (stxe)yP =
((st)*-lXy)B= st(xy)O
= (stxe)ye.
If h is an element of H not in M e , then h maps M e into TMe, while if k is an element of K not in M then k maps M into S M and thus kP maps M e into SMe. But SMeand TMeare disjoint, and therefore H n Kp = M e .
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6
Since the index IH : M e ] = 1 +IT1 exceeds 2 (and the order of S is at least 1) there exist bijections rl/l and rl/z of TS onto ST such that for some t E T we have ( t s y # (ts)? For i = 1, 2, let Bi = ~ ( r l / ~ )Thei . B j is an embedding of K into L extending 0 such that H n Ke' = Me. We have only to show that Kel # K e 2 . The orbit t K = tM u tSM of K containing t corresponds to the orbit
tKei = t M e u (tS)'iMe of Kei containing t. Hence if K"
=
(tS)@lMe= tKel n STM'
But then (tS)$l Kel # Ke2. I
=
K e Z ,then =
tKeZ n S T M e = (tS)$zMe.
(IS)", which contradicts the choice of t. Therefore
We may now prove a much strengthened version of 6.1 c.
6.7 Theorem. If V is any universal group and G is any countabZy infinite, locallyfinite group, then there exist at least 2'O distinct subgroups of V isomorphic to G. Proof. Since G is countably infinite and locally finite, the group G contains a tower (1) = Go c G, c . . . c Gi c . . . c G
uz,
of distinct finite subgroups such that G j = G . For each natural number i 2 1 put di = (Gi : Gi-,l. Clearly we may assume that d, > 3. Let c1 = d , , and for i > 1 define ci = c i - , ! di. Set Ui equal to the symmetric group of order ci!,and for each i embed Ui into U i + by a semiregular representation (i.e. its regular representation repeated di+ times). It follows easily from 6.3 (c.f. the proof of 6.4) that U = Iin~Ui is a countable, universal group. Again identify Ui with its image in U. Suppose that 0 is an embedding of G jinto Uj. Notice that lGi[ = dl d 2 .. .di while
,
lUil = (. . . ( d , ! d z ) !d3 . . . d i - , ) ! di)! > 2dld2 . . . d i ,
since d , > 3. Taking M = G i , K = G i + l , H = Uiand L = U i + l in 6.6 we see that there exist embeddings and O2 of G i , into U i , extending 0 such that
,
e
Ui n G i i l = GB = Ui n GB:,
and
GB:,
,
#
G21.
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DIRECT LIMITS OF SYMMETRIC GROUPS
185
It follows that there exist 2*' distinct sequences {cpi}ieN, where 'pi is an embedding of G , into U iextending cpi-l and satisfying u,-, n GT* = Glpi-l 1-1 such that for any two distinct sequences (cp,} and {1+9~}there existsj such that GY # GP. The sequences {cp,} and {$,} determine embeddings cp and II/, respectively, of G into U and uj n GQ = ~j"'# ~ j "=' uj n G*.
Hence cp # I+9, and there exist 2" distinct subgroups of U isomorphic to G . Finally by 6.1 c the universal group V contains a subgroup isomorphic to the countable group U . I If A and B are isomorphic subgroups of the universal group U one would like to have some additional information about how the embeddings of A and B into U are linked. In general, A and B will not be transformed into one another by any automorphism of U . This becomes obvious if one considers as A any maximal elementary abelian p-subgroup of U for some fixed primep and for B any proper subgroup of A such that IAl = IBI, since then A is a maximal elementary abelian p-subgroup but B is properly contained in some elementary abelian p-subgroup. Thus there cannot be any automorphism of U transforming A into B. We now suggest another way in which the above question may be made more precise. Question VI.4. If U is a universal group and $ A and B are isomorphic subgroups of U, do there exist two endomorphisms and E~ of U with A s U E A and B G U", such that AEAA-IEB = B? What can be said about the group of automorphisms of a universal group? Here we have very little precise information. 6.8 Theorem. The automorphism group A of the countable universal group U has cardinal 2".
Proof. Clearly IAl 5 2 * O . By 6.4 and its proof, the countable universal group U is the union of an ascending sequence ( U i ) i p Nof subgroups Ui which are isomorphic to the symmetric groups of orders nl!, where nl 2 3, I t i + , = n, !, and U iis embedded into U i t l via its right regular representation. Put C,+l = Cui+lUi. Then the subgroup C,+l is essentially the left regular representation of U iin U , , , (see M. Hall [2], p. 86, 6.3.1). In particular, IC i+ I 2 2 for each natural number i. Choose c1 E U , and for every natural number i > 1 choose any element ciE C , . If cpi denotes the inner automorphism of U idefined by
.
xclc2.. ci
9
XE
ui,
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DIRECT LIMITS OF SYMMETRIC GROUPS
[CH.
6
then for every natural number i the automorphism 'piof Ui is equal to the restriction to Ui of the automorphism cpi+ of Ui, Thus the sequence {cpi)iEn determines an automorphism cp of U. Let {dJiSn be a second sequence of elements of U with dl E U,and di E Ci for i > I, and {$JieN the correponding sequence of inner automorphisms of the groups Ui determining the automorphism $ of U . We shall suppose that for some natural number j one has c j # d j , and we shall prove that this entails cp # $. I f j = 1, then (p, # $, since the group U , has trivial centre, and thus cp # $. I f j > 1, then for all elements x E C j one has x(P
=
x(Pj
=
xCICz.,.Cj
- xC'
since clc2 . . . c j - l E U j - l C C u C j .Similarly xs = xdl. Since the subgroup C j is isomorphic to the symmetric group of order n j - l ! its centre is trivial. and thus the inner automorphisms of C j induced by two distinct elements are distinct. Thus x" = xcJ# xdJ= x@ for at least one element x E Cj, and so cp # $. The number of distinct sequences of the sort described is clearly Po. I Observe that in the preceding proof we have in fact exhibited the existence of 2N0automorphisms of the universal group U which preserve the local system { Ui}isNof U . Does every automorphism of the countable universal group U preserve such a local system? In terms of the following definition one can prove a slightly weaker statement. Definition. The automorphism a of the group G is called locally inner if for every finite set F of elements of G there is an element g = gF of G such thatf" = f for every element f E F. The set of all locally inner automorphisms of the group G is a normal subgroup of the group of all automorphisms of G. If G is any locally finite universal group, then every automorphism of G is locally inner. Question VS.5. What can be said about the cardinality, the structure, and the action on U of the group of all automorphisms of an arbitrary universal group U? Question VS.6. Does one have for every countably infinite, locally finite simple group G the equality /Aut GI = 2'O? We now construct a number of other enormous groups; they will also all be direct limits of finite symmetric groups. We begin by considering the restricted symmetric groups S on a countable set (which we take to be N for
CH.
61
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187
convenience: S is then the group of all permutations of N that fix all but a finite number of integers). If we write S,, for the set of those elements of s that fix all the integers exceeding n then S = Si.Clearly S,, is naturally isomorphic to the symmetric group on the set { 1 , 2, . . ., n ) , and so S is a direct limit of finite symmetric groups. In particular S is locally finite and countably infinite. 6.9 Proposition. The restricted symmetric group S on the set N is a countable, enormous locally finite group that does not contain any non-identity radicable subgroups. In particular neither S nor any of its subgroups are universal. Proof. If G is any finite group, then G is embeddable into Slcl and hence into S (using the notation introduced above). In particular S is an enormous group. Let x be any element of S of prime order p . The orbits of x have length 1 or p and the support of x
supp (x) = { n E N; ng
z n},
is by definition of S a finite set. If y is ap'-th root of x in S then the maximal orbits of y have length pr+' and are unions of p r nontrivial orbits of x. Hence each of these maximal orbits is contained in supp(x) and consequently~"' 5 Isupp(x)l. It follows that the element x does not havep'-th roots in S for arbitrarily large r and hence it does not lie in any radicable subgroup of S. Therefore S cannot contain any non-identity radicable subgroups. Since there do exist countably infinite, locally finite radicable groups, 6.1 c implies that no subgroup of S is universal. I Thus the restricted symmetric group S of 6.9 does not contain any Priifer subgroups. However there exist direct limits of finite symmetric groups that contain Prufer p-groups for every prescribed set of primes p. 6.10 Theorem. If n is any set of primes there exists a countably infinite, direct limit G of $nite symmetric groups that contains a Priifer p-subgroup if and only if p E n. Proof. If n is empty then 6.9 provides a suitable example; henceforth we shall assume that 7c is not empty. Let m > 1 be an integer, let S,, denote the symmetric group on {I, 2, . . .,n } and embed S, into S,, by repeating the natural permutational representation of S, m times. That is S,, acts on ( 1 , 2, . . ., mn} by acting naturally on the m sets {kn+ 1 , k n + 2 , . . ., (k+ l ) n } where k = 0, 1 , . . ., m - 1. Clearly every element of S, has an m-th root in S,,,,,.
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6
Now let q be any prime not dividing m and suppose that x is an element of S,, of order q r . The maximal orbits of x have length qr. Suppose that x has s orbits of length qr in its natural representation on {1,2, . . ., n } , so that it has ms orbits of length q' as an element of Smn.If y is a q'-th root of x in S,,, then y will have msq-' orbits of length qr+'. Thus q f divides s and in particular qf 5 n. We now construct our group G. Let {pi}ienbe a sequence of primes from n such that every prime in n occurs in this sequence and occurs infinitely often. Such a sequence exists since n is non-empty. Let no be any integer and let Go denote the symmetric group of order no! If Gi is the symmetric group of order n i ! , put n i f 1 = p i + l n i , let G i + l denote the symmetric group of order ni+1! and embed G i into G i + l via its natural permutational representation repeated p i + l times. Now set G = lim G i . Again it is convenient to identify each Gi with its image in G. Let p E n and let g be any p-element of G. There exists an integer i such that g E Gi and p = pi+1 . Then by the first paragraph of the proof the element g has a p-th root in Gi+ and hence in G. It follows that G contains Priifer p-subgroups. Now let q be a prime not in n,x a q-element of G and y a q'-th root of x in G. Suppose that i is the least integer with x E G i and let y E G j . The embedding of Gi into Gj is simply its natural representation repeated pi+l p i + z. . .pj times. By the second paragraph of this proof t is bounded by the order of Gi and thus x does not have q'-th roots in G for arbitrarily large t. Consequently G does not contain a Prufer q-subgroup. I 6.11 Corollary. The groups G constructed in 6.10 are simple i f 2 E n and contain a simple subgroup of index 2 otherwise. Proof. Each of the groups Gi contains the alternating subgroup Hi of index
2.The embedding of Gi in Gi+ embeds H iinto H i +1 . Hence H =
uz
Hi is a simple subgroup of G of index at most 2. If n i + l = 2ni then the embedding of G i in Gi+ actually embeds Gi into H i +1. Thus if 2 E n then Gi E Hi+ for infinitely many i, and G = H . If 2 $ n then for each i, every 1. odd permutation of Gi is still odd when regarded as an element of Gi+ Hence if t is a transposition of GI, then t 4 H i for every i. Therefore t (-4 H and IG : HI = 2. The same remarks clearly hold for the group S of 6.9 (the I case 71 = 0). 6.12 Corollary. There exist cally finite, simple groups.
2'O
non-isomorphic, countable, enormous, lo-
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61
DIRECT LIMITS OF SYMMETRIC GROUPS
189
Proof. In view of 6.11 the construction of 6.10 certainly leads to 2‘O nonisomorphic, countable, locally finite, simple groups that are direct limits of finite alternating groups. If {nil is any strictly ascending sequence of positive integers an d p is any prime, then the exponent and the derived length of a Sylow p-subgroup of the symmetric group of order n i ! both tend to infinity as i tends to infinity (see M. Hall [2] 0 5.9 and Huppert 111.15.3). The same statement then clearly applies to the corresponding alternating groups and hence each of the simple groups of 6.11 is enormous. I Question VI.7. Are there direct limits G of finite symmetric groups which for a given set E of primes are z-universal, that is, any two isomorphicfinite E-subgroups of G are conjugate, but for no prime p # E is there a Priifer psubgroup in G?
Note. The discovery of the universal groups and their principal properties was made by P. Hall. In particular most of the results 6.1 to 6.7 may be found in Q 1 of P. Hall [3]. The remainder of that paper contains the construction of 2 ” O non-isomorphic, countable, but verbally complete, locally finite p-groups. Much of the residue of this chapter has no doubt been common knowledge for a long time.
Final Comments
Between writing the manuscript and reading the proofs of this book, two papers have come our way which, had we seen them earlier, would have influenced our writing. The first is Sunkov [17]. In this paper Sunkov proves the following: Theorem. Let G be a locally finite group that satisfies min-p f o r every prime p . Then for every prime p the maximal p-subgroups of G are all conjugate. This result should be compared with 3.3,3.4 and the examples of Wehrfritz [71. The second paper is Wilson [3]. It contains yet another proof of Sunkov’s Theorem 3.29, but this proof is a great improvement on earlier ones. Wilson argues by induction on an invariant of his own. However, one can also do it by induction on the p-size of the group and still use Wilson’s basic trick to simplify the proof. It seems nicer to do this, and we indicate it below. The argument uses 4.3 (and thus 4.1 and 4.2) in place of 3.28 (and thus 3.23 and 3.24). If S is any finite group, then S has a unique normal subgroup b,S that is minimal subject to S/b,S being p-soluble. b, is a functorial co-radical rule in the sence of Chapter 1, Section B, and extends smoothly to a functorial co-radial rule defined on the class of locally finite groups. This rule we still denote by b,. If G is any locally finite group, then b,G is the unique normal subgroup of G that is minimal subject to G/b,G being locally p-soluble.
Proposition. Let G be a locally finite group satisfying G = b,G, and suppose that G has a locally p-soluble normal subgroup H such that GIH is infinite and simple, and satisfies min-p. Then G has a local system C of finite subgroups S such that each S E: C has a unique maximal normal subgroup M , and M J ( H n M,) is a p’-group. Proof. Denote the natural projection of G onto G / H by x 190
H
E. There
191
FINAL COMMENTS
exists a finite subgroup P of G such that P is an abelian p-subgroup of maximal rank. P is non-trivial since clearly G is not a p’-group. By 4.3 there is a local system Z of finite subgroups of G containing P such that f o r each S E Z there exists a maximal normal subgroup Ms of S satisfying H n S c M , a n d P n ~ , = (1). Suppose that S is an element of .Z with P c b,S. If S/M, is p-soluble, then b,S E M , and P n b,S c H. This contradicts the non-triviality of P , so S/M, is not p-soluble. It is also simple. Hence S = M,. b,S and M , n b,S is a maximal normal subgroup of b,S such that H n b,S
c M,
n b,S
and P n ( M , n b,S)
=
(1).
Also G = b,G, so {b,S: P c b,S, SEZ} is a local system of G too. Therefore we may assume that S = b,S for every S E Z. Let S E Z, and choose a Sylow p-subgroup Q of the finite group P M , that contains P . If M , is not a p‘-group, then M , contains a non-trivial central element S of Q. But then ( P , 2 ) is an abelian p-group of rank exceeding that of P . This contradicts the choice of P and consequently h?, is a p’-group. Suppose that N, is a second maximal normal subgroup of S. Then S = M,. N,, whence SIN, is p-soluble. But S = b,S and therefore no such N , can exist. The proof is complete. I Proof of 3.29. The ’if’ part of the proof proceeds as before. Let G be a locally finite group with min-p such that every p-element of G has finite range. Suppose that IG : Opt,GI is infinite and that G has minimal p-size subject to this restriction. We seek a contradiction. By 3.16 and 3.17 we may assume that G = b,G. Let N be any proper normal subgroup of G. Then N has smaller p-size than G and hence IN : O,,,NI is finite. Set C = C,(N/O,.,N). If C # G, then as above IC : O,,,CI is finite. But C has finite index in G, so IG : O,,,GI is finite. This contradiction proves that C = G, and consequently shows that N 5 H = OP,,,,G. It follows from 3.16 that H is a p-soluble maximal ncrmal subgroup of G of infinite index. Trivially G contains p-elements not in H, and so by the above proposition and 1.K.2 G contains a p-element with infinite range. This contradiction shows that no such group G can exist and completes the proof of 3.29. I Apart from his proof of Sunkov’s theorem, Wilson’s paper contains several other interesting results. We quote a couple below. Theorem. If G is a locallyjkite group satisfying min-p then there exists
192
FINAL COMMENTS
an integer n depending only on the p-subgroups of G such that each series of G has at most n factors that are not p-soluble.
Corollary. Let G be a locally Jinite group with min-p that is not a p’-group. a If G is simple, then G is absolutely simple.
b If G is characteristically simple, then G is a direct product of a finite number of isomorphic simple groups. In recent work, H. Heineken and J. S . Wilson have constructed a torsionfree, locally soluble group satisfying the minimal condition for normal subgroups. This gives a negative answer to Question 1.6; see “Locally soluble groups with Min-n”, J. Aust. Math. SOC.to appear.
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Index
Absolutely simple group 5 Approximating sequence 116 Ascendant subgroup 5 Ascending series 5 Basis normalizer 27 Carter subgroup 27 Cernikov group 3 1 C-group 147 Characteristically simple group 5 Chief factor 5 Chief series 5 Classically parameterized (very) 120 Class of groups 5 1-class pair 132 Co-finite filter 65 Composition factor 5 Composition series 5 Connected (Dt,-group) 97 Constricted symmetric group 180 Co-radical rule 12 Coupled functors 127 Descendant subgroup Descending series 5 Dihedral group 45 Directed set 55
5
Engel condition (left) 41 Engel group 41 Enormous group 122,177 Factor (of a series) 4 Faithful representation 66 9-equivalence 65 Filter 64 Finite functor 127 Formation theory 27 Frobenius complement 50
Frobenius group 50 Frobenius kernel 50 Hall subgroup 27 Hirsch-Plotkin radical 15 Hyperabelian group 5 Hypercenter 10 Hypercentral group 10 Hyper-Z-group 5 Index category 126 2-infinitely isolated subgroup 153, 144 Inverse limit 54 Inverse system 55 Inverted (by an involution) 75 Involution 45 X-subgroup 89 Large group 122 Limit of simple groups 116 Linear group 59 Linear representation 65 Locally cyclic group 7 Locally dihedral group 47 Locally finite group 2 Locally finite-normal group 4 Locally inner automorphism 186,56. Locally quaternion group 48 Locally H-group 8 Local system 8 Maximal p-subgroup 19 YJtc-group 96, 174 Minimal condition 28 min-p 80 Nearly regular automorphism n-finite group 2 Normal functor 130 209
106
210
Normal index 130 Normal series 4 Normalizer condition
INDEX
Smidt’s problem 68 Stability group 17 Stabilize 17 Stable linear group 119 Strongly embedded subgroup Subnormal subgroup 5 Substitute 118 Sylow basis 27 Sylow subgroup 26, 85 Symmetric group (full) 180
11
Periodic FC-group 4 Periodic group 2 Principal ultrafilter 65 Projective representation Prufer group 30 p-size 92
65
Quasi-dihedral group 48 Quaternion group (generalized) Radical rule 12 Range 99 Reduced product 65 Reducible (p-group w.r.t. a local system) Refinement (of a series) 4 Regular cardinal 69 Restricted symmetric group 186 Rule 12 Self-coupled functor 127 Serial subgroup 5 Series 4 Size (of a p-group) 92 Smaller p-group 92
Tarski monster 30 Three-subgroup lemma Torsion group 2
48
20
17
Ultrafilter 64 Ultraproduct 65 Unipotent element 59 Unipotent group 60 Universal group 177 Weak maximal condition Weak minimal condition Wider category 127 Wider functor 127 X-group of index N X-series 5
131
173 173
14i