On infinite sharply multiply transitive groups

WILLiAM KERBY

Oninfinite sharply multiply transitive groups

VANDENHOECK& RUPRECHT IN GOTTINGEN

Hamburger Mathematische Einzelschriften Neue Folge - Heft 6

Herausgegeben vom Mathematischen Seminar der Universitat Hamburg

ISBN 3-525-40307-0 Gedruckt mit Untcrsi.itzung der Deutschen Forschungsgemeinschaft © Vandenhoeck & Ruprecht, Gottingen 1974. - Printed in Germany.

Ohne ausdriickliche Genehmigung des Verlagsages ist es nich t gestattet, das Bueh oder Teile daraus auf foto- oder akustomechanischem Wege zu vervielfaltigen. Herstellung: Hubert & Co., Gottingen

TABLE OF CONTENTS

Introduction

5

Chapter I. Multiple Transitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

§ 1 . Conventions and Some Basic Results . . . . . . . . . . . . . . . . . . . . § 2 Type Classification of Sharply k·transitive Groups . . . . . . . . . .

8 10

Chapter Il Sharply 2-transitive Groups . . . . . . . . . . . . . . . . . . . . . . § § ·§ '§ §

3 4 5 6 7

Conjugate Classes and Involutions . . . . . . . . . . . Groups of Type 1 . . . . . . . . . . . . . . . . . . . . . Kernels and Complements . . . . . . . . . . . . . . . Near-domains and Sharply 2-transitive Groups. . . Near-fields, Skewfields, Fields, and Sharply 2-transitive Groups . . . . . . . . . . . . . . . . . . . . . § 8 Theorems on Near-domains . . . . . . . . . . . . . . . § 9 Further Theorems on Sharply 2-transitive Groups

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. . . .

. . . .

. . . .

. . . .

12 16 17 21

........... ........... ..........

25 27 29

Chapter Ill Sharply 3-transitive Groups. . . . . . . . . . . . . . . . . . . . . .

46

68

Index of Notation

71

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structure Theorems for Sharply 3-transitive Groups. KT-fields and Sharply 3-transitive Groups . . . . . . . Theorems on KT-fields . . . . . . . . . . . . . . . . . . . Further Theorems on Sharply 3-transitive Groups .

. . . .

. . . .

46 55 58 63

10 11 12 13

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§ § § §

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. . . .

12

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. . . .

. . . .

. . . .

.... ..... ... ................. .......

INTRODUCTION Finite sharply multiply transitive permutation groups have been studied for many years. As early as 1872, C. Jordan [17] discovered that other than the symmetric groups on n letters, for n 4, the alternating groups on m let6, and the Mathieu groups of degrees 11 and 12, there are no ters, form sharply k-transitive groups for k>4 (see Theorem (1.6)). In 1936 H. Zassenhaus determined all finite sharply 2-transitive and sharply 3-transitive groups in the following sense. It can be shown that every finite sharply 2-transitive group is isomorphic to the group of transformations x + mx on a finite near-field (see for example [14] pp. 382-392). Also it can be shown that every. finite sharply 3-transitive group is isomorphic to the group of transformations x

+ m x, with suitable conditions on + n ox

b, m, n, where

' addition and division are those of a properly chosen Galois field and (o) is either the field multiplication or in certain cases a proper near-field multiplication (see [35] or [46]). In [47] Zassenhaus determined all finite near-fields and in [46] he determined all finite near-fields which give rise to sharply 3-transitive groups. J. Tits [39] (1949) also gave a similar proof of Zassen· haus' classification theorem for finite sharply 3-transitive groups, and in 1962 B. Huppert [15] presented a proof of this theorem using group theoretical methods exclusively without resort to the counting processes employed by Zassenhaus and Tits. In the infinite case relatively little has been done. In the papers [40] (1950), [43] (1952) pp. 73-79 of Tits and [13] (1954) of M. Hall the theorem of Jordan was generalized to the infinite case (see Theorem (1.7)) and it was shown that an infinite sharply multiply transitive permutation group must be either sharply 2-transitive or sharply 3-transitive. By placing topological conditions on sharply k-transitive groups, Tits in [41] (1951) generalized a theorem of B. Kerekjarto [30] (1941) and proved that a locally compact connected sharply 3-transitive group is isomorphic to the group of transformations x

a+mx b

.

+ nx, With an

.

bm over the real or the complex numbers. This result

was also obtained at about the same time by H. Freudenthal [10]. In [42] (1952) Tits proved thata locally compact connected sharply 2-transitive group is isomorphic to the group of transformations x a + mx, over either the real numbers, the complex numbers, the quatemions, or the Kalscheue r nearfields(see [18]).

6

Introduction

In 1965 in [20] and later in [21] (1968) H. Karzel introduced the concept of near-domain "Fastbereich" as an algebraic structure suitable for describing infinite sharply 2-transitive groups in terms of transformations of the form a+ mx (see 6). In [27] (1971) it was shown that certain near-domains called are also suitable for describing infinite 3-transitive groups (see 11). The concepts near-domain and KT-field can be considered to be refinements of the concept ofpseudofield due to Tits [39] (1949). In [12] (1963) Gratzer also introduced algebraic structures for describing sharply 2-transitive groups which are not essentially different from the pseudofields of Tits. Near-fields and KT-fields have the advantage over pseudofields that all sharply 2-transitive and sharply 3-transitive groups can be uniquely described by means of near-domains and KT-fields respectively (see Theorem (6.3) and Theorem (11.3)). However, in the case of pseudofields, two non-isomorphic pseudofields may describe one and the same sharply 2-transitive or sharply 3-transitive group. From the definition (see 6) one sees that every near-field is a near-domain. To my knowledge examples of near-domains which are not near-fields are not known, and hence the following question arises. Is every near-domain a nearfield? As Karzel has shown, a near-domain is a near- field if and only if in the sharply 2-transitive group described by the near-domain, the set 1 2 , where J is the set of involutions of the group, forms a subgroup(see Theorem (7.1)). A similar statement holds for KT-fields and sharply 3-transitive groups where of products of two involutions which have the set 1 2 is replaced by the sets a common fixed point a (see Theorem (13.1)). One may now ask, if, in a given sharply 2-transitive group the set forms a subgroup, or in the case of sharp 3-transitivity, if the sets form subgroups. It is the purpose of this paper to present all results known to the authorrelevant to these questions. TI1e structure of infihite sharply k-transitive groups, for k =2,3 is examined with particular stresson the VWUXFWXUH of the sets and Proofs of theorems are given whenever they do not appear, or are not easily accessible in the literature. In chapter I some basic properties of multiply transitive groups are given and a type classification for sharply k-transitive groups is derived which is useful throughout the paper. Chapter II is devoted to the study of infinite sharply 2-transitive groups. A number of conditions are given which are equivalent to the condition .that 1 2 is a subgroup. The concept of characteristic for sharply 2-transitive groups is introduced and it is seen that if a group has characteristic 3, then 1 2 forms a 2 contains subgroup. Also it is shown that every near-domain F with char a largest sub near-field E and ifF then multiplicatively, F is an infinite exten-

Introduction

7

sion of E. Some of the results of§ 9, in particular (9.10), indicate that a sharply 2-transitive group in which J 2 ist not a subgroup must have a very complicated structure. In chapter III we consider sharply 3-transitive groups. It is seen that the sets form subgroups for an infinite number of characteristics, namely for all primes p with p 1 mod 3 (see Theorem (13.5)). A class of sharply 3-transitive groups is introduced which contains all examples of sharply 3-trat1sitive groups known to the author. Some examples of infinite sharply 3-transitive groups are given and it is seen that many characteristic properties of finite sharply 3-transitive groups do not carry over to the infinite case. In closing we note. that the connection between certain geometrical incidence structures and sharply 2-transitive and sharply 3-transitive groups has been discussed under various aspects often in the literature (see for example [1 ], [4], [7], [16], [21]). Some of the terms such as desarguesian and pappian which are applied to sharply multiply transitive groups in this paper derive from these geometrical interpretations. The author would like to thank Helmut Karzel and Heinrich Wefelscheid for their many helpful suggestions and their careful reading of the manuscript.

CHAPTER I. MULTIPLE TRANSITIVITY

§ 1. Conventions and Some Basic Results The notation used in this paper is described on the last page. Many of the properties of transitive groups presented in this section are elementary and will be stated without proof. The proofs are readily available in the literature, for example in [14], [35], or [38]. Throughout this paper M denotes a set of elements, finite or infinite, and Sym (M) denotes the V\PPHWULFgroup of bijections of M onto M. If M is fi. nite, we also write Sym (n) where n M Alt (n) denotes the alternating group on M. A permutation group on M is a subgroup of Sym (M) and we shall denote such groups by (G, M), or simply by G, whenever it is clear that G operates on the set M. If M is finite, then M is called the degree of (G, M). For positive integers k, we define a proper k-tuple to be a k-tuple (a 1, ... , ak) of distinct points ai EM, i = 1, ... , k. A permutation group (G, M) with M k is called k-transitive, if for any two proper k-tuples ...,, ak) and (b 1 , ...,, bk) there exists an element E G such that = fori= 1, ... , k. (G, M) is called sharply k-transitive, if the element is unique. If k = 1, we speak simply of transitivity or simple transitivity and a sharply transitive group is said to be regular. Definition: Let (G, M) be a permutation group and suppose G' is a subgroup of G. A subset M' C M is said to be G'-invariant if (M') C for all E G'. In this case we may consider G' to be a·permutation group operating on G' is called a (sharply) NWUDQVLWLYH subgroup of G if there exists a G'-invariant subset M' Msuch that (G', M') is (sharply) k'-transitive. ... , an) the stabilizer i = 1, .. . ,.n} is a subgroup and an lr

For each proper n-tuple

(ai)

every

=ai,

E G.

=

an= ,

,

G:

(an for

(1.1) Let (G, M) be a transitive permutation group. (a) (G, M) is (sharply) k-transitive IRUN!2 if and only if the groups Ga are

(sharply) (k-1)-transitive on M\{a}.

(b) G is 2-transitive if and only if G = Ga U Ga rGa, for any E G \Ga. In this c_asethe groups Ga, a EM, form a conjugate class of maximal subgroups ofG ([35]pp.J6-17).

§ 1. Conventions and Some Basic Results

9

Definition: A subset H C G of D per.mutation group (G, M) is said to operate transitively on M if, for every pair a, b EM, there exists E H such that =b. If is unique, we say that H operates regularly onM H is said to .operate transitively on distinct points, or regularly on distinct points, if the transitivity, or regularity, of H is asserted only for distinct points a, b EM (1.2) Let {G, M) be a transitive permutation group and let H be a transitive subset of G. Then G HGa = ([35] p. 25). In the study of infinite sharply multiply transitive groups, certain conjugate classes play an important role. Some elementary properties of conjugate classes in transitive groups will be presented here.

(1.3) All elements ofa conjugate class in a permutation group have the same number of fixed points. Proof: Let E G. For p EM we have: (p) = p = Thus induces a bijection from the fixed points of onto the fixed points of We shall consider {id} to be the trivial conjugate class. (1.4) Every nontrivial conjugate class H in a 2-transitive group (G, M) satisfies the following conditions: (a) H operates transitively on distinct points. (b) = H U {id} operates transitively on M, and hence G = HGa =Gall (see (1.2)} (c) If the elements of H have fixed points, then H acts transitively on M, and hence G =HGa =GaR

Proof: Let H be a nontrivial conjugate class, then H = {oao- 1 : 6 E G}, for some a E G, a :f::. id Now let p, q EM with p :f::. q. Since a :f::. id, there exists rEM such that a (r) :f::. r. By 2-transitivity there exists 6 E G such that o(r) = p and 6a(r) = q, and hence 6a6- 1 (p) = q. Thus for~ 6ao- 1 , we have~ E H and ~(p) = q. Parts (b) and (c) follow directly from (a) and the transitivity of G. The group Inn G of inner automorphisms of the per.mutation group G is itself a per.mutation group which operates on the elements of G, and the orbits of Inn G are the conjugate classes of G. The elements of Inn G will be denoted by&, for a E G, where &{3 = afja- 1 , for all~ E G. Let Z (G) denote the center of G. (1.5) If{G, M) is 2-transitive and if IM I> 2, then Z(G) = id, and hence Inn G~G.

10

§ 2. Type Classification of Sharply k·transitive Groups

Proof: Let a; E G \{id} and a EM such that a; (a) =/::a. Since 1M I > 3, there exists b EM with b Et {a, a; (a)}. By 2-transitivity, there exists 6 E G such that o(a) =a and oa:(a) =b. But oa:(a) =band a:o(a) = a:(a) imply oa; =/:: ao, and hence a: Et Z (G). Thus Z (G) = id In conclusion of this section we state Jordan's theorem on finite sharply k· transitive groups, and Hall's generalization of this theorem. We first note that Sym (n) is sharply n-transitive and sharply (n-1)-transitive of degree n, and if n ~ 3, Alt(n) is sharply (n 2)-transitive of degree n (see [3 5] p. 279). The groups Sym (n) and Alt (n) will be considered as being trivial. Theorem (1.6) (C. Jordan [17])Let G be a finite nontrivial sharply k-transitive group of degree n. If k > 4, then we have either k 4, n = 11, and G is the Mathieu group M 1 1! or k = 5, n = 12, and G is the Mathieu group M 12• Titeorem (1.7) (M. Hall [13]) Let (G, M) be 4-transitive, where M is finite or infinite. If the groups Gal> ... , a. are of finite odd order, then G is one of the following groups: Sym(4), Sym(5), Alt(6), Alt(7), or M 11 • In particular, if (G, M) is an infinite sharply k-transitive group where k then k = 2 or k = 3.

> 1,

§ 2. Type Classification of Sharply k-transitive Groups In this section we present a classification of sharply k-transitive groups, for 1 < k ..;;; 5 in terms of the numbers of fixed points which the elements of certain distinguished conjugate classes possess. Theorem (2.1) Let (G, M) be a sltarply k-transitive group. If 1 < k ..;;; 5, then, for any integer i &uch that 0..;;; i..;;; k 2, the set Hk-i of all elements in G of order k i which have at least i fixed points forms a conjugate class of G. In particular, the set Hk of all elements of G of order k forms a conjugate class of G.

-~-~

-··~·--·

Proof: We note thatHk-i ist not empty, since by k-transitivity, for any k distinct points of M, there exists an element in Hk-i which fixes i of the points and acts cyclicly on the rest. We show that, for every a: E Hk-i there exists a EM such that a, a(a), a: 2 (a), ... , ~-,i- 1 (a) are distinct. Now 1 < k..;;; 5. If k i is a prime, then a: generates a simple cyclic group and a, a: (a), ... , a;k- i-l(a) are distinct for ahy a EM with a; (a) =/::a. If k- i = 4, and if a, a(a), a: 2 (a), a: 3 (a) were not distinct for every a EM, then a:(a) =a or a: 2 (a) =a, for each a E G. This implies the contradiction a: 2 = id.

§ 2. Type Classification of Sharply k-transitive Groups

11

Now let a,~ E Hk-i> a~da, b EM such that a, a(a), ... , ak-i- 1 (a) and b, ~(b), ... , ~k- i-1 (b) are distinct. Let ai, bi, j = 1, ... , i be fixed points of a and ~ respectively. By k-transitivity there exists o E G such that q(ai) = bi, for j = 1, ... , i, and oal(a) =~l(b), for /A= 0, 1, ... , k-i-1. Then 8a(bj) = oa(ai) = o(a) = bi, for j = 1, ... , i, and oa(JJI(b)) = [j~l+1(a) = ~/+1(~), for I= 0, ... , k-i-1. By sharp k-transitivity we have~ =oa, since~ andoa agree on b~> .. ., bi> b, a(b), .. ., ak-i-1(b). Theorem (2.1) and (1.3) allow us to make the following type classification of sharply k-transitive groups for 1 < k.;;;; 5.

Definition: Let G be a sharply k-transitive group and suppose 1 < k .;;;; 5. Let

k; be the number o( fixed points of the elements of the conjugate class Hk- i• and define hi = ki- i. Then 0 .;;;; hi .;;;; k- i -1 and G is said to be of type hi> ... , hk- 2).

0o,

ln the case of infinite sharply multiply transitive groups, we have k = 2 or k = 3. If G is a sharply 2-transitive group, finite or infinite, then the set H2 of involutions of G forms a conjugate class and G is of type n, for n E {0, I} whenever the elements of H2 have n fixed points. If G is sharply 3-transitive, then the set H 3 of elements of order 3 and the set H2 of involutions which have at least one fixed point are conjugate classes and G is of type (m, n), form E {0, 1,2} and n E {0, 1}, whenever the elements of H 3 have m, and the elements of H2 have n + 1 fixed points. This is equivalent to saying that G is of type (m, n), if the elements of order 3 have m fixed points and the sharply 2-transitive groups Ga are of type n.

CHAPTER II. SHARPLY 2-TRANSITIVE GROUPS

In this chapter, unless otherwise specified, any group denoted by G will be assumed to be a permutation group which operates sharply 2-transitively on a set M. By § 2, the set J = H2 of involutions of G forms a conjugate class in G, and G is of type 0, whenever the elements of J have no fixed points, and G is of type 1, whenever the elements of J have exactly one fixed point The main purpose of this chapter is to examine the structure of inifinite sharply 2-transitive groups in general, and in particular, to examine the structure of the subsetz J2 = {t.tv : t.t, v E J}. To my knowledge, no examples of sharply 2-transitive groups in which J 2 is not a subgroup are known. However, if G is finite, then J 2 is necessarily a subgroup of G, and in fact G is planar (see § 7). This chapter is organized such that those properties of sharply 2-transitive groups which are readily proven from the defining axioms are presented first (see § 3, § 4, and § 5). These include necessary and sufficient conditions that 1 2 be a subgroup, along with those properties which are required to determine the near-domain associated with a given sharply 2-transitive group. Near-domains are introduced in § 6 and it is shown that all sharply 2-transitive groups are uniquely determined by near-domains. In § 7 we examine groups associated with near-fields, skew fields, and fields. In particular, Theorem (7.1) states that 2 J is a subgroup if and only if the associated near-domain is a near-field. In § 8 a number of theorems on near-domains are stated and several of them including those which do not already ·appear in the literature are proved. Interpreted in terms of the associated sharply 2-transitive groups, many of these theorems+ yield new and stronget theorems on the groups, which are presented in§ 9:

§ 3. Conjugate Classes and Involutions We first present some properties of arbitrary conjugate classes H which operate regulary on distinct points (see § 1). By (1.3) and sharp 2-transitivity the elements of H either have no fixed points, or they each have exactly one fixed point

13

§ 3. Conjugate Classes and Involutions

(3.1) If His a conjugate class in G which operates regularly on distinct points,

then: (a) For a, ~ E H, we have; a a(p) = ~(p)

=~·if and

.

only if there exists p EM such that

(b) If the elements of H have no fixed points, then sharply transitively on M

ii = H

U {id}

operates

(c) If the elements of H have fixed points, then the setH and each set of the form H~, ~ E H, operates sharply transitively on M Further, the pemzuta-

tion groups (G, M) and (Inn G, H) are isomorphic as permutation groups.

Proof: To prove (a) it suffices, by the regularity of H on distinct points, to assume a(p) = ~(p) = p. Let rE M\{p}, then by sharp 2-transitivity we have a (r) =!= r, {3 (r) =!= r, and there exists a unique 'Y E G, such that 'Y(r) = r and "fa(r) = fJ (r). Thus E H and ,Ya (r) = {3 (r) =!= r. By the regularity of H on distinct points we have ,Ya = ~. But 'YCX"f- 1 (p) = {3 (p) p '* CX"f- 1 (p) 1 - 1 (p) '* 1 - 1 (p) = p '* 'Y = id '* a = {3.

ra

Part (b) and the first part of (c) are obtained from (1.4) b, c, the regularity of H on distinct points, and part (a). By regularity of H there exists exactly one ~a E H which has a given point a EM as fixed point. Hence 1/1 : -+ is a bijection.

{M H a -+~a

Now X:

{G-+ ~n Gis an isomorphism by (1.5). 'Y -+ 'Y

Further, -Y~a ~-y(a)• as we have -Y~a E H, and 'Y~a'Y- 1 ('Y (a)) = 'Y (a). Thus (X'Y) 1/1 = 1/J'Y, for all 'Y E G. In the remainder of § 3 we· prove several propositions involving the conjugate class of involutions J, some of which appear at least in part in the literature (for example in [14], [20], [21]) although in differentforms and often with different methods of proof. (3.2) The conjugate class of involutions of a sharply 2-transitive group operates

regularly on distinct points.

Proof: For a, b EM with a=!= b, there exists by sharp 2-transitivity exactly one p. E G such that p.(a) =band p(b) a. But p 2 (a) =a and p. 2 (b) = b implies p 2 id, and hence JJ. E J. Propositions (3.1) and (3.2) imply:

Theorem (3.3) If G is of type 0, then A = i = J U {id} operates regularly on M If G is of type 1, then the set J, and each set of the form A = J w, where w E J, operate regularly on M. In all cases A contains the identity element.

14

§ 3. Conjugate Classes and Involutions

The remaining propositions of § 3 involve the set 1 2 • If a: E J2, say a: =JlV, Jl, v E 1, then for p EM we have: a:(p) =JlV(p) =p ¢> Jl(p) =v (p). But Jl (p) = v (p) ¢> Jl =v by (3.3). Hence, a: (p) = p * a: =JlJl =id, and we have: (3.4) Let a: E 1 2 , then a= id if and only if a: has a fixed point. (3.5) The following propositions are valid in a sharply 2-transitive group G with involutions 1:

=w1, for every w

(i)

1w

(ii)

1 2 = (1w) 2 , for eve1y wE 1

(iii)

a E 1w :;. a:- 1 E 1w, where w E 1

(iv)

E

1

The sets 1w, w E 1, form a conjugate class of subsets of G and

1'YW'Y-1 = 'Y(1w)'Y- 1, for 'Y E G

(v)

J2 c G 1, where G 1 denotes the commutator subgroup of G

(vi)

1 2 operates transitively on M

(vii)

Ji is characteristic,

for each positive integer i.

Proof: (i) and (iv) follow immediately from the fact that lis a conjugate class, and (ii) and (iii) follow directly from (i). Now if Jlrt E J2 \ {id} with /1, rt E 1, then Jl =I= rt and there exists a: E G, a: =I= id, such that arta- 1 =Jl, and hence Jlrt = arta- 1rt. This proves (v). To prove (vi) let p, q EM and rEM\ {p, q} (we assume IM I > 2), then by (3.2) there exist /1, rt E 1 such that rt (p) = r and 11 (r) = q, and hence 1111 (p) = q. Since the automorphic image of an involution is an involution, we have a (1i) C 1i, for every a E Aut G, and hence (vii) is valid.

(3 .6) If 1 n 1 2 =I= f/J, Jhen 1 c 1 2 and G is of type 0. Proof: ~upiJOse 1 n f 2 =I= f/J, then 1 c 1 2 , since 1 is a conjugate class and J2 is invariant with.respect to inner automorphisms by (3.5) (vii). Since by (3.4) /l E 1 n 1 2 has no fixed points, G is of type 0.

Theorem (3. 7) Let G be sharply 2-transitive with 'involutions 1 Then the following conditions are equivalent: (i)

12 < G

(ii)

1 2 is commutative

(iii)

J2 operates regularly on M

(iv)

14 C 12

(v)

1w is commutative for some w E 1

(vi)

13

C

i,

where i

=1 u { id}

§ 3. Conjugate Classes and Involutions

(vii) (viii)

=J, J < G,

12

=Jw, for any w or Jw < G, for any w

I5

if G is of type 0, or 1 2

E J, if G is of type 1

if G is of type 0,

E J, if G is of type 1.

2

2

. Proof: Since idE 1 and 1 contains the inverses of all its elements, we see that (i) and (iv) are equivalent. We now show (vii) => (iv). Suppose (vii) is valid. Then if G is of type 0, 1 4 = J2 = 1 2 U J = 1 2 , since J C 1 2 • If G is of type 1, then 1 4 = (Jw) 2 :::: 1 2 by (3.5) (ii). Thus (iv) follows from (vii). The proof of (i) =>(vii) is given in [20] pp. I32-I33. Thus (i), (iv), and (vii) are equivalent. Also (vii) and (viii) are equivalent; since if (vii) is true, then (i) is valid and we have (viii). Conversely, suppose (viii) is valid. If G is of type 0, J : : J2 = 1 2 U J = 1 2 , since J n J' =I= ~ and hence J c J2 by (3 .6). If G is of type I, then Jw = (Jw) 2 =1 2 by (3.5) (ii). ,We now show (i) => (ii) => (v) => (vi) => (iv). Suppose 1 2 < G. If G is of type 0, then 1 2 :::: J. For JJ., 71 E J we have J.l.71 E J, which implies (JJ.71) 2 :::: id Thus J.l.71 :::: 71/J. and J:::: 1 2 is commutative. If G is of type I, then 1 2 = Jw, for w E J. For a: = J.l.W and {3 = trw, where JJ., 1r E J, we have a:{3 = JJ.Wtrw :::: 71W, for some n E J. But JJ.W'TrW =71W => W7rWf.l = W71 => 1rWf.lW = 71w, and hence cx{3 {3a:. Thus Jw =1 2 is commutative. Obviously (ii) implies (v).

=

If Jw is commutative for some w E J, then by (3.5) (iv), Jw is commutative for every w E J. Let f.l, 71, wE J, then by (3.5) (i) w71, 71W E Jw and we have (JJ.71W) (JJ.71W) = J.l.71 (Wf.l) (1/W) = J.l.7171WWJ.l. = id, and hence ] 3 C J. Now if 1 3 C J, then either idE 1 3 or 1 3 C J. If idE 1 3 , then J n 1 2 =I=~. and by (3.6) J C 1 2 • Thus 1 4 C J} C 1 2 U J 1 2 • If J3 c J, then 1 4 C 1 2 • In both cases we have (iv). We have only to show that (i) and (iii) are equivalent. If 1 2 < G, then part (vii) is valid and by (3.3) 1 2 operates regularly on M. Suppose (iii) is valid. If G is of type I, then 1 2 = Jw, for any wE J, since Jw C 1 2 and by (3.3), Jw operates regularly onM. Let G be of type 0 and let p, q EM with p =I= q. By (3.2) and by assumption, there exist exactly one J.l. E J and exactly one 71W E 1 2 , 71, w E J, such that f.l (p) = 71w (p) = q. Now by (3.5) (vii), J.l.W71J.l. E 1 2 • But J.l.WTJJ.l. (p) = q, and hence by assumption J.l.W71J.l. 71w. Thus J.l.71W = WTJJ.l. and hence f.l71W E/. But f.lTJW(p) = p implies · J.l.71W ::: id, since G is of type 0. Therefore f.l = w71, which implies J n 1 2 =I= ~. and by (3.6) J C 1 2 • This together with (3.3) implies J =1 2 • We see that (iii) implies (vii) and hence (iii) => (i).

16

§ 4. Groups of Type 1

This completes the proof of Theorem (3.7). By Theorem (3. 7) (i), (ii), (iii) and (3.5) (vii) we have: Theorem (3.8) If J 2 is a subgroup of G, then it is a regular abelian characteristic subgroup of G.

§ 4. Groups of Type 1 Except for a note after proposition (4.1), we consider in this section only sharply 2-transitive groups of type 1, i.e. groups whose involutions have a fixed point. (4.1) Suppose (G, M) is sharply 2-transitive of type 1 and let J be the congugate class of involutions in G, then:

(a) (G,M) and (Inn G, J) are isomorphic as permutation groups

=1 2 \{id} is a conjugate classinG (c) J 2 < G if and only if hr =Jw, for a pair 1r, wE J, such that 1r =F w (d) The system U ={h n Jw: n, wE J, 1r =F w} forms a conjugateclass of (b)

JH

subsets of G. Further,

fJ ={12 } if and only if J 2
Proof: Part (a) follows from (3.1)c and (3.2). (b) Let a,~ E J 2 * with a= fJ.V, ~ nw, fJ., v, 1T, wE J, and fJ. =F v, 1l' =F w. By (a) there exists ,Y E Inn G such that ifJ. = 1r and ,Yv = w. Thus ra rfJ.'YV = 2 1TW = ~. which implies that 1 *is a conjugate .class.

=

(c) If J 2 < G, then by Theorem (3.7) J 2 =J1r =Jw. Now suppose l7T =Jw for 1r, w EJ and 1T 1=- w. For fJ. EJ with fJ. =F 1r, let r E InnG such that r(1r) 1r.,and r(w) fJ. (part {a)). Then l7T = r(h) = r(Jw) =]fJ.. Thus l7T = J3 and :! 2 <( G by Theorem (3. 7).

=

(d) For l7T n Jw and ]fJ. n Jv in iJ, let ,Y E Inn G such that ,Y(1r) fJ. and r (w) =v (part (a)). Then r (h n Jw) J/). n Jv' and {]is a conjugate class of subsets. By (c) we have fJ {J 2 } «> J 2
=

=

By (b) we see that the elements of J 2 * all have the s~e order. In § 9 (see (9.2)) we show in fact that the elements of 1 2 * are either of infinite order or their order is some prime p where p =F 2. It is interesting to note that for groups of type 0 it is not known if J2* is necessarily a conjugate class. If this is the case, however, and if J n 1 2 =F 1/J, then by (3.6) J C J2, and we have J 2 * =J, which implies J2 =J and J2 < G by Theorem (3.7).

§ 5. Kernels and Complements

17

In § 9 (see Theorem (9.8)) we prove considerably more about the system [j for the case that the elements of J'l* have finite order. In order to fonnulate the next theorem, we introduce the concept of 2 ply orbit-transitivity for arbitrary permutation groups as follows: Definition: Let (P, N) be a permutation group and suppose A C N and B C N are distinct orbits of P. P is said to operate 2 ply orbit-transitively on (A, B), if for every at. a2 E A and bt. b 2 E B there exists an element o: E P such that o:(at) =a2 and o;(bt) =b2. We observe that (P, N) is 2 ply orbit-transitive on (A, B) if and only if for every a E A and b t. · b 2 E B there exists an element o: E P such that o: (a) =a, and a: (b 1) = b 2 • If at. a2 E A, we have only to choose ~ E P with (3 (a 1) = a2 and set b 3 = ~ (b 1). Then for a: E P with o: (a 2 ) =a2 and o: (b 3) = b2, we have lt[3 EP and c:x(3(a 1) = a2 and ~(b 1 ) = b 2. '

Theorem (4.2) Let G be of type 1. Then the following conditions are equivalent:


(i)

J2

(il)

Inn G operates 2ply orbit-transitively on (J 2tf:, J)

(iii)

J 3 is an orbit in (Aut G, G).

Proof: Suppose J 2 < G, and let w, p E Janda:,~ E J 2 *. By Theorem (3.7) J 2 Jw = Jp and we have o: =Jl.W and·~ vp for properly chosen JJ., v E J with J1. =F w and v =/= p. By Theorem (4.1) a there exists a -? E Inn G such that :YJJ. = v and .Yw = p. Thus .Yo: .YJJ.:Yw = vp = (3, and we see that (i) => (ii). Since J 3 = J 2 *· J, we have (ii) =>(iii).

If J 3 is an orbit in Aut G, then all elements of J 3 have the same order. Since J C J 3 we have J = J 3 and J 2 < G by Theorem (3.7). Thus (iii) => (i).

§ 5. Kernels and Complements In the theory of finite permutation groups, one generalizes the concept of sharp 2-transitivity to that of Frobenius groups (see for example [35]). Two of the concepts from the theory of Frobenius groups, the complement and the kernel, will be extended here to the infinite case for sharply 2-transitive groups. Also, a second kind of kernel will be defined which allows us to define "desarguesian" groups.

18

§ 5. Kernels and Complements

Definition: The subgroups Ga = {aE G: a(a) a}, for a EM, of a sharply 2transitive group G are called the (Frobenius) complements in G. From § 1, 8Ga = G6 (a) for every o E G, and G = Ga U GarGa for any r E G\Ga. By (1.1), (1.2), (3.3) and (3.5) (vi) we have: (5.1) The complements in a sharply 2-transitive group G form a conjugate class

of maximal subgroups of G and each Ga operates regularly on M\{a}. Further G = l 2 Ga = AGa for any complement Ga, and A as defined in (3.3). By Theorem (3.3) we have:

(5.2) lfG is oftype 0, then Ga n J = f/J, for all a EM, and Ga contains no subgroups of order 2. If G is of type 1, then Ga n J = !la• where !la (a) =a, and Ga contains exactly one subgroup of order 2, namely C2 = {1, !la}, and hence C2 < Z (Ga)· Now for each a EM define:

id, if G is of type 0

Ea

l

. lla• where !la = Ga

n J,

if G is of type I.

Then {id, ea} < Z (Ga). For each pair of distinct points a, b EM, define Tla b E J by Tla b (a) = b, and define Tab = Tla b ea. For every c EM\{ a} defi~e c~,b EGa 'by c~,b (b)= c. Since Ga is re~lar on M\{a} by (5.1), c~.b is uniquely defined and{c~b: c EM\{a}}= Ga. Further, define .

.

Since e11 E'z (Ga), then for eve'iy 'Y E Ga, f permutes the elements of Ga\{ea} and ·we may cbnsider r:, b ,Y as a mapping from Ga \{ ea} onto Ga\{id}. We now define

ra,b

= {r EGa : 1-r;:b

= r;,b ·Y}.

(5.3) For every pair of distinct points a, bE M we have Z (Ga)

< ra,b
Proof: ra b is the set of elements 'Y EGa such that .Y centralizes ra\ and hence ra,b id on Ga.

r ::

Definition: The groups ra,b are called the kernels of G. G is called desarguesian, if ra,b =Ga, and pappian, if Ga is abelian. A direct but rather lengthy proof, which will not be included here, verifies the following proposition.

§ 5. Kernels and Complements

19

(5.4) For every a E G and a, b EM with a -=1= b we have M'a,b = ra (a), a (b)· Hence the kernels of G form a conjugate class of subgroups of G and the kernels contained in Ga form a conjugate class of subgroups of Ga. From the definitions and from (5.3) and (5.1) we have:

(5.5) If G is pappian then it is desarguesian. (5.6) G is desarguesian, if and only if its kernels are maximal in G. In § 9 we prove some further results on kernels and desarguesian groups (see

(9.3) and (9.4)). It should be noted here that in the case 1 2 < G one can show Z (Ga) = &ra b, for fixed a, b EM. This is proved using methods of § 6

n

aEGa

' ·

and the fact that the kernel and the center of a near-field satisfy a similar condition (see [2], and [44 ]). The names desarguesian and pappian derive from 'the geometric interpretation of sharply 2-transitive groups as developed, for 'example, by Karzel in [21]. Before defining the Frobenius kernel of a sharply 2-transitive group we note: (5.7) If 1 2

< G,

then G is a semidirect product of 1 2 by Ga, for any Ga.

Proof: By (5.1), (3.5) (vii), and (3.4) we have G = J 2 Ga, Pis normal in G, and 1 2 rl Ga = {id}. Thus if 1 2 < G, then G is a semidirect product of .fby Ga. Now in the finite case, 1 2 can be characterized as the subset of G consisting of the identity together with those elements of G which have no fixed points. This concept is now generalized to the infinite case. Definition: The Frobenius kernel of G is the subset F = {a E G : a (x) =t= x, for all x EM} U {id}. As we have just mentioned, in the finite case F = 12, and F< G. This is not necessarily true in the infinite case even if 1 2 < G, as was shown in [22] and [48] using methods of§ 6. Before considering the situation when F
By definition, F n Ga = {id}. Also by (3.4) 1 2 C F. Therefore, by (3.5} (vi) F operates transitively on M and by (1.2), G = FGa = GaF. Also F contains the regular sets A, where A =J U {id}, if G is of type 0, and A =Jw, for wE J, if G is of type 1 (see Theorem (3.3)). We prove further:

(5.8)

{a} E

F,

for every a E F.

Proof: In general, if a E G and p EM such that a (p) =t= p, and an (p) = p, then a 12 = id, since p and a (p) are distinct fixed points of an. Thus, if a E F then an E F for any integer n.

§ 5. Kernels and Complements

20

(5 .9) f1 is invariant with respect to inner automorphisms. Further, if G is of type 1, then F is characteristic. Proof; Obviously the first part of the theorem is true. Now suppose G is of type 1. Let a E F, a =I= id, and a E AutG, then a (a)- 1 = a (a- 1). Assume a(a)(p)=p, forsomepEM. Letp.PEJwithJJ.p(p)=p. Now a (a- 1)(p)=p and a (a)p.p a (a- 1)(p) p. Thus a (a) f.!p fl (a- 1) = f.!p which implies a a-I (fJ.p) a- 1 = a -I(p.p). Let q EM such that a -I (p.p) (q) = q, then a a- 1(P.p) a- 1(q) q and a - 1 (p.p) a- 1 (q) = a- 1 (q). This implies a- 1 (q) q which contradicts a EF\{id}. Therefore a (a) E F. (5.10)

J 2
Proof; Obviously J 2

cF.

< G => J 4 C F,

since J 4 C J 2 and J 2 C

F.

We note that r E J 4 for every T E J 3 , since forT fJ.V1T, Jl, v, 1T E J, we have T2 = JlV1TJlV1T = {J.VTrJlV. Now suppose c F, then J 4 () Ga = {id}. If G is of type 0, we show J 2 = J als follows. Let JJ., 1T E J, Jl =I= 1T, a EM fixed, and p E J such that p (a) = fJ.1T (a). Then PJl1T E G0 and (PJl1T) 2 E J 4 n G0 • Hence (pJ11T) 2 = id, which means pJl1T id, since PfJ.1T has a as fixed point. But PJl1T id => fJ.1T p => fJ.1T E J. Thus J n J 2 =I= f/1 and J 2 c J. By (3.6) we have J 2 = J. Suppose G is of type 1, and let p., 1T E J, Jl =I= 1T. Choose a EM and let wE J with w (a) =a. We show that Jl1T E Jw and hence J 2 = Jw. We assume the nontrivial case p., 1T =I= w. For p E J with p (a) = fJ.W1T (a), we have pp.wn E J 4 n Ga, hence pp.wn id. But pp.wn = id => p.pp.w Jl1f => Jl1T E Jw. 2

r

By Theorem (3.7) J 2 is a subgroup in either case. Definition: G is called planar if F < G. The name planar d~rives again from geometrical considerations. The "geometry" d~scrlbed by a sharply 2~l:ransitive group in the sense of Karzel [21] is an affine plane• if and only if the group is planar. The following proposition is due to Karzel [20], [21]. (5.11) If G is planar, then ofF by G0 .

F = J 2• Hence J 2 < G and G is a semidirect product

Proof: If a E F \{id} and a EM, then for a' E J 2 with a' (a) =a (a) (see (3.5) vi) we have a- 1 a' E P, since F < G, and a- 1 a' (a) = a. Thus a = a'. (5.12) The following conditions are equivalent:

(ii)

F< G F is commutative

(iii)

F operates sharply transitively on M

(i)

§ 6. Near-domains and sharply 2-transitive groups

(iv)

21

F =A,

where A = J if G is of type 0, or A = Jw, w E J, if G is of type 1.

Proof: The equivalence of (i), (iU), and (iv) is shown in (21 ]. IfF< G, then and F is commutative by (3.7). IfF is commutative, then 1 2 is commutative and J2 < G by Theorem (3. 7). Let a; E F, a EM, and a:' E 1 2 with a:' (a) = a: (a), then a- 1 a:' (a) =a and a;- 1 a' a (a) = a' (a) = a; (a). But a(a) =I= a=> a- 1 a'= id =>a= a.'. Hence F 1 2 and F
F =J 2 by (5.11)

In § 7 we prove that if G is desarguesian, then it is planar. We also show in § 7 that G is pappian if and only ifF is equal to the commutator subgroup of G (see (7.3) and (7.4)).

, § 6. Near-domains and sharply 2-transitive groups It is the purpose of this section to show that sharply 2-transitive groups can be completely characterized by means of "one-dimensional affine" transformations x-+ a+ m · x on a suitable algebraic structure. The structures considered here will be called near-domains, although this term has no fixed usage in the literature (see [11 ]). Near-domains, as defined here, were first introduced by Karzel in [20] and [21] under the name "Fastbereich". Definition: A near-domain is a triple (F, +, ·) where F is a set on which two binary operations(+) and 0 are defined which satisfy the following axioms: Fb 1:

(F, +) is a loop (with neutral element 0)

Fb2:

a+ b

Fb3:

(F*, ·)is a group (with neutral element 1, F* = F\ {0})

Fb4:

0 ·a

Fb5:

a· (b +c)

Fb6:

For every pair of elements a, b E F there exists an element da,b E F*, such that a+ (b + x) =(a+ b)+ da,bX• for every x EF.

= 0 => b +a= 0 0, for every a E F a· b +a· c, for all a, b, c E F

Karzel actually defined near-domains by Fb 1, Fb3:-Fb6, and Fb6': If a+ b = 0, then da,b =I. But Fb6' implies the left inverse law a + (b + x) = x, for a + b = 0, and this in turn implies Fb 2. On the other hand, if Fb 2 is valid and a + b 0, for a =I= 0, then a= a+ (b +a)= (a+ b)+ da,ba = da,b a, which implies da,b = 1. Hence the definitions are equivalent.

22

§ 6. Near-domains and sharply 2-transitive groups

Now Fb 1, and Fb2 imply that every element a E F has a unique left and right additive inverse which will be denoted by -a and we write b + (-a) b -a. A near-field is a near-domain in which (F, +)is a group. Hence a near-domain is a near-field if and only if da,b = 1 for every pair a, b E F. Define the set:

T2 (F)

= {(a, m>

: (a, m> : x

-+

a + mx, a, m

E;:.

F, m =f::. 0}

of "one-dimensional affine transformations" over F. It is easily seen that T2 (F) is a subgroup of Sym(F) and for a, bE F, a =f::. b, (a, -a+ b) is the unique element in T2 (F) which maps 0 onto a and I onto b. Thus we have:

(6.1) The group T2 (F) of one-dimensional affine transformations on a neardomain (F, +, ·)operates sharply 2-transitively on the elements of F. In T2 (F) we have: (i) id =(0,1), (ii) = (a +rna', da,ma' mm'), for all a, a', m, m' E F with m, m' =f::. 0. Consider now an arbitrary sharply 2-transitive group G. By (5.1) we have G = AGe =GcA, for any Gc where A = J, if G is of type 0, and A Jw, w E J, if G is of type I. In the latter case choose w E J such that w (c) =c, (see (3.3)). Since A is a set of permutations which operates regulary on M and which contains the identity, and since Gc can be taken to be a regular permutation group bn M \{c}, we obtain a loop addition on M with neutral element c 0 and a group multiplication on M\{0} with neutral element 1 EM\{0} as follows: For every a EM (or a EM*, M* M\{0}) define the permutation a+ (or a') to be the unique element in A (or G0 ) which satisfies a+ (0) = a (or a' ( 1) = a). Addition and multiJ>lication are defined as follows:

.

·~

. '+' ; M ?< M-+ M and · : M* X M* (a, b)-+ a+ (b)

-+

M*

(a, b)-+ d (b)

Multiplication is extended to M by 0 . X =X • 0 = 0, for all X EM Now a+ b = o =>a+ (b)= 0 => (a+r 1 (0) =b => (a+t 1 b+, since (a+r 1 E A (see (3.5) (iii)), and since A operates regularly on M. Thus a + b = 0 => b + a b+ (a)= (a+t 1 (a) = 0, which implies Fb2. Since J is a conjugate class and since by theorem (3.3) awa- 1 = w, for all a E G0 , if G is of type 1, we have al.l.a- 1 =A, for all a E G0 . Now for a, b EM, a =f::. 0, we obtain (a't 1 (a· bf a· EA. Further, (a')- 1 (a. bf a' (0) = a- 1 • a. b = b, and again by (3.3) (a')- 1 (a · bf a' = b+, which implies (a· bt d a' b+. This means a· b +a· c =a. (b +c), for every c EM, and Fb5 is satisfied.

§ 6. Near-domains and sharply 2-transitive groups

23

Finally for a, b EM define 6a b =((a+ bt)- 1 a+b+, then oa bE G0 , since oa b(O) = ((a+ bn- 1 (a +b)= 0. Ford~ b = oa b (I) we have (d0 b)';, oa b• and a+b+,; (a+ br
.

'

In an earlier attempt to describe sharply 2-transitive groups in terms of "affine" transfonnations, Tits in [39) used essentially the same procedure as that described here, except that addition is defined with the aid of the set j rather .than the set A. Since j operates sharply transitively on M only if G is of type ·0, the addition defined by Tits is not necessarily a loop if G is of type 1. Also from any given group G of type 1, one can derive non-isomorphic structures which satisfy. the axioms of Tits and yet describe one and the same sharply 2transitive group G. This difficulty does not arise in the case of near-domains. Let us call two near-domains (F, +, ·)and (F', +', 0) isomorphic if there exists a bijection ofF onto F', which is a loop isomorphism from (F, +) onto (F', +' ), and a group isomorphism from (F*, ·) onto (F' *, 0). Then, as was shown in [25), we have: Theorem (6.3) Let G and G' be pe1mutation groups which operate sharply 2transitive/y on sets M and M' respectively. Further, let (Mo,t. +,·)and (M0•, 1•, +', 0) be the associated near-domains with respect to distinct points 0, I E M and 0',1' EM', according to (6.2). Then (G, M) and (G', M') are isomorphic · as permutation groups, if and only if the near-domains (Mo,t. +, -)and (M~·. 1• +', 0) are isomorphic. In particular, if G G' then theorem (6.3) tells us that all associated neardomains are isomorphic and we may speak of the near-domain associated with G. Near-domains serve as a valuable tool in the study of sharply 2-transitive groups. Hence it is of value to express the important subsets and subgroups of a sharply 2-transitive group in tenns of the mappings (a, m). This is done in (6.5) and (6.6). We first prove:

(6.4) Let (F, +, ·)be a near-domain and let G T2 (F) be the associated sharply 2-transitive group. For the complements G0 , we have Ga ~ (F*, ·).If x -1 is the solution of the equation I + x = 0, then -1 E Z (F*, ·) and y -a= (-I)a = a(-l) is the solution of a+ y = 0. Further z 2 == 1 if and only if z =·1 or z =-I. Also G is of type 0, if and only if -l = 1.

24

§ 6. Near-domains and sharply 2-transitive groups

Proof: Since ab' (1) =a' (b) =db' (1), we have ab' =a' b', for all a, b EM\ { 0 }, and hence ·= {F*-+ ~0 is an isomorphism a-+ a

Let v E J, where v (0) = 1, then:

v , if G is of type 0, and 1+ =

{ Vflo, where

flo E

J, with

flo

(0) = 0, if G is of type 1.

Now let x = e0 (1), where e0 is defined in § 5. Then:·

1 + x = 1+(x) ==

v(l) = 0, if G is of type 0, and

j

Vfloflo(l) = 0, if G is of type 1.

Hence x = e0 (1) = -1 and -l E Z(F*, ·),since e0 E Z(G0 ). From a+ a(-1) = a (1-1) = 0, we have -a =a (-1) = (-l)a. Further, z2 = 1 if and only if z' 2 = id, which is true if and only if z' E {id, e0 }. Thus z2 = 1 ~ z = 1, or z = -1. Now T2 (F) is of Type 0 if and only if eo id, and this occurs if and only if -1 = 1.

(6.5) Let G be a sharply 2-transitive group and (F, +, ·)the associated near-

domain. Then we may assume G = T2 (F) and we have: 1

[{
l

{(a, -D: a E F} , if G is of type 1.

A ={
! 2 = {: a, be F*} GD= {(0, m); mE F*} .

_. " F. = {
Proof: Now (a, m) E J ~(a, m) i= (0,1) and (a+ ma, da,mam 2 ) = (0,1) ~ (a, m) i= (0,1) and a+ ma = 0 and m 2 = 1. The·expression for J follows now from (6.4). Now the expressions for A= J U {id} and 1 2 are obviously correct if G is of type 0. If G is ~J type 1 then A = ]fl 0 , where flo = (0, -1> and we have (a, -1}(0, -D =(a, da,O (-1) 2 ) =(a, 1), and hence A= {, (-b, .:....1) E J, (a, -1}(-b, -1} (a+ b, d0 b>, and hence 1 2 = {: a, b E F*} U {(-a2~ 1 , :.._1) = (aT 1 + a2- 1, d02 -• 02 -,) =(a, 1) Thus {: a, b EF*}. The repres'entations for G0 andFfollow directly from the definitions of these sets.

§ 7. Near·fields, Skewfie!ds, Fields, and Sharply 2-transitive Groups

25

Now let F and G be as in (6.5) and let f be the group which is the inverse image in F* of the kernel r 0 , 1 under the isomorphism·: a-+ d. f will be called the kernel of (F, +, ·). (6.6) Let f be the kernel of a near-domain (F, +, ·), then Z (F*, ·) < f (F*, ·)and t ={kEF*: (a+ b)k = ak+bk, for all a, bE F}.

<

Proof: By (5.3) we have Z(F*, ·) < f <(F*, ·). Now f = {k E F* : k. E

ro' .} ;: : {k E F* : i(;· Ttl = Tt' 1 R}.

.

0,1) since To 1 (0) = 1 and To 1 (-1) = 0, and hence Ttt(x·) = , ' ' t'"" ' [T0 , 1 (x)]" = (1 + x)" (0, 1 + x). Thus f ={kEF*: (0, k)(O, 1 + x)(O,k- 1)= (0, 1 + kx k- 1), for all x E F*\{-1}}= {kEF*: k(l + x)k- 1 = 1 + kxk- 1 , for all x E F*} . But

Tot=

•Replacing k by k- 1 we obtain:

'f ={kEF*: (1 + x)k = k + xk, for all x E F*} ;::: {kEF*: (a + b)k =ak + bk, b =ax, for

an a, X

E F*}.

= {kEF*: (a + b) k = ak + bk, for all a, b E F}.

§ 7. Near-fields, Skewfields, Fields, and Sharply 2-transitive--Groups As was mentioned at the beginning of this chapter, it is not known if in a given sharply 2-transitive group the set J 2 must in fact be a subgroup. This question can now be interpreted in terms of near-domains. In fact, from the following theorem, which is due to Karzel [20], we see that groups in which J 2 is not a subgroup exist if and only if near-domains exist which are not nearfields. Theorem (7.1) Let G be a sharply 2-transitive group and let J be the set of involutions of G. Then f 2 forms a subgroup of G if and only if the associated near-domain is a near-field In this case J 2 is isomorphic to the additive group of the near-field. Proof: From the definition in § 6, addition is a group if and only if the set . A, where A = i for type 0, and A = Jw, w E J, for type 1, is a subgroup. One now applies Theorem (3. 7) (i) (vii) (viii:). In view of the definitions of the Frobenius kernel F and planar groups (see § 5), and considering the expression for Fin (6.5), it is natural to call a neardomain (F, +, ·)planar, if for every a, m E F, m =I= 0,1 there exists an x E F such that a+ mx =x. By sharp 2-transitivity, x is unique. One can readily

26

§ 7. Near-fields, Shewfields, Fields, and Sharply 2-transitive Groups

show that the class of planar groups is the class considered by M: Hall in [14] Theorem 20. 7.1. Part [a] of the following theorem is essentililly due to Hall, although the more recent concept of near-domain was not available to him. Theorem (7.2) Let G be a sharply 2-transitive group and (F, +,·)its associated near-domain. Then the conditions listed below under [x] are equivalent, for x = a, x = b, and x = c: [a] (i) (ii) (iii)

[b] (i)

G is planar (F, +, ·) is planar (F, +, ·) is a planar near-field. G is desarguesian

(ii)

(F, +,·)is right distributive, i.e. (a+ b)c

(iii)

(F, +, ·)is a skewfie/d

[c] (i) (ii) (iii)

=ac +be, for all a, b, cEF

G is pappian (F, +, ·) is commutative with respect to multiplication (F, +, ·)is a commutative field.

Proof: [a] If G is planar, then by (5.11) and Theorem (3.7) F =1 2 =A, and by (6.5) F = {(a, 1): a E F} = {(a, m): a + mx ::/= x, for all x E F}. Thus (a, m) has a fixed point whenever m ::/= 1, which means that (F, +, ·) is planar. If (F, +, ·)is planar, then the assumption da,b ::/= 1 implies that (at b, da,b >EJ2 has a fixed point, which contradicts 1 2 c F. Thus da,b = 1, for all a, b E F and hence (F, +, ·) is a planar near-field. Further, if (F, +, ·) is a planar nearfield, then 1 2 < G'by Theorem (7.1) and F 1 2 by (6.5). Therefore G T2 (F) is plan~r. ' ., [b] Suppo'se G is desarguesian. Then ro, 1 = Go, which implies f = F*, and by (6.6), (F, +, ·) is right distributive. Now if (F, +, ·) is right distributive and m ::/= 0, 1, then (a, m> has the fixed point x = ya, where y- 1 + m = 1, since a+ mO = 0, for a= 0, and (at mx)x- 1 = ax-1 + m =y- 1 t m = 1, for a::/= 0. Thus (F, +, ·) is planar and hence by part [a] a right distributive near-field, which is a skewfield If (F, +, ·) is a skewfield, then f =F* by (6.6) and Po, 1 =G0 • Thus G = T2 (F) is desarguesian. [c] It is readily seen that (i), (ii) and (iii) are equivalent, using pa1t [b] and the fact that commutativity of multiplication implies right distributivity. Before proceeding to the next theorem, we note that all finite sharply 2-transitive groups are planar and hence all finite near-domanis are planar near-fields.

27

§ 8; Theorems on Near-Domains

This result, stated indifferent terms, was proved by H. Zassenhaus in [47], where, as was mentioned in the introduction, a complete determination of all finite near-fields and hence of all finite sharply 2-transitive groups is presented (s.ee also [14] pp. 390-391). Now since every skewfield is a planar near-field we have by Theorem (7.2) [a], [b]: Theorem (7.3) If a sharply 2-transitive group is desarguesian, then it is planar. Theorem (7 .4) A sharply 2-transitive group is pappian, if and only ifF= J 2 = G I, where G 1 denotes the commutator subgroup of G. Proof: IfF= J 2 = G 1 , then by (5.7) G = G 1Ga is a semidirect product of G 1 by Ga and Ga ~ G / G 1 • Thus Ga is commutative and hence G is pappian. On the other hand, if G is pappian, then by Theorem (7.2) the associated near4omain (F, +, ·)is a commutative field, and G is planar. Thus by (5.11)P = 1 2 , and G = J 2 G0 is a semidirect product of J 2 by G0 , which implies G0 ~ G/J2. Since G0 is commutative, we have G 1 < f 2 • But J 2 c G 1 by (3.5) (v), and hence 1 2 = G 1 •

§ 8. Theorems on Near-Domains Here we present a number of theorems, which give some information on the structure of infinite near-domains. Throughout this section we assume (F, +, ·) to be a near-domain. (8.1) The following identities are valid in near-domains. (i) (iii)

(v)

(ii)

da, 0 = 0 ,a = 1

a

(iv)

da,a = l

(vi)

dab-'=dba ' ' x +a b=>x

+b

= da,b (b +a)

cda b C-l = dca cb I

I

(vii)

da+b,b = da,b

(ix)

da,b b =(a +b) -a

(xi)

da,b = d-b,b+a

(xiii) (xv)

a

a· 0 = 0

da,b =d-da,b b,a (1 + a-ltl = 1 (1

(viii)

+ at

1

-a+ dv.,bb

(x)

da,b db+a,c =da,b+c db,c

(xii)

dab= da,cdc+a' -c+bd-c, b '

(xiv)

da.~

(xvi)

If a + b

= c + d,

d

da,b = dc,d·

+c

db,a «>db=l a, «>

then b

+a =

Proof: (i) through (xiv) are proved in [25], [26], [27], and [28]. By (ix),(iii), and Fb5 we have (1 +a)·- l = d 1 a a= (1 + a)(a + l)- 1 a = (1 +a) [a- 1 (a+1)r 1 '

28

§ 8. Theorems on Near-Domains

(1 + a)(1 + a-'r' which implies (xv). (xvi) follows from the equations da,b (b + a) =a + b =c + d = dc,d (d +c), (see (iii)). The following two propositions concerning the additive loop of a near-domain are verified in [25]. (8.2) The additive loop (F, +)of a near-domain (F, +, ·)is commutative if and only if (F, +, ·) is a near-field. (8.3) The additive loop of a near-domain is power associative, i.e. the subloop generated additively by a single element is a (cyclic} group. By (8.3) we may define a multiplication a · n of elements a E F with integers n E Z in the usual way without regard to associativity. Further, we have dan, am = 1 for all a E F and m, n E Z. In the proof of (8.2), the commutativity of addition for near-fields was used. That the additive group of a near-field is commutative was first proved (in more general tenns) by B. H. Neumann [33], and has been discussed often in the literature (see for example [20], [32], [50]). Here the commutativity of addition for near-fields is a consequence of Theorem (7.1) and Theorem (3.7) (i), (ii). It is interesting to note that if (F, +, ·)is a near-domain, then an addition can be defined on F such that (F, w, ·) satisfies the following axioms:

(i)

(F, w) is a commutative loop (with neutral element OJ

(ii)

(F*, ·)is a group

(iii)

a· (b

til

c) =a. b

til

GJ

a. c.

We have only to define a GJ b =a+ d_ 0 b · b. By (8.1) (viii), x =a GJ b is the solution of.x-a =b. Thus -x._: b =-a 'and by (8.1) (viii), -x -b + db,-a(-a). Since db -a= d_b a by (8.1) (v) and (6.4), we have x b + d_b a· a= b Ell a. Ob;iousiy 0 4 x ,:: x til 0 = x, for every x E F. Further, a til x ='b has the solution x = b a, since a Ell (b- a) =a + d-a, b-a (b- a) =a + d-a, b (b- a) = a + (-a + b) =b by (8.1) (vii), (vi), (iii). Further, x is unique, since a e x =b implies d-a, x · x = -a + b, and by (8.1) (ix), the mapping x -+ d-a,x · x = (-a+ x) +a is bijective. Thus (F, +)is a commutative loop. Also by (8.l)(v) we have for a =I= 0: a · (b til c)= a (b + d-b,c c)= ab + ad_b,c a- 1 ac = ab + d-ab, ac · ac = ab Ell ac. We see that (F, til, ·) satisfies axioms (i), (ii) and (iii). Also, if (F, w, ·) is itself a near-domain, then by (8.2) it is a near-field. Now we note -b til b = -b + db,b · b = 0 by (8.1) (iv), and da_a,bb,a

=d_d_a,bb,-a. = d-a,b

(8.1) (v) (xiii). Thus we have a= (-bEll b)

til

a= -b

til

by (6.4) and

(bEll a)= (a

til

b) Ell -b =

+

29

§ B. Theorems on Near-Domains

(a Ell b) -d0 ebb · b, and (a Ell b) -b =(a+ d_ 0 b · b) -b =a+ (d_ 0 b · bdd_a,bb,a ·b)'= a+ d-a,b (b- b)= a, ((8.1) (vi)). This implies damb:b =' 1, for all a, b E F, and hence (F, +, ·) is a near-field.

Hence the following conditionsare equivalent:

(l)

(F,

Ell, • )

is a near-domain

(2)

(F,

Ell, ·)

= (F, +, ·)

(3)

(F, +, ·) is a near-field.

By (8.1) (iii), d 1,1 l, ·and . hence I+ (1 + 1) =(I+ 1) +I. An inductive argument verifies the following theorem (see [25]). Theorem (8.4) Every near-domain contains a uniquely determined prime field, namely the sub near-domain generated by the multiplicative identity element. f[ence we may speak of the characteristic of a near-domain. Theorem (8.5) Let (F, +, ·)be a near-domain and f its kernel (see (6.6)). If I f I > I, then the following conditions are equivalent:

(i)

(F, +, ·)is a near-field

(ii)

The set f u {0} forms a sub skewfield of(F, +, ·)

(iii)

There exists k E f such that I + k E f

(iii)'

There exist k, k' E f such that k + k' E f .

In particular these conditions are equivalent if char F

* 2.

Proof: One readily verifies (i) => (ii), see for example [I], and obviously (ii) => (iii), if If I> I, which is satisfied if char F i= 2 since 1 E f by (6.4). (iii) and (iii)' are equivalent since k + k' = k(l + k- 1 k') and k- 1 k' E f. To prove (iii) => (i) we first show k, I + k E f => dk,l = l. Suppose k, 1 + k E f . Then we have by Fb6, FbS, (8.1) (iii), (v), (vi): 1 + [k +(I + k)] = (1 +

k) + dl,k (1 + k) = (1 + dl,k) (I+ k) =(I+ dl,k) +(I+ dl,k)k = 1 + [dl,k + ddl,k•1 (1 + dl,k)k] = 1 + [dl,k + (dl,k + l)k] 1 + [d1,k + (dl,k k + k)] = 1 + [(dl k + dl k k) + dd k d kk · k] '

'

1 • 1

1 + d1 k [(1 + k) + k], and hence •

k + (1 + k) d 1 k [(1 + k) + kJ. But Fb6, FbS, and (8.1) (iii) imply k + (1 + k) = (k + 1) + d; 1 k = dk 1 [(1 + k) + k]. Therefore, either k -2-I, and hence dk, 1 = 1, since 2~ 1 is in the prime field, 'or dk, 1 = d 1,k, which implies dk, 1 = 1 by (8.1) (xiv). Now dk,l = 1 and (8.1) (v) imply dxk,x =.dx.xk = 1, for all X E F*. Let a, bE F* and set a'= ak- 1 and b' b (1 + kt 1 • Then (a' + b') (1 + k) =d.' (1 + k) + b' (1 + k) =(a'+ a'k) + b =a'+ (a+ da',a'k b)= a'+ (a+ b), and by (8.1)

30

§ 8. Theorems on Near-Domains

(iii), (a'+ b')(1 + k) =(a'+ b') +(a'+ b')k a'+ (b' +db' a' (a'+ b')k) a'+ (b' + (b' + a')k) =a' + (b' + (b' k +a' k)) =a'+ ((b' + 'b' k) +db', b',ka) = a'+ (b +a). Thus (F, +)is commutative and by (8.2), (F, +, ·)is a near-field. The following two propositions are immediate corollaries of (8.5). (8.6) Let (F, +, ·)be a near-domain with char F =I= 0,2. Then (F, +, ·)is a near-field if and only if a multiplicative generator of its prime field is contained in f .

Theorem (8.7) A near-domain (F, +,·)with char F

=3 is a near-field.

By taking k = 1, and observing that 1 + 1 = -1 if char F = 3, we see that Theorem (8.7) is a consequence of (8.5). However, Theorem (8.7) is also a consequence of the following proposition.

(8.8) Let (F, +, ·)be a near-domain with char F =I= 2. Then (F, +, ·) is a nearfield if and only if 3 E f U {0}. Proof: By (8.5) we have 3 E

t

U {0}, if (F, +, ·)is a near-field.

We first note for a, b E F: (a+ b)· 2 =(a+ b)+ (a+ b)=a +(b + db,a(a +b)) =a+ (b + (b +a))= a+ (b2 +a), by Fb6, (8.1) (vi), (iii), (iv).

Thus we have: (a+ b) 3=(a +b)+ (a+ b)2=a+ [b +db,a(a + b)2] =a+ [b + (b +a}2] =a+ [b + (b +(a2+b))] ::Ea+[b2+(a2+b)] . .,

=

No\\! assu!{le 3,E f U {0}, and set a= a' 2, and b b' 2, then (a'+ b') 3 = a'+ [b'2 + (a'2 + b')] =a'+ [b +(a+ b')], and (a'+ b')3 = a'3 + b'3 = (a'+ a'2) + (b'2 + b') =a'+ [a+ (b + b')]. Thus 3 E f U {0} implies a+ (b + b') = b +(a+ b'). By interchanging a' and b' we obtain a + (b +a') = b + (a +a'). If a= b, then a + b = b +a. If a =I= b, then a' and b' are two distinct points on which the mappings (a + b, da, b> and (b + a, db, a> agree. Thus (a + b, 4:. b> =(b + a, db) and hence we have a + b = b + a. By (8.2), (F. +, ·) is a near-field. The setsD and D 1 defined below play an important role in the study of neardomains:

-v·;;; {da, b: a, bE F},

and D 1

{dl,b: bE F}.

§ 8. Theorems on Near-Domains

31

(8.9) The sets D and D 1 satisfy the following conditions: (i) n- 1 = D, (ii)D! 1 =D 11 (iii)DCD 3, (iv)D=U xD 1 x- 1.IfcharF=I=2,thenDCD2• xEF*

Proof: By (8.1) (vi), (8.1) (xii), and (8.1) (v) we have parts (i), (iii), and(iv) respectively. Again by (8.1) (ix) the mapping b--+- d1 bb = (1 + b)-1 is bijective. For a given a E F, let b E F such that -d1 b · b ' =a. Then by (8.1) (vi), (xiii) we obtain d 1 b =d_d b 1 = da 1 = di 1a: which implies (ii). Now if char '

1b •

•

'

F =I= 2, then -1 =I= 1, and 2 '=1= 0, where 2 = 1 + 1. For any b' E F, define b E F by b' = b + b = b 2. In (8.1) (x) let c =b and apply (8.1) (iv), then . 2 dab'= dab db+a b• and hence dab' ED . '

,

J

'

The following two propositions are verified in [26]:

(8.10) Let (F, +, ·)be a near-domain. For each da b ED, either da b = 1 or [(F*, ·), (t (da b)] is infinite, where Gl.: (da b) den~tes the centraliz.er {x E F*: xda bx-1 = da ~}. ' '

.

(8.11) A near-domain is a near-field if and only if Dis finite. We noted earlier that finite near-domains are near-fields. (8.11), and (8.13) which follows, can be considered as generalizations of this result. Consider the following classes of (infinite) groups:

[FC] - The class of groups with finite conjugate classes.

[FD] - The class of groups with a finite commutator group. [FYZ]- The class of groups whose center is of finite index. The class [FC] was introduced in [3 ], and a result of [34] states: ( *)

[FC] ::J [FD] ::J [FYZ].

We first prove:

(8.12)(a)DnZ(F*)=D 1 1\ f =1. (b)N<:JF*andNfinite~NnD=l. (c) A< F*, A E [FC], and [F*: A] finite~ AnD= 1. Proof: D n Z(F*) = 1 and part (b) follow directly from (8.10). Let d1 b E D 1 n f and let c + d 1,b = 1, then d 1,b c + d 1,b 2 = cd1,b + d1,b 2 = d 1,b,' and hence d1 b c = cd1 b· If c =I= 0 then 1 + d 1 b c- 1 = c- 1 , which implies (a+ b)+ da+b, bc.!1 = c- 1 , f~r a+ b = l. Thus by (B.1) (vii), (a+ b)+ da, b c- 1 = c- 1 , i.e. (a+ b, da, b> (c- 1) = c- 1 which contradicts (3.4) and (6.5). Thus c = 0 and d1,b = 1. To prove (c) suppose A E [FC] and da,b EA. Since [F*: A] and l{xda,b[ 1: x EA}I are finite, then i{yda,bY- 1 : y E F*} I is finite which implies da, b = 1 by (8.1 0).

32

§ 8. Theorems on Near-Domains

(8.13) A near-domain (F, +, ·), which satisfies the condition (F*, ·) E [FC] is a near-field In particular, a near-domain which has a finite commutatorgroup is a near-field. Further, a near-domain whose center is of finite index is either a commutative field or a finite near-field. Proof: The first part of the theorem follows from (8.12)c and(*). Suppose [F* : Z (F*)] is finite, then (F, +, ·) is a near-field, by (8.12)c and ( *), and also [F*: f ] is finite, since Z (F*) < f, (6.6). By (8.5), f U {0} is a skewfield. If f = F*, then F is a commutative field, since [F*: Z (F*)] is infinite for proper skewfields (see [38] ch. 14). If f :f:: F*, then for x E F*\ f we 1 1 have for k, k E f, (x + k) f= (x + k ) f #- k =k', and hence.[F*: ~If I. This implies that f, and hence F, is finite. Using known properties of near-fields and finite groups of linear transforma· tions without fixed points, one proves the following lemma ([45], (3.7)).

n

(8.14) Let (F, +, ·)be a near-field and N a finite subgroup of (F*, ·), then:

(a)

lfN ={x} :f:: {1} is cyclic, then r

(b)

I N I :f:: p, where p

=char F

\IN I #-1 +x + ... + x'- 1 = 0

(c)

If IN I =l q, where p and q are prime numbers (which are not necessarily distinct), then N is cyclic

(d)

The Sylow subgroups of N are cyclic or generalized quaternion groups

(e)

If the 2-Sylow subgroups of N are cyclic, or if at least one of them is a direct factor of N, then N is metacyclic.

1

Theorem (8.15) Let (F, +, ·) be a near-domain and char F = p, where p Define the set:

(x

f:!1' ={x EF..: 1 + + (x 2 +. ··:.. + (xP- 2 + xP- 1) ... ) (F, +, ·)~is a near-field if and only if & = {1}.

.

.

.

= 0}.

> 2.

Then D 1 c f:!l'and

Proof: Suppose F is a near-domain with char F =p, p > 2. Let J be the set of involutions of T2 (F) (6.1), then the elements of J2* all have order p (see (9.2) p. 41, or 3.1 in [25]). Forb EF* determine a E F such that a+ b =1, and define /1,11 E J by J1 =-(a, -D, 11 (-b, -1) (6.5). Then we have by (8.1) (vii) and (6.1) (iii) 1111 =(a+ b, da,b> = 0, da+b,b> (1, d 1,b>· An induc.tive proof using (6.1) (iii) shows that for every i = 1, 2 ... we have: (1, d1 b) 1 = (1 + (d1 b + 2 • 2 . 1 ' ' (dl, b + · .. + (dt b + d'l, b) ...), d), where d is a product of elements in D.

But ord JlT/ ord (1, d 1, b) =p =? (1, d 1, b)P = id = <0, 1). Thus a1• b E f:!1' for every bE F and we obtain D 1 C f:!l'. If we assume f:!1' ={1}, then D 1 ={1}, w~ich by (8.1) (v) implies da,b =a- 1d 1,a-•b a = 1, for all a, b E F, a :f:: 0, and F lS a near-field.

§ 8. Theorems on Near-Domains

33

Now let F be a near-field; then by (8.14)a,b we have f!J = {1}. Now definie the set:

={x E F: x + 1 = 1 +X} = {X E F: d1, x =1} ={x E F: dx, 1 ,;, 1}

E

(see (8.1) (iii), (xi)). By definition E contains every sub near-field of F. The following two propostions involving the set E are proved in [28]. (8.16) For a, bEE and c E F, we have: (i)

-a EE

(ii)

a- 1 E E, for a =I= 0

(iii)

an

(iv)

c E E $>a+ (c +a) E E

(v)

(a+b)2EE.

E

E, for every integer n

'

(8.17) The following statements are equivalent: (i) (ii)

(E, +, ·) is a sub near-field of (F, +, ·) (E, +, ·)is a sub near-domain of (F, +, ·)

(iii)

a, b E E

(iv)

a, b E E =?abE E.

=?

a+b E E

Theorem (8.18) Let (F, +, ·) be a near-domain with char F > 2. Then F contains a largest sub near-field, namely E = {x E F: x + 1 = 1 + x}, and we have

either E

= F, or [(F*, ·): (E*, ·)] is infinite.

Proof: Suppose a, bEE, then by (8.16) (iii) (iv), (a+ b)2n EE, fc:ir every 1 integer n. For n = p ; we have a + b E E which by (8.17) implies that (E,

+, ·) is a sub near-field of (F, +, ·). By definition E is the largest sub near-field of F. We prove the latter part of the theorem as follows: By (8.15) we have D 1 n E = {1}, since Eisa near-field and D 1 C f1JJ. This implies DEn E = {1}, for DE = { da, e : a E F, e E E}, since da, e E E, e E E =? e- 1 da, e e E E =? de-• a, 1 E E =? de-• a 1 = 1, and we have da e = ede-•a 1 e-1 = 1. Now suppose E =I= F, and a E F\E. Fore, e E Ewe can now prove: ,

,

J

,

da, e E*=da, e·E*$>e=e'. Assume da,eE* = da,e•E*, then by (8.1) (vi) de•, ada, e E E*. By (8.1) (xii) we have de', ada, e = de· +a,_ e'+e = de• +a, e", for e" = -e' + e. Thus de'+·a, e" E DE n E, which from the above result implies de'+a, e" = 1. Now the assumption e =I= e'

34

§ 8. Theorems on Near-Domains

implies de"-• (e'+a), 1 = 1 => e"- 1 (e' +a) E E =>a E E. This contradicts our assumption. Hence l{da eE*: e E E}l = IE 1. '

Now [(F*, ·): (E*, ·)];;;;;. l{da, eE*: e E E}l and the result follows immediately if E is infinite. If E is finite the result is obviously valid, since every finite neardomain is a near-field. As a corollary of Theorem (8.18) we have:

Theorem (8.19) Let (F, +, ·) be a near-domain with char F > 2, E its largest sub near-field, and (F, +) its additive loop. The sets aE= {ae: e EE} = {x EF: dx, a= I}, for a E F*, are commutative subgroups of(F, +). These groups along with their subgroups are the only subgroups of (F, +). Further we have:

U

(i)

F =

(ii)

aE

(iii)

!{aE: a E F*}l = [(F*, ·): (E*, ·)].

aEF*

aE,

n bE = 0, or aE = bE, for a, b

E

F*,

From the last theorem we see that the additive groups aE distinguish the "commutative parts" of the loop (F, +), whenever char F > 2. In the following three propositions we examine the sets:

Obviously, Ea = E if and only if a E E, and in this case, Ea is a commutative group. Theorem (8.21) below states that for near-domains (F, +,·)with char F > 2, any set Ea where a Et E, is "totally non-commutative" with respect to addition. (8.20) F01: the subsets Ea of a,near-domain we have: (i) _ "

x. E ~a

(ii)

x E Ea => x + 1 E Ea and x' E Ea, where x' + 1 = x

(iii)

a Et E => da, 1 Et Ea

(iv)

x, y EEa => y-x = (y + 1)- (x + 1), and ely, -x = dy+1,-(x+l)·

¢:.

(x + 1) - x

= da, 1

¢:. -

(x + 1) + (x + 2) = da, 1

Proof: By (8.1) (ix) we have da, 1 = dx, 1 = (x + 1)- x, and by Fb6, x + 2 (x + 1) + dx, 1 and hence da, 1 = dx, 1 = -(x + 1) + (x + 2).

=

Suppose x E Ea. By (8.1) (vii) one obtains da, 1 = dx, 1 = dx+1,1 = dx'+1,1 =

dx',1·

Now assume da, 1 E Ea. Then for x = da, 1 we have x = dx, 1 and by (8.1) (iii) (vi) (xiv), x + 1 =x (1 + x) =x + x 2 => x 2 = 1 => x = 1, which implies a E E.

§ 8. Theorems on Near-Domains

35

Applying Fb6 and (8.1) (vi) (iii) we obtain, (y + 1)- (x + 1) = y + [1d1,y(x+1)]=y + [1-d1,x(x + 1)] = y + [1 (1 +x)] = y-x, and analogously (x + 1) -(y + 1) = x y. Now (8.1) (xvi) implies dy,~x = ~+1,-(x+l)'

Theorem (8 ..21) Suppose (E, +, ·)is a sub near-field of (F, +, ·),(in particular this is true if char F > 2), then: If a E F \ E and x, y E E0 , then x

+y

=y

+ x if and only if x = y.

Proof: Now x + y = x(1 + x- y) = y + x = x(x- 1y + 1) imply x- 1y EE and y =xe, for some e EE. Now x- 1y EE => -x- 1 y EE => d_x-'y,l 1 => d-y,x = 1 => dv -x 1. By (8.20) 1 = dy -x = dy+l -(x+l)' This implies, as above, 'I I ' , t y + 1 = (x + 1)e, for somee EE.Again by (8.20)(iv),x(e-1) = y-x =(y+1)-(x+1)= (x + 1) (e' -1), and hence e- 1 = (1 + x- 1 )(e' -1). This implies either e = e' 1,orl+x-1 EE.But1+x- 1 EE=>xEE=>d~ 1 =d~ 1 =1=>a which _contradicts the assumption a E F \E. Therefore e = e' = 1 andy = x. 1

.

As a consequence of Theorem (8.21) we have: (8.22) If (E, +, ·)is a sub near-field of (F, +, ·), E =F F, and a E F\E, then:

(i)

E0 n aE =a

(ii)

The mappings E0 --+ {xE : x E En} and E--+ {Eae : e E E} are bijective x-+xE e-+Eae

(iii)

[(F*, ·): (E*, ·)]~I E0 I. for each a E F\E

(iv)

ID 1 I ~ I E I If char F =F 2, then x E E0 => -x E!: E0 •

(v)

Proof: (i) and (v) follow from (8.21) and the commutativity of aE.

(ii). If x, y E E0 and xE yE, then by (i), x =Ex n xE = Ey n yE = y. If Eae Eae', then ae = Eae n aeE = Eae n aE Eae' n ae' E = ae', and hence

e

e. I

(iii) and (iv) follow from (ii). (8.23) If (E, +, ·)is a sub near-field of (F, +, ·),and char F =F 2, then a+ x = y + a and x + a = a + y => x = y. Proof: It suffices to prove: 1 + x = y + 1 and x + 1 = 1 + y => x = y. By (8.1) (xvi) (vi) we have dl,y d1,x = 1, and by FbS-6, and (8.1) (iii) we have: 2 +x 1 +(1 +x) = 1 +(y + 1) =d1,y[(y + 1) + 1] =d1,y [(1 +x) + 1] =d1,y dl,x[x+(I+l)]=x+2.But2+x=x+2=>dx, 2 = 1 =>dz-'x,l 1 =>2- 1x E E=>x EE=>x =y.

36

§ 8. Theorems on Near-Domains

The following theorem has an important application in chapter III ..

Theorem (8.24) Suppose (F, +, ·)is a near-domain and (E, +, ·) is a sub nearfield. If (E*, ·)contains a subgroup of order 3, which is normal in (F*, ·), then F is a near-field with char F =I= 3. Proof: Suppose C3 = {1,m,m 2 } is a subgroup of E* which is normal in F*. Since by assumption E is a near-field, we have by (8.14)a, b 1 + m + m 2 = 0, and char F = char E =I= 3. Denote the centralizer of C3 in (F*, ·) by [ (C3 ). Since C3
Suppose [F*: [ (C3 )] =2, then xmx- 1 = m ~ x E Q: (C3 ) and xmx- 1 = m 2 ~ x Et [ (C3). Now let x E [ (C3) and x =I= -1. We show, by means of contradiction, that 1 + x E [ (C3) ~ x + 1 E [ (C3 ). Suppose x + 1 E Q: (C3 ) and 1 + x Et Q: (C3 ). Since 1 + m = -m 2 and 1 + m 2 = -m we have by Fb S-6, (8.1) (iii) (v) (vi). (i) (ii)

(x + 1) (l + m) = (x + 1) + (x + 1) m = x + [ 1 + (l + x) m]

=x

+ [1 + m 2 (1 + x)]

=x

=x

+ [(l + m 2 ) + m 2 x]

+ (-m + xm 2 )

(x + 1) (1 + m) = (l + m) (x + 1) = (1 + m)x + (l + m)

=x (l

+ m) + (l + m)

= (x

+ xm) + ( 1 + m)

=x

+ [xm + dxm, x (l + m)]

= x + [mx + (1 + m)] = x + (mx- m 2 ).

Therefore, -m + xm 2 = mx - m 2 , and hence -1 + xm plication by m 2 • Similarly one obtains:

=x- m

by left multi-

~

(iii)

(iv)

(x-+ 1) (1 + m 2 ) = x +,(-m 2 + xm), and

+ .

(x·+ 1~ (1 + m 2 ) = x + (m 2 x- m).

But x + (-m 2 + xm) = x + (m 2 x-m) =? -m 2 + xm = m 2 x-m =?-1+xm 2 = x- m 2 • Thus the mappings (-1, x) and· (x, -1) agree on the distinct points m and m 2• This implies the contradiction x = -1. Thus we have: x E Q; (C3 )\{-1} and x + 1 E [ (C3) =? 1 + x E [ (C3). But 1 + x E [ (C3) =? x- 1 + 1 E [ (C3) =? 1 + x- 1 E [ (C3) =? x + 1 E [ (C3), and hence 1 + x E [ (C3 ) ~X + 1 E [ (C3). Now let X E [ (C3 ), X =I= -1, then either 1 +X, X+ 1 E [ (C3 ), or 1 +X, X+ 1 Et (£ (C3). If 1 + x, x + 1 E [ (C3 ), then we have: ( 1 + x) (1 +'m) =(1 + x) + ( 1 + x) m = 1 + [x + (x + 1) m]

= 1 + [x + m (x + 1)]

= 1 + [x + (xm + m)] = 1 + [(x + xm) + dx,xm m]

§ 8. Theorems on Near-Domains

37

= 1 + [x(1+m)+m]=1 +(-xm 2 +m),and

(1 + x) (1 + m) = (1 + m) (1 + x) = (1 + m) + x(l + m) = 1 + (m-xm 2 ) Therefore, m -xm 2 = -xm 2 + m =? 1 - xm = -xm + 1 =? -xm E E =? x E E, since by assumption m E E. If 1 + x, x + 1 Et !r (C3 ), then we have: (1+x)(l+m)=(1+x)+(l+x)m =1+[x+(x+1)m]=1+[x+m 2 (x+1)] = 1 + [x (1 + m 2 ) + m 2 ] = 1 +(-xm + m 2 ), and

(1 + x) (1 + m) = (1 + x) (-m

2

)

= -m (1 + x) = (1 + m 2 ) (1 + x)

= (1 + m 2 ) + (1 + m 2 )x = 1 + (m 2

-

xm).

2

But 1 + (-xm +m?) = 1 + (m 2 -xm) =?-xm+m =m 2 -xm=?-xm 2 +1= 1 - xm 2 =? -xm 2 E E =? x E E. )hus we have proved !r (C3 ) C E, and hence [(F*, ·): (E*, ·)] ~ 2, since [F*: (f (C3 )] = 2. But [F* :E*] ~ 2 =? E = F, since if [F*: E*·] = 2, then for x E F\E we have x- 1 + 1 E F\E and x (x- 1 + 1) = 1 + x E E, and hence the contradiction x E E. As a consequence of this theorem, we have:

=

(8.25) Let (F, +, ·) be a near-domain with char F = p and p 1 mod 3. Let denote the prime field of F. If n* <J F*, then F is a near-field.

1T

=

Proof: If p 1 mod 3 then 3llrr* I and hence rr* contains a unique subgroup C3 of order 3. If rr* <J F*, then C3 <J F*, and by Theorem (8.24) (F, +, ·)is a near-field. By generalizing the dickson procedure for constructing finite near-fields (see [8]), Ellers and Karzel. in [9] and later Karzel in [19] introduced the concept of dickson near-field as follows: A near-field (F, +, ·) is called a dickson near-field, if there exists a multiplication * on F such that (F, +, *) is a skewfield and the mapping a'P : x -+ a- 1, * (a· x) is an automorphism of (F, +,*),for all a E F*. Here a- 1, denotes the inverse of a with respect to(*). One also speaks of the quadruple (F, +, ·, *) as a dickson near-field and we say that (F, +, ·)is derived from (F, +, *). In [9] it was shown that all finite near-fields constructable by the dickson method, and hence according to Zassenhaus [47] all finite near-fields with seven exceptions, are precisely the dickson near-fields according to this definition. Several classes of infinite near-fields both planar and non-planar appear in the literature (see for example [7], [19], [22], [23], [37], and [48]). To the author's

38

§ 8. Theorems on Near-Domains

knowledge all known examples of infinite near-fields are in fact dickson nearfields. A result of Wiihling's [45] states in part, that a near-field (F, +, ·) is a dickson near-field derived from a commutative field, if its multiplicative group contains a normal abelian subgroup A such that A = A · {1, -1} generates F as a near-ring. We use this result to prove part of the last theorem of this section. Theorem (8.26) Let (F, +, ·) be a near-domain and suppose A< F* such that AnD= 1, where D ={da,b: a, bE F}. lf any one of the following conditions is satisfied, then (F, +, ·)is a near-field:

=0, and [F*: A] 2 char F = 2, [F* : A] = 2, and IE I > 2 char F =p, p > 2, and [F*: A]< IE I (in particular, char F

(i) (ii) (iii)

[F*:A]
this is valid if

For FC subgroups A < F* for which [F* : A] is finite the condition A n D =1 is superfluous (see (8.12){c)i For normal subgroups A <J F*, AnD= 1 is equivalent to An Dt =1 (see {8.9) (iv)). Further if A is commutative and [F* : A] =2 and IE I > 2, then F is a dickson near-field derived from a commutative field. Proof: We first prove:

(*)If A< F* with AnD= 1 and [F*: A]= 2, then Eisa sub near-field of F. To prove(*) we let c

=1 in (8.l)(x), then

db+a, 1 =db.IJda, b+l db,l· For b ~Ewe have:

(I)

. db~p,1 ~ db,ada,b+1·

Also for a, bEE we have:

(II) Case 1.

1 + (b +a)

=(1 + b) +a= (b

+ 1) +a

= (b +a)+ db.a= d1' b+a [(b +a)+ 1]. (II) db, a = 1 => 1 + (b + a) =(b + a) + 1 =>

b + (1 +a) = b + (a + 1) b + a E E.

(I)

Case 2. da, b+l = 1 => db+a,1 =db, a => 1 + (b + a) da,b [(b + a) + 1] =>

a +b

.E. = 1 +(a+ b)

(a+ b)+ d0 b

=b + a => db ,a ;::: db+a,,1 =1 => b + a E

Case 3. db, a ::f. 1 and d0 , b+l ::f. 1 =>db, a, d0 , b+l El: A, since AnD= 1. Thus

§ 9. Further Theorems on Sharply 2-transitive Groups

39

db+a, 1 = db,.a d~. b + 1 E A, since [F* : A] = 2, and this together with AnD= 1 implies db+a, 1 = 1, and hence b +a E E. (*)now follows from (8.17). Now suppose IE I > 2 and c E E, c =I= 0, c =I= 1. By (8.1) (xii) we have d0 1 = da, c dc+a, -c+1 , for any a E F*. ' Case 1. d0, c = 1 => dc_, a, 1 = 1 => c-1 a E E =>a E E by ( * ). 1

Case 2. dc+a, -c+ 1 = 1 => d(-c+l)-' (c+a), 1 = 1 => ( -c + 1)- (c +a) E E => a EE by(*). Case 3. da, c =I= 1 and dc+a, -c+ 1 =I= 1 => d0 , 1

= 1 as

above, and hence a E E.

Thus we have E = F, if (i) or (ii) is satisfied. !';Tow suppose (iii) is valid and further suppose E =I= F and a E F\E. Suppose ~. e' E E such that da,e A = da, e' A. By (8.1) (xii) we have d0 , e = d0 , e' de'+a,-e'+e d_e' e = d0 e'de'+a -e'+e· Thus de'+a -e'+eEA, which implies de'+a -e'+e = 1, sine~ A n = 1. Thus either a E E ~r -e' + e = 0 since by (8.18),'E is a nearfield. Since a Et E we must have e = e'. This implies the contradiction [F* : A] ;;;. IE I. Therefore, E =F.

b

Now suppose A is abelian, [F*: A] = 2, and IE I > 2, then F is a near-field. LetA' be the sub near-ring generated by A· .{1, -1}, then either A'= AU {0} or A'= F, since [F*: A]= 2. Suppose A'= A U {0} and let x E F* \A, then 1 + x E xA. But 1 + x E xA => x- 1 + 1 E A => x-1 EA.=> x EA. This contradicts x E F*\A and we see that A satisfies the conditions ofWahling. Therefore(F,+,·) is a dickson near-field derived from a commutative field.

It is known that the multiplicative group of a proper skewfield does not contain an abelian subgroup of index 2 (see for example [38] Ch. 14). This is not necessarily so in the case of near-fields. In connection with sharply 3-transitive groups we present in § 12 a class of proper infinite near-fields which do in fact contain abelian subgroups of index 2.

§ 9. Further Theorems on Sharply 2-transitive Groups In § 6 we saw how sharply 2-transitive groups are completely characterized by near-domains: We now use some of the theorems on near-domains from § 7 and § 8, which were proved primarily using algebraic methods, to prove further theorems on sharply 2-transitive groups. We first prove a generalization of a result of Cohn [6].

40

§ 9. Further Theorems on Sharply 2-transitive Groups

Theorem (9.1) A group His isomorphic to a complement of a sharply 2-transitive group if and only if there exist an element e E Z (H) with e2 =1, and an involution e E Sym (H*), where H* =H\ {id}, which satisfies the following conditions: (i)

For every a E H*, there exists aunique x E H such that e (ex)= xa

(ii)

e(y- 1 x)

=e(y- 1 ) e((e(y)r 1 e (x)), for all x, y E H, x =F y.

Proof: Suppose G is a sharply 2-transitive group such that H 2:! Ga. Let (F, +,·)be the near-domain associated with G (see theorem (7.1)), thenG e:!n.(F) and without loss of generality we may assume H = F* and ll =F (then H* = F* \ {1}). Fore= -1 we have e E Z(F*) and e2 =1, by (6.4). Further e = (1,-1) is an involution on R Since (F, +)is a loop there exists for every a E H*, a unique y E F such thaty + 1 =a. For x = y- 1 we have e(ex) = 1 + x = xa, and (i) is satisfied. To prove (ii) we first prove: (*) d1,-Cl-y)-• (1

Yr

1

= -(1- Y- 1r 1Y- 1• for ally E R

(8.1) (v) (xi) (vi) (iii) imply: 1

1

1

d1,-(1-y)-l (1- Yt =(1-y)- d1-y,-1 =(1-Yt d1,-y =0 y)-~(1-y)(-y + 1 r~ =[-y(l-y·lw1 -(I- y·lrly-1. Now by FbS, Fb6, (8.1) (xv), and(*) we obtain:

e((e(y))' 1 e (x)) =1- (1- y)' 1 (1- x) =1- [(1 =[1 (1-ytll+d1,-0-ytl (1- Ytlx=(I ;:::: (1 - y-1 r~(1- y-1 x) =(e (y-1)t1 e(y-lx).

-yt

1

-

(1 - yr 1 x]

y""~r~-(1-y-l)-1y-1x

Now suppose' H, e, and e satisfy the conditions of the theorem. Set ll = H lj {0} + and extend e-toHby e(O) = 1 and e(l) =0. We define addition onH by 0 + b =b +0 = b, for all bEll, and a+ b = ae(ea- 1 b), for alia, b EH, and/we extend multiplication to H by Oa aO 0, for all a E H. If (il, +, ·) is a neardomain, then for G = 12 (H) we have H 2:! Go by ( 6.4) and the theorem is proved. Multiplication is obviously a group multiplication. Left distributivity is proved as follows: a(b +c)= abe(eb- 1 c) = abe(e(ab)' 1 ac) =ab + ac, for all a, bE H. The unique solubility of the equations 1 + x a and x + 1 =a, for all a E H follows from the fact that e is a bijection which satisfies (i). This together with the left distributive law implies that (H, +)is a loop. Further, for a, b E H: a+ b = 0 => ae(ea- 1 b) = 0 => e(ea- 1b) = 0 => ea- 1 b = 1 => eb- 1 a = 1 => E(eb- 1a) = 0 => be(eb- 1 a) = 0 => b +a= 0. To prove Fb6 it suffices to prove for every a Ell there exists d 1,a E H such that 1 +(a+ b)= (1 +a)+ d1,a b.

= =

,

§ 9. Further Theorems on Sharply 2-transitive Groups

41

Fora= 0 or 1 +a= 0 define d 1,a,; 1, and define d 1,a = e(ea) (e(ea- 1 )t1 a- 1 , for all other a E H. For a, b E H, 1 +a =I= 0, let y- 1 = ea and x = e (ea- 1 b), then y = (eat 1 = ea- 1 and e (x) = ea- 1 b. Now by (ii) we have: 1 +(a + b) = e (eae(ea- 1b)) = e(ea) e((e(ea- 1)t1 ea- 1 b) = e(ea) e (e (e (ea)t 1 e(ea) (e (ea- 1)t1 a-1 b)= (1 +a)+ d 1 0 b. Therefore (H, +, ·)is a near-domain and the proof is completed.

'

It is natural to make the following

Definition: Let G be a sharply 2-transitive group and (F, +, ·)its associated near-domain, then we define char G, the characteristic of G by: char G = char F, (see (8.4)). By (6.4), G is of type 0 if and only if char G = 2. Also, if G is of type 1, say charG = p, p =I= 2, then by (4.1)b J 2 * is a conjugate class, and hence all of its elements have the same order. For a= (1, 1), we have a E P, since a= (1, -1} (0,-1), and an= 2. If we adopt the convention ord a = 0, if a generates an infinite group, then we may state: (9.2) Let G be a sharply 2-transitive group, then we have:

(i)

G is of type 0

(ii)

If G is of type 1, then {a} C J 2 , and ord a =char G, for every a E J 2 *. Further, if char G = p =I= 0, then each complement G0 contains a cyclic subgroup of order p - 1.

-:?

char G = 2, and

Again denote the kernels of G by ra b and let e0 , r0 b• and r; b be defined as ' ' ' in § 5. (9.3) If I fa, b I > 1, then the following conditions are equivalent:


(i)

J2

(ii)

rd, b (ra, b \{ e

(iii)

0 })

= ra, b \{id}, for a pair a, b EM, a =I= b

There exists an element rEG, such that r Era b a11d ' for a pair a, b E M, a =I= b.

r;' b (r) E fa' b•

Proof: We write a= 0, b = 1, and let (F, +, ·) = (M0,1, +,·)be the neardomain associated with G with zero element 0 and multiplicative identity 1, (6.2). Then·G = Y;(F), ro,l = (1, 1), and ro,l = {(O,k): kEf} (6.6). Further, for 'Y E rO,l• say 'Y =(0, k), k E f, we have r0,1 ('Y) =[(1, 1) (k)] = (0, 1 + k). Also we see that f U {0} forms a skewfield if and only if rQ', 1(r0, 1\ { e0}) = {(0, 1 + k): k E f \{-1}} = r 0 1\{id}. Thus (9.3) is a consequence of Theorems (7.1) and (8.5). ' Proposition (9.3) is an extension of a theorem of Zemmer [49].

42

§ 9. Further Theorems on Sharply 2-transitive Groups

(9.4)SupposeP
. .

Pro of: Write a = 0, b = 1 and interpret (F, +, -) as in the preceeding proof. Since 1 2 < G, the set T = f u {0} = r0 1(1) u {0} forms a sub skewfield ofF, 2 (7.1) and(8.5). NowA 0, 1 = {
= 3,

then

(9.6) If the complements Gain a sharply 2-transitive group satisfy the condition Ga E [FC], then J2 < G. In particular, if the commutator subgroup of ~~~~~mJ2
their centers, then G is pappian or finite.

(9.7) If 1 2 < G, then any finite subgroup 9f a complement Ga satisfies the conditions (b), (c), (d), and (e) of (8.14), where F is replaced by Gin (b). Theorem (9.8) Let G be a sharply 2-transitive group with char G> 2, (F, +, ·) its associated near-domain, and E the largest sub near-field ofF (theorem (8.18)).

ihr ()

(i) The class V = .fW : n, w E J, 1f =F w} of ( 4.1 )d forms a class of conjugate commutative subgroups of G. 1n fact, each U = hr n Jw in Vis isomorphic to· (E, +) ana U = ~(a) = { r E G: .Ya = a}, for any a E U*. Also if U, U' E D, then either U = U' or U n U' = {id}.

(ii) Let have:

1f

E J, and a EM such that n (a) =a. For any· U E V with U C hr we

hr=

U

"(EGa

,YU, and{VEU:Vnhr=F{id}}={,YU:rEGa}.

Also I{.YU: r E Ga}l = [(F*, ·): (£*,·)],and the groups subgroups are the only subgroups of G contained in ln .

. (iii) For any U E 12

=,u

~EG

U we have: €u=

u_ u'.

U'EU

.YU.

r EGa. and their

43

§ 9. Further Theorems on Sharply 2·transitive Groups

Proof: (i) Since by (4.1)d the elements of Dare conjagate, it suffices to show that U1 = lJ1 0 n Ji, where Jlo, i E J with Jlo(O) = 0, and i(2- 1) = 2- 1, is isomorphic to (E, +)and that U1 = 'i&'(a), for a E Ut. Now Jlo = (0, -1>, i = (-1, -1>, lJ1 0 ={a= (a, 1): a E F}, and Ji ={{3 = <x + 1, dx,t>: x E F}. Thus {3 E U1 {} x E E, and we have U1 = {(e, 1) : e E E}. Since (e, 1) (e', 1> = (e + e', dee'>= (e + e', 1>, we see that e-+ (e, 1) is an isomorphism of (E, +) onto U1• Now let a E U1~ say a= (e, 1), e E E*, and 'Y E G, say 'Y =(a, m). Then "fa= a1 {}(a +me, da, mem> =(e +a, de, am>{} a +me = e +a and dame= de a{} a +me = e +a and me +a =a + e {}me = e, by (8.2:3), and ' a E E {} 'Y E U1 •

.

We now prove the following lemma:

(*)

U1 =JpnJp'{}p,p'E{(e,

l):eEE}andp=Fp'.

For any pair p, p' E J of the form p = (e, -1), p' = (e', -1>, where e, e' E E and e =Fe', we have p = :YJ1o and p' = :Yi, for 'Y = (e2- 1, e- e'). Since e2"' 1, e- e' E E, we have U1 = :YU1= :Y (JJ1 0 n Ji) = J p n J p'. Conversely, if p E J, say p =(-a, -1), and Jlp E U1 n Jp, with J1 = (x, -1>, x =F-a, then J1P = <x + a, dx,a> = (e, 1>, for some e E E. Thus X+ a= e and dx,a = dx+a,a = de,a 1, and hence a E E. This proves p E {(e, -D: e E E}. Thus if U1 = Jp n Jp', then p, p' E {(e, -1>: e E E}. Now suppose U, U' ED and U n U' =F {id}. By (4.l)d we may assume U = U1 without loss of generality. Now U' = lJ1 n Jrr, for some pair Jl, n E J, J1 =fo rr. But U1 n U' =F {id} ~ U1 n lJ1 =F {id} and U1 n Jrr {id}. By the above argu· ment we have Jl, rr E {<e, -1>: e E E}, and hence by(*) U' = U1•

*

(ii) By (3.5) (iv) it suffices to prove (ii) for the set lJ10 = {(a, 1): a E F}. Now G0 {(0, c): c E F*}, and for U:::. Ut. we have U C lJ1 0 • For every a E F*, define a* = aE* and write (F*, E*) {a*: a E F*}. Then for Ua* = {
. -yEG 0 b*E(F*,E*) b* -+ Ub* is a bijection of (F*, E*) onto {Ub* : b E F*}. Now b -+ (b, 1) is a bijection of (F, +) onto lJ1 0 , and hence the last sentence of (ii) is a conse· quence of theorem (8,19). The relation {V E D: V n lJ1 0 =F {id}}= { :YU: 'Y E G0 }, for U E U with U C lJ1 0 , is a consequence of the last part of (i) and the statement lJlo :::. :YU.

U

-yEG0

(iii) By ( 4.1 )d

U

'yEG

:YU := U~ U', for any U E U'EU

D.

Since U C !

2

,

we have

44

§ 9. Further Theorems on Sharply 2-transitive Groups

U :Yu c 1

2

•

But if V1T E 1a, then for U' = l1T n 11Tv1r, we obtain U' E [J

'YEG

and V1T E U'. Thus J2 E

U

U'EU

U'.

Theorem (9.9) Let G be a sharply 2-transitive group with char G > 2, (F, +, ·) its associated near-domain and E the largest sub near-field of F. Also let a, bE M and J.1,1TE1, with J.l.(a)=a and 1r(b)= b. Define lfa,b =1J.l. n l7T, La,b ={[a(a)]~,b: a E Ua* b} (see§ 5), and M' = {a(a): a E Ua b}. Then Lab is a subgroup of Ga, and th~ subgroup of G generated by lfa, b a~d La, b is a ;emidirect product Ua,bLa,b• and ~ b La b is a sharply 2-transitive subgroup operating on M'. Further (lfa bL~ b• M') and (T2 (E), E) are isomorphic as permutation groups. Consider the' syst~m {L} of 2-transitive subgroups L of G; which have the following property:

< L, where ft = 1 n L denotes the set of involutions of L. The groups lfa, b La, b• for all a, b EM, a -=I= b, are conjugate maximal elements in the system {L}. ( *) 1l

Proof: Write a= 0 and b = 1, then from Theorem (9.8) (i), U0 1 = {<e, I>: e E E} is a commutative subgroup of G. Also L 0 1 = {[(e, 1) (0)] ~ 1 : (e, l)E U 1} = {(0, e): e E E*} and M' = {<e, 1)(0): (e, E U0 , 1}= E. Thus L 0, 1 is a subgroup of Go, in fact, L 0, 1 ~ (E*, ·) and U0, 1 L 0 , 1 = {<e, e'>: e, e' E E, e' -=I= 0}. Thus we see that U0 , 1 L 0, 1 is a semidirect product of U0, 1 by L 0 , 1 and that (Uo 1 Lo 1 , M') ~ (T2 (E), E) as permutation groups. Since (E, +, ·)is a nearfield, w~ have by theorem (7.1): U0 1 L 0 1 E {L}. Further, U0 1 L 0 1 is maximal in {L}, since any larger 2-transitive ~ubgroup in L would dete~in~ a sub nearfield of (F, +,·)larger than (E, +,·),which contradicts theorem (8.18). One also easily verifies &Ua,bba,b = Ucx(a),cx(b)Lcx(a),cx(b)• for a E G, and hence the subgroups~ Ua:b La, b are conjugate~

0,

i>

Gbe a sharply 2-transitive group with char G > 2. The set 1 2 operates either sharply transitively on M, i. e. 1 2 < G (see Theorem (3. 7)), or for every pair of distinct points p, q EM, tliere exist an infinite number of permutations a E 1 2 , such that a (p) = q. Theorem (9.10) Let

Proof: Let (F, +, ·) be the near-domain associated with G and (E, +, ·)its largest sub near-field. Suppose that 1 2 does not operate sharply transitively on M, then by theorem (3. 7) and theorem (7.1), E -=I= F. If E is infinite, then by (8.22) (iv) the set D 1 = {d1,b : b E F} is infinite. Thus the set 1 = {a = (1, d1 b) : dn E D 1} is infinite. But a (0) = 1, for every aE 'Also, for a E with 'a + b = 1, we have by (8.1) (vii) a= (1, d1,b> = . 1

16,

J6

F

________ jt!_~~· da+b,b> =(a+ b, da,b>, and hence by (6.5) 16, 1 C 1 2 • Now for p,qEM,

§ 9. Further Theorems on Sharply 2-transitive Groups

p =I= q, choose 'Y E G, such that 'Y (0). =p and 'Y (1) finite, q C 2 , and (3 (p) = q, for every (3 E q.

1;, 1

1;,

=q.

45

Then ~~ q = rJ6, 1 is in-

Now suppose E is finite. Then U0, 1 = {<e, 1): e E E} is finite, and as. we see from(*) in the proof of theorem (9.8) (i), the number of sets Jp, p E J, such that there exists a p' E J, such that U0 1 ,;, Jp n Jp', is finite. This is obviously true for any U E fJ. Now let p, q EM: p =f=. q, and let J.1. 1 E J be fixed. By theorem (3.3) there exists exactly one a 1 E ]J.l.I> such that a 1 (p) = q. Further, a 1 E Ul> for some U1 E fJ with U1 C ]J.1. 1• (U1 does not necessarily equal {<e, 1): e E E}). Now by theorem (9.8) (ii) and theorem (8.18), the number of distinct sets ]J.l., J.1. E J, is infinite. Thus there must exist a ]J.1. 2 , such that U1 =I= ]J.1. 2 n Jw, for every w E J. Now there exists exactly one a 2 E ]J.1. 2 , such that a 2 (p) = q. Since a 1 E U1 and U1 n ]J.1. 2 = {id} by theorem (9.8) (ii), we have a2 =I= a!. But a2 E u2, for some u2 E D with u2 c ]J.1.2· Arguing as above, tliere must be a set ]J.1. 3 , J.1.3 E J, such that U1 =I= ]J.1.3 n Jw and U2 =I= ]J.1.3 n Jw, for all wE J. But there exists exactly one a 3 E ]J.1. 3 , such that a 3 (p) = q. Also, as above, a 3 =I= a2 and a 3 =I= a~> and a 3 E U3, for some U3 E D with U3 c ]J.1.3· Continuing in this manner, we construct an infinite set of distinct elements ai E 1 2 , i EN, such that ai(P) = q.

CHAPTER Ill. SHARPLY 3-TRANSITIVE GROUPS Throughout this chapter, any group denoted by G will be assumed to be a permutation group which operates sharply 3-transitively on a set M We consider the question which arises in a natrural way from chapter II: when do the sets of products of two involutions of G which have a common fixed point form subgroups? It is shown in § 13 that these sets do in fact form subgroups for an infinite number of characteristics, namely for those primes p, such that p =: 1 mod 3. This means that sharply 3-transitive groups of characteristic == 1 mod 3 can be described in terms of near-fields in the sense of § 11. In§ 10 it is shown that there are no groups of type (1,0), and that a group G is of type (1, 1) if and only if char G = 3. In § 11 we show generally that sharply 3-transitive groups can be uniquely described in terms of structures deiived from near-domains which we call KT-fields. Several theorems on KTfields are presented in § 12. Also in § 12, a class of KT-fields is introduced which contains all examples of these structures known to the author. In § 13 some of the results of § 12 are interpreted in terms of the associated sharply 3-transitive groups and examples of both finite and infinite non-pappian sharply 3-transitive groups are given. These examples include in fact all non-pappian finite sharply 3-transitive groups as determined by Zassenhaus [46]. It is seen that many characteristic properties of finite sharply 3-transitive groups do not carry over to the infinite case.

§ 10. Structure. Theorems for Sharply 3-transitive Groups ., + .

As in§ 1, we define G0 ={a E G: a(a) =a} fora EM Then by (1.1) G = G0 U G0 rG0 , for any rEG\ G0 , lJG0 = G6 (a)• for any o E G, and in fact: (10.1) The groups G0 form a class of maximal conjugate 'subgroups of G and each G0 operates sharply 2-transitively on M\ {a}. Further, define K = H 3 ={a E G: ord 0! = 3}, and let J and 11 = H2 denote the set of involutions and the set of involutions which have fixed points respectively. Then from § 2, K and ft are conjugate classes in G and G is of type (m, n), mE{0,1,2}, nE {0, 1} if and only if the elements of K have m, and the elements of ft have n + 1 fixed points. Define 10 = J n G0 = ft n G0 , for a EM.

47

§ 10. Structure Theorems for Sharply 3-transitive Groups

Then Ja is the set of involutions in Ga and &Ia = JCY.(a)• for all a EM. Thus we have &1} = 1;;(a)• for all a EM and the sets la2 , a EM, form a class of conjugate subsets of G. If we think of Ga as being a sharply 2-transitive group operating on M \{a}, then is the set of product of two involutions in this group and we may state the main question of Chapter II here for sharply 3-transitive groups as follows:

1;

In a given sharply 3-transitive group G, do the sets 102 fonn subgroups of G? It is useful to define 1/j =

U

aEM

102 •

.fc} is the

set of all products of two involu-

tions in G which have a common fixed point. We note .fc} n G0 = Ja2 • We may think of J~ as playing the role in sharply 3-transitive groups which 1 2 played in the study of sharply 2-transitive groups. We note, that by (3.4), fu2 n J?i = {id} for a =I= b. By (10.1) it is meaningful to define char G =char G0 . . From (9.2) and (4.1)b we obtain: (10.2) Let G be a sharply ]-transitive group, then we have:

(i)

G is of type (m, 0), formE {0, 1,2} '¢?char G = 2.

(ii)

If G is of type (m, 1), formE {0, 1,2}, then 1\{id} is a conjugate class in G and we have: ord a= char G, for every a E 1\{id}. Further, for any a EM we have {a} C for every a E 1;.

IJ

1;,

IJ

We now examine the sets J, ft, and K in more detail.

(10.3) Let Jl, 17 E J, and suppose J1 (a) =17 (a) and J1 (b) = 17 (b ),for distinct points a, b EM, then either J1 = 17 or J1 (a) = 17 (a) =b. Proof: For a'= J1 (a)= 17 (a), we have Jl(a') = 17 (a')= a. If a'= a, then /1, 17 E Ja and J1 (b) = 17 (b) implies J1 = 17 by (3.4). If a' =I= a, and a' =I= b, then J1 = 17 by sharp 3-transitivity. (10.4) Let (a, b, c) be a proper triple, and let a E K, such that a (a)= b, a(b) = c, and a(c) =a. Further, let 7i Eft, i = 1, 2, 3 such that 7 1 (a) =a, 7 1 (b) = c; 7 2 (b) = b, 7 2 (a) = c; 7 3 (c) = c, 7 3 (a) =b. Then:

(iii)

If gEM is a fixed point of a, then 7 1 (g)= 7 2 (g) =7 3 (g), and h = 7 1 (g) is also a fixed point of a, which is distinct from g if and only if a has two fixed points.

§ 10. Structure Theorems for Sharply 3-transitive Groups

48

Proof: We note that (ii) implies (i). Now 71r 2(a) =7s7 1(a) = 7zrs(a) =b; 71 72 (b) = T371 (b) = Tz73(b) = C, and TtTz(C) =7s71 (C) =T273(c) =a. By sharp 3-transitivity a =r 1 7z = 73r 1 =7z 73.

(iii) Suppose a(g) =g, then by (ii) r 1 72(g)

=r 3 r 1 (g) =r 2 r 3(g) =g.

Thus

r 1 (g) = 72 (g) =r 3(g). For h = 71 (g) = 72(g) =73 (g), we have 7t(h) = 7z(h) = r 3 (h) =g, and by (ii) and (i), a(h) = 7 1r 2 (h) = 7 1 (g)= h. If h =g, and if g' were a second fixed point of a, then as above 71 (g') = 72 (g') = 73 (g') and by (10.3) we would have r 1 (g') = 7 2 (g') = 7 3 (g') =g, which implies the contradiction g' = h. Thus g is the only fixed point of a, if h =g. As a consequence of (10.4) we have:

(1 0.5) If a E K has two fixed points g, h EM and J1 E Jr such that J1 (g) then Jla E Jr and JlaJl = a 2 •

=h,

Proof: Suppose a EM such that J1 (a) a. By 3-transitivity, there exists 7 E Jr such that 7a(a) = a(a) and r (g)= h. By sharp 3-transitivity and (10.4) (ii) (iii) we have, for r 1 and 7 2 as defined in (10.4), J1 = rl> r 7 2 , a= JlT, and hence 11a E Jr and JlCXJl =TJl =a- 1 = a 2 .

Now for a E K, we have by (10.4) (ii) a =7 1 r 2 , for proper elements 7I> TzEJ[' But a- 1 =a 2 = r 2 71 and a= (a2 ) 2 =72 71 r 2 r 1 [72 , rt]. Thus a is a commutator. This together with (10.4) (ii) implies:

(1 0.6) K c

Jj n G 1,

where G 1 denotes the commutator subgroup of G.

The following two theorems answer our main question concerning the structure of for groups of type (1,n), for n E {0, 1}.

1;

Theorem (10.7) (a) There are no sharply 3-transitive groups of type (1, 0). (b) The following conditio~ns are equivalent: ~

(i)

(ii)

em)

·~

G i~ of type (.1, 1)

char G

f;

=3

K = 1\{td}

Proof: (a) Suppose G were of type (1,0) and a E K, and a EM such that a(a) ::f. a (we assume IM I> 3). Write b = a(a) and c =a 2 (a), then by (10.4) (ii) a=r1 r 2 where 7t. 72 E~ with r 1 (a) a, 71 (b) =c and 72 (b) =b, 7:z(a) =c. Now a would have exactly one fixed point p EM and p Et {a, b, c}. Thus by (10.4) (iii) we would have 7 1 (p) = 7 2 (p) = p which contradicts the assumption that the elements of Jr have only one fixed point. (b) Suppose G is of type (1, 1) and let a E K. Again by (10.4) (ii), a= Tt 72, for proper elements 71 , 72 E Let p EM be the single fixed point of a, then

Jr-

§ 10. Structure Theorems for Sharply 3-transitive Groups

49

by (10.4) (iii), r 1 (p) := r~(p) =p. Thus a E 1;, and hence, by (10.2), char G = ord a = 3. If we assume char G = 3, then by (10.2), 1\{id} C K which implies J;f\{id} = K; since 1\ {id} and K are conjugate classes. Finally, if .fc} \ {id} = K and a E K, then a E f;, for some a EM By (3.4), a is the only fixed point of a. Therefore, G is of type (1, 1) or (1,0). From part (a) we conclude that G is of type (1, 1).

1;

J-;

As a corollary of Theorem (10.7) and Theorem (9.5) we have:

Theorem ( 10.8) If G is a sharply 3-transitive group of type ( 1, 1), then Ji = (K 11 Ga) U {id}, for each a EM, and each 1; forms a subgroup of G. From Theorem (10.8), Theorem (3.7) (i), (iii), and the equatiop Ji 11 Jg = {id} we obtain: (10.9) If G is of type (1, 1), then for any three distinct points a, b, c EM, there exists exactly one a E K such that: '

a(a) =a, and a(b) =c.

The following proposition is analogous to (10.9) for groups of type (2,n), for

n E {0, 1}. It is, in fact, a generalization of a theorem of Tits [39].

(10.10) If G is of type (2, n), where n E {0, 1},then, for any a, b, c EM such that a =I= band a =I= c, there exist exactly two distinct elements a; E K, i = 1, 2, such that:

a;(a) =a, and a;(b) =c. Proof: Since the elements of K have order 3 we have:

( *) If a, {3 E K and a (a) = {3 (a) and a 2 (a) = {3 2 (a) for an element a EM such that a (a) =I= a, then a = {3. By 3-transitivity it is clear that there exists a E K such that a (a) = a and a(b) = c, for any a, b, c EM with a =I= b and a =I= c. Now consider the special case b =c. Let a, {3 E K such that a(a) = {3(a) =a and a(b) = {3(b) =b. For any c E M\{a, b}, there exists by 3-transitivity JJ.E G such that JJ.(c) = c, JJ.a(c) = {3(c), and JJ.a 2 (c) = {3 2 (c). Then {3 = JJ.aJJ.- 1 by(*), and hence aJJ.- 1 (a)= JJ.- 1 (a) and aJJ.- 1 (b)= JJ.- 1 (b). Since a and b are the only fixed points of a, we have either J1. = id , or J1. (a) = b and J1. (b) = a. In the first case {3 = a, and in the second, J1. E 11' and {3 = a 2 by (1 0.5). Therefore, a and a 2 are the only elements in K which fix a and b. Consider the case b =I= c. Let a E K such that a (a) =a, and a (b) = c and let a' be the other fixed point of a. Also let o E G such that o(a') =a, o(b)= b, and o(c)= c. Then for {3 = 6a we have {3 E K, {3(a) =a, and {3(b) =c. Further,

50

§ 10. Structure Theorems for Sharply 3-transitive Groups

~ = a implies ao- 1 (a') = o- 1 (a'), which implies li (a) =a', and hence li E Jf. By

(10.5) we have the contradiction~= a 2 • Thus~ =I= a. Note that a and li(a) are the fixed points of~ and that a = o- 1 ~ll. Now suppose ~~ E K such that ~~(a)= a and~' (b)= c. Foro' E G such that o'(b) = b, li'~(b) =~'(b), and li'~ 1 (b) = ~' 2 (b) we have by(*),~~= 8~. From o'~o'- 1 (a) =~'(a)= a, we obtain ~o'- 1 (a) = o'- 1 (a) which implies o'(a) ;::: a, or o'- 1 (a) ;::: li (a). Now c = ~'(b)= o'~o'- 1 (b) = li'~(b) = li'(c). Therefore, by sharp 3-transitivity, if o'(a) ;::: a, then o' = id and ~/ = ~. and if o'- 1 (a) ;::: li (a), then o'-l = li, since o'- 1 (b) = b = li(b) and o'- 1 (c) = c = li(c), and we have~'= o- 1 ~8 =a. Thus a 1 = a and a 2 = ~ are the only elements inK which fix a and map b, onto c. (10.11) Let G be of type (2, n), where n E {0, 1}, and let a, b, c EM be distinct points. Also, let a and ~ be the two distinct elements in K such that a (a) =~(a) =a, and a(b) =~(b) = c, according to (10.10). For p E lt such that p (a)= a and p(b) = c, we have: pap= ~ 2 , p~p = a 2 , and p(a') = b', · where a' and b' are the other fixed points of a and.~ respectively. Proof: Since a 2 (c) = ~ 2 (c) = b, we have pa 2p(a) = p~ 2p(a) =a and pa2 p(b)= p~ 2p(b) =c. Therefore, by (10.10), pa 2p E {a,M and p~ 2p E {a,~}. But if pa 2 p =a, then p~ 2p = ~. since a =I=~. and we would have p (a') ;::: a' and p (b') = b'. This implies the contradiction p = id. Thus pa 2p;::: ~and p~ 2p =a. Now p~ 2p (a') ;::: a (a') = a' and hence ~ 2p (a') = p(a' ), which implies p (a') E {a, b'}. But p (a);::: a, and hence p (a') = b'. We shall call the groups Ga, b• for a, b EM with a =I= b, the complements in G. We have: char G = 2

'* Ga, b

has no subgroup of order 2

char G =I= 2 ~ Ga, b has ex;ctly one su~group of order 2, namely

n Jd n lb ;::: Ga, b n ]a = Ga, b n Jb ={id, Pa, b}, ~

Ga, b

,

where ll.a, bE ] 0 with Pa, b(b) =b. We note that Pa, b E Z (G0 , b)· Now define

id, if char G = 2, and Ea, b =

{

Pa, b, if char G =I= 2.

Then we have Ga, b Z(Ga, b)·

n j ;::: Ga, b n fr

= Ga, b

n la

= {id, Ea, b}, and {id, Ea, b} C

The next t11eorem is a slight generalization of a theorem of Huppert [15], Hilfsatz 7.

§ 10. Structure Theorems for Sharply 3-transitive Groups

51

Theorem (10.12) Let N(Ga,b) denote the normalizer.of the complement Ga, b in G. Then N(Ga,b) = Ga,b UTGa,b,for any T E G such that r(a) =band T(b) =a, and hence [N(Ga b): Ga b1 = 2. For any T E Jf such that T(a) = b, . we have, for a EGa, b• fa::, a if dnd only if a E {id, ea. b}· Proof: The equation N(Ga,b) =Ga,b U TGa,b follows directly from TGa, bT- 1 = Gr(a),r(b) = Gb,a = Ga,b· We note thatN(Ga,b) ={~ E G: {~(a), ~(b)}= {a, b}}. Suppose r E Jf and T (a) b. Let {c, c'} be the set of fixed points of T (c = c' occurs whenever char G 2). Now, for a E Ga. b we have fa = a => m (c) a (c) and m(c') = a(c') => {a(c), a(c')} = {c, c'} =>a= id, or a(c) = c' and a (c') =c. But a(c) = c'' and a(c') = c implies a E Jf and hence

ea, b· On the other hand, fea, b = er(a),r(b) = eb, a = ea, b· (We note that ea, b interchanges the fixed points of any T E (N(Ga, b)\ Ga, b) n Jf if char • G i= 2).

a

' Since obviously &N(Ga, b)= N(Get.(a),Ot(b)), for any a E G, we see that the normalizers of the complements form a class of conjugate subgroups of G. Now define

Aa;:::

J

la U {id}, if char G = 2,

l law,

for some w E la, if char G i= 2.

Then by Theorems (3.3) and (1.2) we have, for any bE M\{a}, Ga = Ga,bAa = AaGa,b· Also, for any T E G\Ga, we have G Ga U GaTGa. 1fT EN(Ga,b)\ Ga, b, then TGa, b = Ga, b T. Thus we may write GaTGa = GaTGa, bAa =GaGa, bTAa GaTAa = AarGa, and we have:

=

(10.13) G = Ga U GaTAa = Ga U AarGa, for any T E N(Ga, b)\ Ga, b• where

a, b EM with a i= b.

Again suppose rGa,b is the coset of Ga, b in N(Ga,b), then since [N(Ga, b): Ga, b] = 2, we have (rGa, b) 2 =Ga, b· This implies K n TGa, b = 0, for if a E K TGa, b, then a2 E K n Ga, b• which implies the contradiction a E Ga, b· Thus K n N(Ga, b)= K n Ga, b· This together with (10.10) implies:

n

Theorem (10.14) In a sharply 3-transitive group G the following conditions are equivalent: (i)

G is of type (2, 0), or (2, 1)

(ii)

Ga, b contains a subgroup of order 3

(iii)

Ga,b contains exactly one subgroup of order 3, namely Ca, b = K n Ga,b• and hence Ca, b
52

§ 10. Structure Theorems for Sharply 3-transitive Groups

(iv)

N(Ga, b) contains a subgroup of order 3

(v)

N(Ga, b) contains exactly one subgroup of order 3.

We also note that if G is of type (2,0) or (2,1 ), then 8ca, b = C0 (a),o (b)• for any G and the subgroups or order 3 form a class of conjugate subgroups of G.

oE

An obvious consequence of(10.5) and the relation {id, ea,b} C Z(Ga,b) is: ( 10.15) Suppose a E Ca, b and 11 E Jp then P.a = a 2 if and only if 11 EN (Ga, b)\

Ga,b· (10.16) Suppose G is of type (2,0) or (2,1) and Cab is the unique subgroup ofGa,b of order 3. Then N(Ca,b);, N(Ga,b) and [N(Ga,b): [ (Ca,b)] = 2.

Further, [ (Cab) = Ga b• if and only if Cab C Z (Gab), and if Cab
Proof: Let Ca,b = {id, a, a }. Now SCa,b = Co(a),o(b) imp!iesN(Ca,b) = N(Ga,b). Further, [ (Ca,b) = {o EN(Ga,b): oa = ao}, and obviously: [ (Ca, b) = Ga, b implies Ca, b C Z (Ga, b). Also, if Ca, b C Z (Ga, b), then Ga, b C \r(Ca b). But if o E rGa b• where T E (N(Ga b)\ Gab) n 11, then by (10.15), Ba =· ra = a 2 , and henc~ (£ (Ca, b) = Ga, b. If ca, b (Ga, b), then there 2 exists a p E Ga, b such that pa = a • But ca, b < Ga, b and Ica, b I = 3 imply Gab = Ba b U pBa b· Thus [Gab : Ba b] = 2. Also for 0 E rGa b• say o = r{3, ~e have; 5a =a·~ f~a = a ;. ~a= 'fa= a 2 ~ {3 E pEa, b· Th~refore,

a: z

\r (Ca, b)= Ba, b U rpBa, b· Consider an arbitrary sharply 3-transitive group G and two distinct points a, b EM. We define:

flla,b =(N{Ga,b)\Ga,b)nJt={p.EJf: 11(a) =b}, and

.?a b"= (N(Ga b)\ Gab) n J ={11 E J: 11(a) = b}. '

'

'

Suppose 77 E Jf Va (71 E J\ Ja ). There exists 11 E Ja, where la = Ja U {id}, such that 11 (b) = 77 (a). Thus 117111 E ~a. b (]17111 E ~.b), and we have: (10.17) Jf

=Ja u [

u 11~a. bl1]· and J =Ja u [ 11Ela u 11.9'a, b11].

!J.E]a

From the definitions we have 8~a.b = ~o(a),o(b) all o E G, and hence N(~a b)= N(~ b)= N(Ga for every c EM\{a, b}, th~re exists e;actly one and hence flla, b = {am- 1 : a E N(Ga, b)}= {(3r{3- 1 since Gab operates transitively on M\ {a, b}, Also __ _ __ _{r{3 :_{3 EGa, b and 7{3 = {3- 1}.

and B~,b = 95(a),o(b)• for b). By sharp 3-transitivity, Eflla b such that {3(c) = c, : {3 E 'ca, b}, for any T E -~a. b• by definition we have ~ b =

d

'

§ 10. Structure Theorems for Sharply 3-transitive Groups

53

Now, for any T E &£a, b we have T 9ih, b C T .?a, b C Ga, b. We note that T81?a, b = {[r, a:] = To:T0:- 1 : a: E Ga, b} = {[a:, T] : a: E Ga, b}. Further, a(r 81?a, b) = a(r) 81?a, b and a(r Yd. b) = a(r) .?a, b• for a: E Ga, b• and hence T81?a, b = /J.B1?a, b (or T Y'a, b = IJ.fi'a,b) for all T, /). E f3fa,b• if and only ifr&i'a,b (or Tfl'a,b) is normal in Ga,b• for some T E &?a, b· In the finite case one can show that the sets T ff9a, b and T fl'a, b are normal in Ga, b· In fact both sets form subgroups and [Ga, b : T81?a b] = 2, if char G 1= 2 (see § 13). This is not so in the infinite case. Based on ~xamples given in § 13, we show that the set T fl'a, b is not necessarily a subgroup nor is it necessarily normal in Ga, b. (10.18) The sets r81?a, b and T fl'a, b satisfy the following conditions: (a)

:(b) (c)

r81?a, b and Tfl'a, b contain the subgroup generated by any one of their elements Let Aa, b C T.?a, b• then Aa, b < G ~ fAa, b tative

= Aa, b and A a, b is commu-

If G is of type (2,0) or (2,1), and Ca, b is the subgroup of Ga, b of order 3, then Cab C T81?a b• for every T E &la b· '

'

'

Proof: (a) is obviously valid forT .?a, b ={~EGa, b : f~ = ~- 1 }. Now let a: E f!lla, b, then To: E rf!lla, b and we have:

(i)

(ro:t 1 =

(ii)

(ro:) 211

(iii)

(ro:) 2n+l = r [(o:r) 11 o:(o:rr 11 ], for integers n;;;;. 1.

=;

O:T

=

T

(ro:r),

r[o:(ro:) 11 - 1r(ro:)-(n-l)o:], and

SinceN(~a.b) =N(Ga,b), we obtain (ro:)iET&i'a,b• for any integer i. LetAa,b C

Y'a, b•

then for a:, ~ E Aa, b we have f (a:) = o:-1, f~ = ~- 1 , and f~ = o:- 1 ~- 1 . Thus 1 wehave:A-a,b CAa,b ~fAa,b =Aa,b and~EAa,b ~f(~)=(~t 1 ~o:- 1 ~- 1 = 1 (~t ~~=~a:, and (c) is verified.

T

If G is of type (2,0) or (2,1), then by (10.15) fa:= o: 2 , for a: E Ca b• and hence a: = o:4 = To:T0: 2 = TO:T0:- 1 . Thus ca b c T81?a b· ' '· ' Here as in the case of sharply 2-transitive groups one can define pappian groups. We call a sharply 3-transitive group G pappian if the groups Ga are pappian. Hence G is pappian if and only if the yomplements Ga, b are commutative. As we shall see in § 13 Theorem (13,2), it is not necessary to define desarguesian groups in the sharply 3-transitive case. Also in § 13 we see that pappian groups are characterized by the projective groups PGL (2, F) over fields F. With this in mind one sees that the equivalence of (i) and (iii) in the following theorem is basically a result due to Staudt and Petkantschin [36].

54

§ 10. Structure Theorems for Sharply 3-transitive Groups

Theorem (10.19) Let G be a sharply 3-transitive group. The following conditions are equivalent:

(i)

G is pappian

(ii)

r Yo b = G0 b• for ,

,

7

E &l 0 b• or equivalently '

Si; b = 7G0 b• i.e. the coset J

of G0 , b inN (G0 , b) consists entirely of involutions

(iii)

,

For every quadruple of distinct points a, b, c, d EM, there exists a 11 E J such that 11 (a) = band p. (c) =d.

Proof: The implication (i) => (ii) is proved later using methods of§ 11 and § 12 (see theorem (12.3) and the note after theorem (13.2)). Now G0 b and hence 7G0 , b operates sharply transitively on M\ {a, b}. Thus (ii) and (iii) are equivalent. Now 79;;,b = Ga,b implies that Ga,b is commutative by (10.18)b and hence (ii) => (i). Consider the mapping '.fJ : Ga, b -+ 7&la, b· Then 1,0(a) = 1,0(~) ~ ~ E {a, €0 , b a}, since a -+ [a, 7] by Theorem (10.12) 1,0(a) = 1,0@) ~ afa- 1 = ~f~- 1 ~f(a- 1 ~) = a-1 ~ ~ a- 1 ~ E {id, €0 b}. In particular, if char G = 2, then 1,0 is injective. We see that if G is. finite 'and char G = 2, then 1,0 is bijective and by theorem (10.19) G is pappian. This result was first proved by Zassenhaus in [46]. In [27], H. Wefelscheid presented examples of infinite non-pappian sharply 3-transitive groups of characteristic 2 (see § 13). Now if G0 b is commutative, then by theorem {10.19) fa= a- 1 , for every a E Ga,b· 'Thus 1,0 is the homomorphism·a-+ a 2 and we have: (10.20) If G is pappian, then 7&l0 b is the subgroup of squares in G0 b· ~ '

.

The following tlu~,orem gives a necessary."and sufficient condition under which a sharply 2-transitive group is imbeddable, in a certian sense, into a sharply 3transitive group.

Theorem (10.21) Let (G', M') be a sharply 2-transitive group. Then G' is isomorphic as a permutation group to a group G0 in a sharply 3-transitive group G, if and only if, for an element oo El: M', there exists an involution 7 E Sym (M), where M = M' U {oo}, such that:

(i)

7(oo)7'=oo

(ii)

7(c) = c, for some c EM'

(iii)

fGoo,r(oo) = Goo,T(oo)• where aIM' E G~(oo)}• and

Goo,T(oo)

={a E Sym (M): a(oo) =

00

and

55

§ 11. KT-Fields and Sharply 3-transitive Groups

(iv)

For 11 E Sym (M) d.efined by Jl(oo) =co and 111 1 such that J1 (c)= 7(oo), we have (711) 3 = id.

lrf' = /1

1 ,

where /1 E J' 1

Proof: Suppose G is sharply 3-transitive and co EM, then G' =Go, operates sharply 2-transitively onM' =M\{oo}. Let b EM', 7 E[J£a,b• and c EM with 7(c) =c. Then Goo,b = Goo,r(oo) and, for 11 E Joo with 11 (c) = 7 (co) we have by (10.4), 7/1 E K and hence (7/1) 3 = id. Thus 7 satisfies the conditions (i)-(iv). Conversely, suppose G' is sharply 2-transitive and 7 E Sym (M) satisfies the conditions of the theorem. Define Goo= {a E Sym(M): a(oo) =co and aIM' E G'}. Then Goo< Sym (M) and by (l.l)(b) Goo= Goo,r(oo) U Goo,r(oo)llGoo,r(oo)' for '/1 E Goo \Goo, r(oo)· We take 11 to be the involution defined in (iv). Now define 'G = Goo U G 7G 00

00

,

For a proof that G is a 3-transitive group we refer to the more general theorem (10.6.16) in [38]. Now obviously G~ ~Goo. The sharp 3-transitivity of G follows from (l.l)(a). § 11.

KT·Fields and Sharply 3-transitive Groups

Analogously to § 6 we show here that shaqjly 3-transitive groups can be completely characterized by means of transformations on a suitable algebraic structure. The structures introduced here are derived from the pseudofields defined by Tits in [39] and form the near-domains of Karzel, and hence the name KT-field. Definition: A KT-field is a quadruple (F, +,·,a), which satisfies the following axioms: KT 1:

(F, +, ·) is a near-domain

KT2:

a is an involutory automorphism of the multiplicative group (F*, ·) which satisfies the jUnctional equation

a(1 + a(x)) = 1- a(l + x), for all x

E F\{0,

-1}.

The characteristic of (F, +, ·, a) is defined to be the characteristic of (F, +, ·). Now let co El: F and extend (+)and 0 to co by a+ co= co and moo= co, for all a, mE F, m =I= 0. Further, defme a (0) =co and a(oo) = 0. Let M = F U {co} and for each pair a, m E F, m =I= 0, define (a, m) E Sym (M) by (a, m) (co) =co and (a, m) (x) =a + mx, for x E F.

56

§ 11. KT-Fields and Sharply 3-transitive Groups

From (6.1) we know that the set Goo = {, then by (6.5), J1 E 1 Now KT2 implies tWJ1 (x) = 1- a(1- x) = a(1 + a(-x)) = a(1- a(x)) = apa(x), for all x EM, and hence (ap) 3 = id. 00

•

Therefore, a satisfies the conditions of Theorem (10.21) and we have, for 13(F) =Goo U GooaGoo:

(11.1) The set 13(F) forms a group (the group of generalized fractional affine transformations on the KT-field (F, +, ·,a)), which operates sharply ]-transitively on the elements ofF U {oo}. The name "generalized fractional affine transformations" derives from the fact that if Tj(F) is pappian, then (F, +, ·)is a field, a (x) = x-\ for all x E F*, and

T3 (F) is the group of fractional afftne transformations X bm =I= 0, on F U {oo} (see Theorem (13.2)).

-+

ab::' where an -

Consider now the converse situation. Suppose G is a sharply 3-transitive group operating on a set M. Select a point oo EM and let Moo = M\ {oo}. Then by (10.1), Goo operates sharply 2-transitively on. Moo and, for any two distinct points 0,1 E Moo we may construct the associated near-domain (MQ, 1, +, ·) according to (6.2). Then the groups Goo and T2 (Moo) as permutation groups on Moo are identical. Now~let r E 1t such that r (1) = 1 and r (0) = ""· For every x EM\ (oo, 0} deftne x· E Goo,O ·by x· (1) = x (see also § 6). Then fx' E Goo,O and fx' (1) = r(x), and hence fx' = r(x)'. Therefore, r(x · y) = rx· (y) = fx'r(y) = r(x)'r(y) = r(x) · r(y), for all x, y E M\{oo, 0}. Thus a E Aut (M"" *, ·), where a = r I M\ {oo, 0}. Also for J1 E 1.,;. such that J1 (0) = 1 we have by (10.4), (rp) 3 = id, and hence J1TJ1 = rpr. But 111 Moo= (1, -1), and we obtain the functional equation a(1 + a(x)) = 1- a(1 + x), for all x EM""\ {0, -1}. Thus (MQ,l, +, ·,a) is a KT-fteld and T3(MQ,1) =Goo U Goo r Goo= G and we have: ( 11.2) Let G be a permutation group which operates sharply ]-transitively on a set M For any triple oo, 0, 1 of distinct elements of M, determine the near· domain (MQ, 1 , +, ·)associated with G Further, let r E 1t where r (1) = 1 and r (0) = oo and define a = r I M\{oo, 0}. Then (M01 , +, ·, a) is KT-field and the ______ permuJation groups G and T3 (Mo~1 ) are identical 00

•

§ 11. KT-Fields and Sharply 3-transitive Groups

57

We shall call two KT-fields (F, +, .,·a) and (F', +', o, a') isomorphic, if there exists a bijection I{) : F -+ F' which is an isomorphism of the near-domains (F, +, ·)and (F', +', o) and which satisfies the condition l{)al{)- 1 =a', for I{) and 'P- 1 restricted to F* and F'* respeCtively. One readily verifies the equation I{)T3 (F)I{)- 1 T3 (F') where lf'(oo) oo', oo E!:F, oo' E!:F' and with the aid of (6.3) (see also [27]) one obtains:

=

=

Theorem (11.3) Let G and G' be permutation groups which operate sharply

]-transitively on sets M and M' respectively. Further, let (Mo\, +, ·, a) and (M0":''1,, +', o, a') be the associated KT-fields with respect to distinct points oo, 0,1 EM and oo', 0', 1' EM' according to (11.2). Then (G, M)and(G',M') are isomorphic as permutation groups if and only if the KT-fields (M0\, +,·,a) and (MQ-.' 1', +', o, a') are isomorphic. · '

·In particular, if (G, M) and (G', M') are identical, we see by Theorem (11.3) 'that the KT-fields associated with (G, M) are all isomorphic and we may speak of the KT-field (F, +, ·, a) associated with (G, M). For (F, +, ·, a) we have G0 ~ T2 (F) and Ga,b ~ (F*, ·),for a, b EM, a =I= b. Now let (F, +, ·, a) be a KT-field and define:

R ={xa(x- 1 ): x E F*}, and S = {x E F*: a(x) = x- 1}. (11.4) Let G be a sharply ]-transitive group and (F, +, ·,a) the associated KTfield. Then we may assume G = T3 (F) and we have: G ={
rm?co,O = {(O,r): r E R}

rs-:,o = {
1

= f{
1

ForK("")= {(b, (b- c)a (-c +b)- 1)r(c,· -1): b, c E F, b =I= c} we have: K = .

K(""), if G is of type (0, 0) or (0, 1), K("") U {(a, 1): a E F*}, if G is of type (1, 1), and K(co) U {(a, axa- 1): a, x E F*, and 1 + (x + x 2 ) = 0} U {
I

58

§ 12. Theorems on KT·fields

Proof: The expressions for G, r&i'.. ,o. r Z,o• 11, and J follow directly from the definitions and propositions (6.5), (10.13), and (10.17). ·

One easily verifies for a =(b, (b- c) a (-c + bt 1>r (c, -I> E K(=): a(oo) = b, a(b) = c, and a(c) = oo, Thus K("") is the set of elements inK which do not fix oo, If the elements of K have fixed points, then K n G,. =I= f/1. If G is of type (1,1), then (K n G,.) U {id} = J~ is a subgroup of G by theorem (10.8) and hence K =K(=) U {: a E F*}, by Theorem (3.7) (i) (vii) and (6.5). If G is of type (2, 0) or (2, 1), then by (10.10) there are exactly two elements a, ~ E K n G= such that a (O) =~ (O) =1. Now a =(1, t), for some t E F*. But a 3 = (1 + (t + t 2 ), dl,t+tz td1, 1 t 2 ) =(0, 1) => 1 + (t + t 2 ) =0 and d1, 1 t- 3 • Now 1 + (t + t 2 ) =o => r 2 + (t- 1 + 1) = o => r 1 + 1 = -r 2 => 1 =-(t- 1 + r 2 ) => 1 + (t- 1 + r 2 ) = 0. Also d1, 1 =r 3 => dt, 1 = t 3 => r 1 d1, 1 t = t 3 => d1,r• = t 3 • Therefore,~= (1, r 1 >. Now (O,a)a(O,a- 1 ) =(a, ata- 1 ) and (0, a)~(o,a- 1 ) = (a, ar 1 a- 1 ), for a E F* and we see that (a, ata- 1 ) and (a, ar 1 a- 1 ) are the two distinct elements of K n G"" which map 0 onto a. Now by (1 0.1 0), and also Theorem (10.14), there exist exactly two elements v and v2 in F* of order 3, and hence (0, v) and (0, v2 ) are the two elements of K n G"" which fix 0. Thus for x E {t, t- 1} andy E {v, v2 } we have the expression forK.

§ 12. Theorems on KT-fields Here we present some theorems on KT-fields and we shall also introduce a class of KT-fields which includes all examples of these structures known to the author. (12.1) Let (F, +,·,a) be a..KT-field and (n, +,·)the prime field of(F, +, ·), then a(n) = n- 1, fer every n En*, and. hence (n, +,·,a i1r) is a sub KT-field (see [27]).. • " · . Obviously we have: (12.2) Every field forms a KT-field with a(x)

=x-

1 ,

for all x E F*.

The next theorem is due to Benz-Elliger [S] and it can be stated here in our terminology as follows: Theorem (12.3) Let (F, +, ·, a) be a KT-field, then the following conditions are equivalent:

(i)

(F, +, ·) is a field

(ii)

(F, +, ·) is a skewfield

(iii)

a(x) = x- 1, for all x E F*.

§ 12.' Theorems on KT-fields

59

The implication (i) => (iii) is relevant to Theorem (10.19) and we present a proof of this here. Suppose (F, +, ·) is a commutative field and S ={x E F*: xa(x) =1}. Now xya(xy) = xa(x)ya(y) = 1, for all x, yES, and for any z E F*, a(za(z))= a (z)z =za (z), and hence za (z) = ±I. This implies that Sis a subgroup ofF* with [F*: S].;;:;; 2. If x, yES and x + y =I= 0, then xa(x + y) = xa(x)a(I + x- 1y) = 1- a (I+ xy- 1 ) = 1- a(y- 1 )a(y + x) = 1- ya(x + y), and hence (x + y) a (x + y) = 1. Thus S U {0} forms a sub field of (F, +, ·) which, together with [F*: S].;;:;; 2 implies S = F*. In the following theorem we show that the near-domains associated with KTfields are in fact near-fields for an infinite number of characteristics (see also [24 ]). Theorem (12.4) Let (F, +, ·, a) be a KT-field. If char F

is -a near-field.

=1 mod 3, then (F, +, ·)

Proof: If char F =p and p = 1 mod 3, then In* I = p- 1 and 3lln* I, where n is the prime field of F. Thus n* contains a subgroup C3 of order 3. By (I 1.1) and Theorem (10.14) we have C3
a -)- a'P Theorem 2.1 of [27] states: (12.5) Let (F, +, ·,*)be a dickson near-field and let a be the mapping x-)- x- 1~

Then (F, +, ·,a) is a KT-field if and only if (F, +, *)is a commutative field and a'P = (a- 11)'P, for all a E F*.

Now suppose (F, +,*)is a skewfield and.lfJ:F-)-Aut(F, +,*)is a mapping, a -)- a'P

then define a· b =a * a'P(b), a =I= 0, and 0 · b = 0. One readily shows that (F, +, ·, *) is a dickson near-field if and only if (x * x'P (y))'P = x'Py'P, for all x,y E F*, i.e. '{J is a homomorphism from (F*, ·)into Aut(F, +, *),(see [19]). The group r = '{J(F*) is called the D-group of (F, +, ·, *). In many examples

60

§ 12. Theorems on KT-fields

of dickson near-fields op is also a homomorphism with respect to the field multi· plication ( *). Let us call such near-fields strongly coupled From ( 12. 5) we have (12.6) Let (F, +, ·, *) be a strongly coupled dickson near-field and let a : x -+ x- 1, and suppose (F, +, *)is a commutative field. Further, denote the group of squares of (F*, *) by Q and let U == Kern op ={x E F* : x

(i)

{F, +, ·, a) is a KT-field

(ii)

The D-group

r (iii)

is an (abelian) group of involutions. In particular, if n is finite, then r =X C2 =elementary abelian group of order 2n.

r

Q < u.

We may now prove Theorem (12.7) Let (F, +, *)be a commutative field and let a: x-+ x- 11.

Suppose A< (F*,. *)such that: (i)

Q
(ii)

There exists a monomorphism n: F/.4-+ Aut(F, +, *).Let f= 1T

(iii)

r (x) E xA, for all r E rand all x E F*.

(F~)-

De,[ine 0 · b =0, and for a -=f:: 0, a· b a * a

field and (F, +, ·)is derived from (F, +,*)by the method described in theorem ( 12. 7). (12.6) and (12.7) allow us to make the following class distinction. For each natura1 number n define:

61

§ 12. Theorems on KT-fields

[KT, 2 11 ] - The class of all KT-fields ~onstructable by the method of theorem ( 12. 7) and which satisfy the condition I r I < 2 11 , i.e. [F*: A] < 2 11 • Further, define

[KT, oo] - The class of all KT-fields determined by ( 12. 7). By (12.2), the class [KT, 1] is essentially the class of all commutative fields, and by Zassenhaus' results [47], all finite KT-fields are contained in [KT, 2] (see also examples I, § 13). If there exists a KT-field (F, +,·,a) E [KT, 2 11 +1 ] such that [F*: A]= 2 11 +1 , then (F, +,·,a) El: [Kt, 2 11 ], since A is the kemel of a mapping 1fJ and, with the exception of the proper near-field with 9 elements, is as such a unique maximal normal abelian subgroup of (F*, ·)(see [45], 5.2, 5.6, 5.10). In§ 13 (examples IV) we show that for any n EN there exist KTfields in [KT, 2 11 ] which satisfy [F*: A]= 2 11 , and hence [KT, 2 11 ] ~ [KT, 2 11 +1], fGr all n EN. Further, we show that there exist KT-fields (F, +, ·, a) E [KT, oo], for which [F* : A] is infinite, and hence (F, +, ·,a) El: [KT, 2 n], for any n E N .

It is of interest to determine the sets R and S in KT-fields in [KT, oo]. We first note that 1 = x · x- 1 = x *X (x- 1 ) => x (x- 1 ) = x- 1, and x- 1 = x (x- 1t) = <(J <(J <(J (x

X E F*}, and X· a(x-

=X· a((x <(J (x)t 1t) =X· X<(J (x) =X* X<(J X<(J (x) = x * x = x\ and hence R = Q, which means R
)

Further, for any z E F* we have z·x2t·z-1 =z:tcz (x 2t*z- 1 )=z*Z (x 2t)*z (z- 1 )=z*(Z (x)) 2t*z- 1t= <(J <(J <(J <(J (z (x))l Thus R <J (F*, ·). <(J

Summarizing these results we have: R = Q, R


C S, and S = {s E F* : s E

F;; }. <(J

We now restrict our considerations to the class [KT, 2]. In general we know, that if (F, +, ·, a) is a KT-field and A is a subset ofF*, such that A c S, then by (10.18) (b) and (11.4), A is a subgroup of (F*, ·)if and only if a(A) =A and A is abelian. However, if (F*, ·)contains an abelian subgroup of index 2, then as is shown in [29] Satz 2.5, (F*, ·) contains an abelian subgroup A of index 2, such that A c S. In fact, if (F*, ·) is non-abelian, then except for the case of the proper near-field with nine elements, (F*, ·) contains at most one abelian subgroup ofindex 2.

62

§ 12. Theorems on KT-fields

We now introduce the class [KT,

[KT, 2]'

2]' of KT-fields as follows:

Then class of KT-fields (F, +, ·, a), such that (F*, ·) contains an abelian subgroup of index ,;; 2.

Obviously we have [KT, 2] C [KT, 2]'. Now define [ND, 2) to be the class of all near-domains F which contain an abelian multiplicative subgroup A of index,;; 2. If [F*: A]= 1, then F is a commutative field. Suppose (F, +, ·) E [ND, 2] with char F =I= 2, F* is non abelian, and A < F* such that A is abelian and [F*: A] =2. By theorem (8.26), (F, +, ·) is a dickson near-field derived from some commutative field. (F, +,*).Further, our assumptions imply Z(F*)
[F* :A]= 2. Thus we have: F=A U {cg +ch: c EA}U{O}={

i; Gt: Gt E A, n E

Accor;i:~

{1, 2}l. One can show, in fact, F ={a + b: a, b EA}. to Wahling [45) (5.1), (F, +, ·) is derived from a field (F, +, *) where x * y =ax+ bx,

for x, y E F, y =a+ b, with a, b E A, and x · y =x * x"' (y), where the mapping x :a+ b-+- xax- 1 + xbx- 1 is an automorphism of (F, +, *),for every xEF*. Since A is abelian and [F*: A] :::: 2, we have:

"'

id, ifx EA x =

"'

{

r, ifx El: A

,wherer:a+b-+xax-1 +xbx- 1,forxEF*\A. .

Obviously r is well-defined. Also r 2 (a +b):= x 2ax-2 + x 2 bx- 2 =a+ b, for all a+ b E F, and hence r is aJI involution in Aut (F, +, *).Therefore by (12.7), (F, +, ·, a), where 6 (x) = x- 1, is a KT-tield in the class [KT, 2]. We define: . [ND, 2]' The class of all KT-fields which are either commutative fields or are derived from near-domains in [ND, 2] by the procedure duscribe~ above.

..

By the above result we have [ND, 2]' C [KT, 2) for characteristic =I= 2. Now suppose (F, +,·,a) E [KT, 2]', then (F, +, ·) E [ND, 2). Assume char F =I= 2, F is not a field, and let A be the abelian subgroup of (F*, ·) of index 2 with A C S. Further, let (F, +, *)be the field and (F, +, ·, a') the KT-field obtained from (F, +, ·)by the procedure described above. Then x * y =ax + bx, for x, y E Fand y =a+ b, with a, b EA, and also a' (y) y- 11, for y E F*. Now by KT2 we have: aa(a +b)= aa(a)a(1 + a- 1 b)= 1- a(1 + ab- 1 ) = 1- a(b- 1 )a(b +a)= 1 ba(a +b), and hence aa(a +b)+ ba(a +b)= 1. Thus

63

§ 12. Theorems on KT-fields

a(y) * y = 1, for ally E F, and hence a= a'. This implies [KT, 2]' C [ND, 2]' for characteristic =F 2. In fact we have also proved that the following mapping is bijective:. [ND, 2] -r [ND, 2]', where (F, +,·,a) is the KT-field of (12.2), if(F, +,·)is a (F, +, ·) -4- (F, +, ·, a) field, or is derived from (F, +, ·) by the procedure described above, if(F, +, ·) is not a field.

It is easily shown that two near-fields in [ND, 2] are isomorphic if and only if their corresponding KT-fields in [ND, 2]' are isomorphic (see [29]). Thus we have:

Theorem (12.9) Consider only near-domains of characteristic =F 2. Then we have: [KT, 2]

= [KT,

2]'

= [ND,

2] I '

and the mapping [ND, 2] -r [ND, 2]' is bijective. Further, (F, +, ·) ~(F', +', ·')
Several examples of KT-fields in the klass [KT, 2] are given in [27]. Some of these examples are also discussed in § 13.

§ 13. Further Theorems on Sharply 3-transitive Groups We now use the results of the last two sections to derive further theorems on sharply 3-transitive groups.

Theorem (13.1) Let G be a sharply ]-transitive group and (F, +, ·, a) its associated KT-field. The sets form subgroups of G if and only if (F, +, ·) is a near-field.

1;

Proof: The theorem follows from Theorem (7.1) and the fact that G0 for a EM.

~

T2 (F),

Theorem (13.2) A sharply 2-transitive d_esarquesian group G' is imbeddable in a sharply ]-transitive group G in the sense of Theorem ( 10.21 ), if and only if G' is pappian. In this case, if (F, +, ·, a) is the KT-field associated with G, then (F, +, ·) is a commutative field, a (x) =x-I, for all x E F* and G is isomorphic as a permutation group to the one-dimensional projective group PGL (2, F) on the projective line over F.

64

§ 13. Further Theorems on Sharply 3-transitive Groups

==

Proof: We have G' G0 T2 (F) and the theorem now follows from Theorem (12.3). One readily verifies in this case that T3 (F) is the group of transformations a +mx of the form x -? b + nx' an- bm =I= 0. Note that, forTE Jf with T IF* = a we have T ~. 0 ={13 E Goo,O: f{3 ={3- 1}. But fx" = r(x)" = a(x)" =(x- 1 )" = x· -1, for all x E F*, where x· E G 0 such that x· (1) =X. Thus T Y'oo,O = Goo,O whenever G is pappian as was st;ted earlier in Theorem (10.19).

Theorem (13.3) Let G be a sharply ]-transitive group and (F, +, ·, a) its associated KT-field. Then Yo b = N(G0 b)\ G0 b if and only if (F, +, ·) is a commu-

Ya, b = N (G

tative field, and flA! a, b = element ofF is a square.

0

,

b~ \

G b if and only if every non-zero 0,

Proof: By (11.4) we have T flA!oo,O = {<0, r): r E R}, where T E 11 such that r(O) = oo and riF* =a, and R ={xa(x- 1): x EF*}. Now rflA!oo,O C T~,o and the theorem follows from theorem (10.19), (10.20), and theorem (13.2). From .theorems (8. 7), (10. 7), (1 0.14), and (13.1) we have:

Theorem (13.4) Let G be sharply ]-transitive and let (F, +, ·, a) be the KT-. field associated with G. Then G is of type {1,1) if and only if (F, +,·)is a nearfield with char F = 3, and G is of tpye {2,0) or (2,1) if and only if (F*, ·) contains a subgroup or order 3.

Theorem (13.5) If G is a sharply ]-transitive group and char G = 1 mod 3, then G is of type (2,1}, 1; < G, and (F, +, ·)is a near-field, where (F, +,·,a) is the associated KT-field.

Proof: If char G = 1 mod 3, then the multiplicative group of the prime field of (F, +, ·) contain~ a subgroup of order 3, and hence by theorem (13.4) G is of type (2,1). Type (2,0) is excluded since char G =I= 2. By theorem (12.4) and theorem "(13.1) we' have 1; < G and (F, +, ·) is a near-field. ,

=

=

Since p 1 mod 3 or p -1 mod 3 for every prime p quence of theorem (13.5) is:

> 3,

an obvious conse-

(13.6) If G is of type (0, n), n E {0,1}, then charGE {0,2} u {p: p a prime and p = -1 mod 3}. Now consider the class [S3T, 2] of sharply 3-transitive groups G which have the property that each complement G0 , b contains an abelian subgroup A0 , b such that [G0 b : A0 b] < 2. By § 11 this class is completely described by the KT-fields in the cla~s [KT, 2]' of § 12. Further, define [S2T, 2] to b~' the class of sharply 2-transitive groups whose complements contain abelian s~bgroup of jn~~2\-S_ 2. Then by theorem (12.9) and § 11 we have: .

§ 13. Further Theorems on Sharply 3-transitive Groups

65

Theorem (13.7) (a) If G E [S3T, 2]; then Ga E [S2T, 2]. (b) If G' is a sharply 2-transitive group in [S2T, 2] of characteristic =I= 2, then G' can be imbedded in the sense of theorem (10.21) up to isomorphism into exactly one sharply ]~transitive group G. FUrther, G is in the class [S3T, 2], and two groups in [S2T, 2] are isomorphic as permutation groups if and only if their corresponding groups in [S3T, 2] are isomorphic as permutation groups. By identifying isomorphic groups we may assert that the mapping [S2T, 2] --? [S3T, 2] is bijective. G' --? G We now present examples of non-pappian sharply 3-transitive groups in [S3T, 2] which we obtain by constructing proper KT-fields according to (12.7). By (12.7) we obtain such KT-fields from commutative fields (F, +,*)which contain subgroups A < F* such that [F* :A] = 2 and for which there exists r E Aut(F, +, *), T =I= id, r 2 = id, such that r(A) =A, then r= {id, r}. Further~ from § 12 we know that S =A U F7 , where F7 denotes the field of fixed points of r, R = Q =group of squares of (F*, *), and R 2 and letT be the automorphism x --? xPn. Further, let A be the group of squares in GF (p 211 ). Define a multiplication(·) on GF(p 2n) according to (12.7). Then (GF(p 2n), +,·,a) where a(x) =x- 11is aKT-field. According to Zassenhaus [46], all finiteKTfields which are not fields are obtained in this manner. We note that I GF(p 2 n)* I= p 2 n- 1 = (pn + 1)(pn -1) and hence 311 GF(p 2 n)* I, if p > 3. Thus every finite KT-field which is not a field either has characteristic 3, or it contains a multiplicative subgroup of order 3. Consequently, according to § 11 and Theorem (13.4), every proper finite KT-field describes a sharply 3-transitive group of type (1,1) or (2,1). Therefore, all finite sharply 3-transitive groups of type (0, 1), (0,0) or (2,0) are pappian. II. Let F = Q(i) where Q denotes the field of rational numbers and i = yCI. Further, let T be the automorphism T: a + ib --?a- ib. For any fixed prime p define A= {a+ ib E Q (i): a 2 + b 2 = 2e, 3e 3 ••• peP ... , where ep = 0 mod 2}, where the product indicates the decomposition of the rational number a 2 + b2

66

§ 13. Further Theorems on Sharply 3-transitive Groups

in prime powers, positive, negative, or zero. These examples were presented · as examples of infinite near-fields by Karzel in [19]. We note that these KT-fields describe sharply 3-transitive groups of type (0, 1). Since char F = 0 El: {2,3}, we have only to show by Theorem (13.4) that (F*, ·) does not contain a subgroup or order 3. Now if x E A, then x 3 = x 31 =I= 1 since iQ (i) does not contain the cube roots of unity. If x El: A, then x 3 E A • x. III. Let (K, +, *) be a field and (K (t ), +, *) the field of rational functions in the indeterminate t with coefficients in K. Each element of K (t) has the form

f = / 1 * fi 11 =~where / 1 and f2 are polynomials in t with coefficients in K. Now for f = f 1 * fi 11 define B (f)= deg / 1 - deg/2, and

A ={/E K(t)*: B(f)

= 0 mod 2},

where deg denotes degree. If T E Aut (K (t), +, ·) is an involution and B (r (f)) = B (f), for all f E K (t), then A and T satisfy the conditions of ( 12. 7). In particular, if we take

where a E Aut(K, +, *)and a is an involution, then forcE K with a(c) = c, the element ct is in the fixed point field K (t) T ofT but ct El: A. . Therefore, the setS = {! E K (t)* : fa (f) = 1} =A U K (t)r lies properly between A and K(t)*. Furthermore, forb E K with a(b) =I=± b, we have a(b 2) =I= b 2 and (bt) · (ct) · (btt 1 = bt* T (ct * T (T (btt 11))·= bt * 0! (c) ta (b) C 11 = cb ( cb • ) - ca (b) cb cb a(b) t. ButT Ct.{b) t - -b- t =I= a(b) t Therefore, a(b) tEl: Au K(t)T, and hence S is not riomial in (K (t)*, ·). This means, in terms of the associated sharply 3-transitive group, that the sets rSa, b. are not necessarily sugroups nor are they necessarily normal in the complement Ga, b (see § 10). We also note by taking T : t-+ -t + 1 that (K (t), +, ·, a) is a proper KT-field for any base field K. Therefore, there exist infinite non-pappian sharply 3transitive groups in [S3T, 2] of characteristic p, for p = 0 and for every prime p. In conclusion we show, as previously mentioned, that for each n E N there exist KT-fields (F, +, ·, a) (of arbitrary characteristic) with the property [F*: A]= 2 11 and that there exist KT-fields such that [F*: A] is infin,ite (see (12.7)). In terms of the associated sharply 3-transitive groups these examples show that

67

§ 13. Further Theorems on Sharply 3-transitive Groups

the class of groups derived from the KT-fields of [KT, oo] is much larger than [S3T, 2]. The following class of examples obviously contains the examples of III. IV. Let I be an arbitrary index set (finite or infinite). If I is finite and III= n we take I= {1, 2, ... , n}. Further, let K be a commutative field and F = K (ti), i E I the field of rational functions in II I transcendental indeterminates ti, i E I. Also let ri E Aut (F, +, *), i E I, such that ri =I= id, rf = id and TiTi = T/i• for all i, j El. We define r = h: i E I}= {IT 7fi: i E I, ni EN, and 1{'4: ni = 1 mod 2}1 is finite}. Now let B: F*-+ (Z, +)(direct sum) and let Bi> i E I, be the component functions of B. Note that the direct sum means that for each f E F*, Bi (f) = 0, for all but a finite number of indicies i E l

r

Further assume:

B = Br, for all r E

• (i)

and

r,

B (f *g) = B (f) + B (g).

(ii)

Now define <.p: F*-+

f

r,

where/ =IT r 1fliif>, iEI, andf-g = f * f (g), for ~

-+[~

~

[=I= 0, and 0 · g = 0. Since Bi(f· g)= Bi(f *!~(g))= Bi(f) + BJ~(g) =

B1• (f) + Bi (g) = B1• (f * g), for all i E I, we have f ~ g~ =IT r~i(f) nrfli(g) = I I IT rfli(f)+Bj(g) =IT rfliif>~
I

I


'

~

~

Thus (F, +, ·,*)is a strongly coupled dickson near-field and by (12.6) or (12.7), (F, +, ·, a) is a KT-field, where a Ex-+ x- 1,, We note that A = Kern <.p ={/E F*: Bi(f) = 0 mod 2, for all i EI}, and [F*: A]= 21 1 1 if I is finite, and [F* : A] = II I, if I is infinite . In particular we may choose ri E Aut(F, +,*)such that ri(c) = c, for all c'EK,

ri(~) = ~ forj =l=i,

and ri(ti)=·-ti+ 1 andB·(f)=B· f 1 (ti) = deg .[1 '

I

lf2(fj)

I

-

degi/2 , where degi ft, I = 1; 2, denotes the degree of the polynomial [z with respect to ti.

BIBLIOGRAPHY [ 1) [ 2) [ 3) ( 4] [ 5) [ 6] [ 7] [ 8] [ 9] [10]

(11] [12] [13] [ 14] [ 15] [16] [17) [18] [19] [20] [21)

[22) [23]

Andre, J.: Projektive Ebenen iiber Fastkorpern, Math. Zeitschr. 62 (1955) 137-160. Andre, J.: Ober eine Bcziehung zwischen Zentrum und Kern endlicher Fastkorper, Arch. Math. 14 (1963) 145-146. Baer, R.: Finiteness properties of groups, Duke Math. S. 15 (1948) 1021-32. Benz, W.: Permutations and plane sections of a nded quadric, lstituto Nazionale di Alta Mat. Sym. Math. Vol. V (1970) 325-339. Benz, W. and Elliger, S.: Ober die Funktionalgleichung /(1 + x) + /(1 + f(x)) = 1, Aequationes Math. Vol. 1 (1968) 267-274. Cohn, P.M.: On the em beddings of rings in skew fields, Proced. London Math. Soc. 11 (1961) 511-530. Davis, E. 1-I.: Incidence systems associated with non-planar near-fields, Canadian J. of Math. 22 (1970) 939-952. Dickson, L. E.: On finite algebras, Nachrichten der Kg!. Ges. d. Wiss. z. Gottingen Math.-phys. Klasse (1905) 358-393. Ellers, E. and Karzel, H.: Endliche Inzidenzgrnppen, Abh. Math. Sem. Univ. Ham· burg 27 (1964) 250-264. Freudenthal, H.: La structure des groupes a deux bouts et des groupes triplement transitifs, Nederl. Akad. Wetensch. Proc. SerA. 54 lndagtiones Math. 13 (1951) 288-294. Graves, J. A.: Near domains, Doctoral dissertation, Texas A and M University 1971. Gratzer, G.: A theorem on doubly transitive permutation groups with application to universal algebras, Fund. Math. 53 (1963) 25-41. Hall, M. Jr.: On a theorem of Jordan, Pacific J. Math. 4 (1954) 219-226. Hall, M. Jr.: The theory of groups, Macmillan, New York 1959. Huppert, B.: Scharf dreifach transitive Permutationsgruppen, Arch. Math. 13 (1962) 61-72. Jonnson, W. J.: Doubly transitive groups, near-fields, and geometry, Arch. Math.17 (1966) 83-88. Jordan, C.: Recherches sur les substitutions, J. Math. Pures Appl. (2) 17 (1872) 351-363. .. . Kalscheu"er, F.:· Die 'Bestimmung alter stet~gen Fastkorper i.ibcr dcm Korper der reellen Zahlen als Grundkorper, Abh. Math. Sem. Univ. Hamburg 13 (1940) 413-435. Karzel, H.: Unendliche Dicksonsche Fastkorper, Arch. Math. 16 (1965) 247-256. Karzel, H.: Inzidenzgruppen I, lecture notes by I. Pieper and K. Sorensen, Univ. Hamburg (1965) 123-135. Karzel, H.: Zusammenhange zwischen Fastbereichen, scharf 2-fach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom, Abh. Math. Sem. Univ. Hamburg 32 (1968) 191-206. Kerby, W.: Projektive und nicht·projektive Fastkorper, Abh. Math. Sem. Univ. Ham· burg 32 (1968) 20-24. Kerby, W.: Angeordnete Fastkorper;--AlJh. Math. Sem. Univ. Hamburg 32 (1968) 135-146.

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[24] . Kerby, W.: Sharply 3-transitive Groups ofCharacteristic 1 mod 3, J. of Algebra 32 (1974}. [25] Kerby, W. and Wefelschei~, H.: Bemerkungen iiber Fastbereiche und scharf 2· fach transitive Gruppen, Abh. Math. Serri. Univ. Hamburg 37 (1972} 23-32. [26] Kerby, W. and Wefelscheid, H.: Conditions of finiteness on sharply 2-transitive groups, Aequationes Math. Vol. 8 (1972) 287-290. [27] Kerby, W. and Wefelscheid, H.: Ober cine scharf 3-fach transitiv~n Gruppen zu· geordnete algebraische Struktur, Abh. Math. Sem. Univ. Hamburg 37 (1972) 225235. [28] Kerby, W. and Wefelscheid, H.: The maximal sub near-field of a near-domain, J. of Algebra 28 (1974} 319-325. [29) Kerby, W. and Wefelscheid, H.: Ober eine Klasse von scharf 3-fach transitiven Gruppen, Crelles Journal 264 (1974). [30] Kerekj arto, B. De: Surles groupes transitifs de Ia droite, Acta Univ. szeged Sect. Sci. Math. v. 10 (1941). [31) Landau, E.: Vorlesungen iiber Zahlentheorie, Chelsea, New York 1950. )32] Ligh, S. Mcquarrie, B. and Slotterbeck, 0.: On near-fields, J. London Math. Soc. 5 (1972) 87-90. [33] Neumann, B. H.: On the commutativity of addition, J. London Math. Soc. 15 (1940) 203-208. [34] Neumann, B. H.: Groups with finite classes of subgroups, Proc. London Math. Soc. 1 (1951) 178-187. [35] Passman, D.: Permutation groups, Mathematics lecture note series Benjamin, New York 1968. [36] P e tka n tsch in, B.: Axiomatischer Aufbau der zweidimensionalen Mobius'schen Geometric, Annuaire Univ. Sofia, Fac. Phys.-Math. 36 (1940) 219-325. [37] Pokropp, F.: Dicksonsche Fastkorper, Abh. Math. Sem. Univ. Hamburg 30 (1967) 188-219. [38] Scott, W. R.: Group Theory, Prentice-Hall, Englewood Cliffs, N.J. 1964. [39] Tits, L: Gereralisations des groupes projectifs, Akad. Roy. Belgique, Cl. Sci. Mem. Coli. 5e Ser. 35 (1949) 197-208,224-233,568-589, 756-773. [4p] Tits, J.: Groupes triplement transitifs et generalisations. Algebre et Theorie des Nombres, Coli. Int. du Centre Nat. de Ia Rech. Sci. no. 24 (1950) 207-208. [41] Tits, J.: Surles groupes triplement transitifs continus; generalisation d'un theoreme de Kerekjarto, Compositio Math. 9 (1951) 85-96. (42] Tits, J.: Surles groupes doublement transitifs continus, Comm. Math. Helv. 26 (1952) 203-224. Corrections et comlement in Comm. Math. Helv. 30 (1955) 234-240. [43]

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INDEX OF NOTATION id

<
\ F* U"'

J

Identity mapping Is a subgroup of Is a normal subgroup of Isomorphic to Set difference

F\{0}

U\{id}

J u {id}

~

IMI IM

Cardinality of M Function a restricted toM

fl~

llall- 1

Factor group Index of H in G Commutator xyx- 1y-• S Subgroup generated by set S Sym (M) Symmetric group on set M Sym (n) Symmetric group of degree n Alt (M) Alternating group on set M Alt (n) Alternating group of degree n Aut (G) Automorphism group Inn (G) Group of inner automorphisms (t (S) Centralizer of set S N (S) Normalizer of set S Z (G) Center of group G Let G be a sharply 2-transitive group operating on M Ga Complement, Ga ={a E G: ~(a) a}, a EM P Frobenius kernel, P ={a E G: a:(x) if' x, for all x EM} u {id} ra b Kernel, ra b h E Ga: +r~. b b .Y}, a, bE M, a* b (seep. 18). Let F be a near-domain E = {x E F: x + 1 = 1 + x} G!H

[G : H] [x, y]

T:

Hamburger Mathematische Einzelschriften Neue Folge Herausgegeben vom Mathematischen Seminar der Universittit Hamburg.

1

Peter Roquette Analytic theory of elliptic functions over local fields 1970. 90 Seiten, kartoniert

2

Reinhold Baer Gruppen mit abzahlbaren Automorphismengruppen Friedrich Bachmann zurn 60. Geburtstag gewidmet. 1970. 122 Seiten, kartoniert

3

Wilhelm Junkers Mehrwertige Ordnungsfunktionen 1971. 90 Seiten, kartoniert

4

Hans Joachim Arnold Die Geometric der Ringe im Rahmen allgemeiner analytischer Losungen 1971. 86 Seift)n, kartoniert

5

.

Hanno Rund Invariant theory of variational problems on subspaces of a Riemannian manifold 1971. 55 Seiten, kartoniert

Vandenhoeck & Ruprecht Gi:ittingen und Zurich

1111111