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contains a minimal member BNIN, say, where B E d. Let g = (A E d ;AN = BN}. Since N E Min, the set {A n N; A E g}has a minimal member C n N, say, with C E 97. Clearly, C E d .Let A be an element of d such that A G C. By the choice of B, one has AN = CN = BN, and hence A E 39. Thus, by the choice of C one has A n N = C n N. Therefore, by Dedekind‘s modular law, C = AN n C = A ( C n N ) = A, and consequently, C is a minimal member of d.1 Clearly, the infinite cyclic group does not satisfy rnin, and so l.E.l entails the following observation.
1.E.2 Lemma. Every group satisfying rnin is periodic. By considering an infinite, elementary abelian p-group, one realises that Min # LMin (otherwise, of course, the class L s of all locally finite groups would be contained in Mm). But fortunately, a slightly restricted local theorem still holds.
1.E.3 Lemma. If every countable subgroup of the group G satisjes min, then G E Min.
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Proof. If this were not so, then in the group G there would be an infinite, strictly descending chain {Gi}, Gi 3 G i + l , of subgroups. For each natural number i choose an element gi E G,\G,+ . Put H = ( g i , i E N). This subgroup H is countable and therefore satisfies min by assumption. But by construction one has a strictly descending chain { H n Gi} of subgroups of H. This contradiction shows that the assumption, that such a chain {G,} exists, cannot be maintained. I Examples of groups in Min are easily found: every finite group satisfies min. There is one further basic class of groups satisfying min, namely the class of Prufer groups. Let p be any prime. For each natural number i the cyclic group Cpi+lof order pi+’ contains a unique copy of the cyclic group Cpjof order pi.Choosing an embedding of Cpiinto Cpi+ I for each i we obtain a direct system. The direct limit of this system is called a Cp,-group or a Prufer group, after H. Prufer (1896-,1934). It is easily seen that such a group is uniquely determined up to isomorphism by the prime p, that is, it is independent of the choice of the embeddings. Alternative definitions of a Prufer p-group describe it as an infinite, locally cyclicp-group (this is essentially the above definition) or as the p-primary component of the factor group Q/Z; the exponential map shows that the group Q/Z is isomorphic to the group of complex roots of unity. Every known group satisfying min has a finite sequence of subgroups
,
(1)
=
.
G o E G, G . . E G,
=
G with G i 4 G i + l for 0 S i < n
such that each factor Gi+i / G i either is finite or a Prufer p-group for some prime p. Notice that according to 1.A.2 all these examples are locally finite groups. We shall see below that every such group is even abelian-by-finite, being an extension of a direct product of finitely many Prufer groups by a finite group. Do these examples exhaust the class Min? Question 1.5. Is every group G E Min abelian-by-finite (and hence, in particular, locally finite)? A. Tarski is said to have suggested that there might exist an infinite group G such that every proper subgroup of G has prime order. Such a group obviously would have to be simple, and it would satisfy min as well as the maximal condition for subgroups (every chain of subgroups in fact, consists of three terms at most). Since any two involutions would generate a dihedral subgroup, it is clear that such a group could not contain involutions. The existence of such “Tarski monsters” G is still an enigma, and it seems hard to imagine how one might tackle this problem, even in the case in which the group G has prime exponent.
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THE MINIMAL CONDITION ON SUBGROUPS
31
In view of Cernikov's prolonged concern with Question 1.5 and his partial solutions to this question, we shall call a group G a C'ernikou group if it is abelian-by-finite and satisfies min. (This slightly extends the use of the name in KuroS [I], 9 64, where only primary groups with this property are called Cernikov groups.) For a wide class of groups an affirmative answer to Question 1.5 is given in 1.E.5. That locally finite groups satisfying min are Cernikov groups has been shown in Sunkov [12] and in Kegel and Wehrfritz [I 1. We shall obtain this result in a considerably more general context in Chapter 5 (see 5.8 or 5.10).
1.E.4 Proposition. If the set of all normal subgroups of finite index of the group G satisfies the descending chain condition, then the intersection JG of all subgroups offinite index in G has itselffinite index in G. Proof. If H is a subgroup of finite index in the group G, then the intersection H galso has finite index in G. Thus the intersection JG of all subgroups of finite index in G coincides with the intersection of all normal subgroups of finite index in G . By the descending chain condition on normal subgroups of finite index G has a minimal normal subgroup of finite index. Let Kand L be two such minimal normal subgroups of finite index in G, then the intersection K n L is also of finite index in G. Thus K = K n L = L, and JG is of finite index in G. I Now we shall discuss locally soluble and locally nilpotent groups satisfying min. The following statement is an immediate consequence of 1.B.4 and 1.B.8.
nsEti
1.E.5 Lemma. Let G be a group such that the set of all normal subgroups If G is locally soluble, then it is hyperabelian; i f G is locally nilpotent, then it is hypercentral. of G satisJies the descending chain condition.
After these formalities we come to the first (and for almost twenty years only) general result pointing in the direction indicated by Question 1.5. It appears in Cernikov [14].
1.E.6 Theorem. If in the locally soluble group G the set of subnormal subgroups satisjies the descending chain condition, then G is a Cernikov group; in particular G is soluble. Proof. By 1.E.4 (and l.E.l) we may assume that G is infinite and does not contain any proper subgroup of finite index. If H is any normal subgroup of G then the commutator quotient group HIH' satisfies min. Hence it is periodic (1.E.2) and only finitely many of its primary components are nontrivial; also every elementary abelian subgroup of H/H' is finite, and so each
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primary component of the abelian group H/H’ has finite rank. Therefore HIH’ is the direct product of some finite abelian group and a finite number of Prufer groups. In particular, if H/H‘ # . Then one has: a The subgroup C is normal and of index 2 in G. b If the dihedral group G is non-abelian, then C i s a characteristic subgroup of G. c Every element of G \ C inverts every element of C, that is, g E G \ C implies C' = c- l . d Every element of G \ C is an involution. e The set G \ C is a single conjugacy class if and only if the order ICI is finite artd odd; it is the union of two conjugacy classes otherwise.
Proof. a From c
= xy
one has
cx = x - l c x = x(xy)x = y x = y - ' x - l
=
c-1 .
Similarly cy = c-'. Hence C = (c) is a cyclic normal subgroup of G = ( x , y). Clearly, C n ( x ) = (1) and C(x> = G, SO IG : CI = 2.1 b If G is non-abelian, then [CI > 2. Once we have estalished d, then the uniqueness of C is clear. I c Since G = C ( x ) , it follows that every element g E C G \ has the form g = c l x , c1 E C. Clearly one has cg = cx = c-'. I d The element g E G\C has the form g = clx. Hence g 2 = c l x c l x = c1(c1)-' = 1, and g is an involution. I e For any integer r one has by c (C2rX)=
and (C2'-1X)cr
=
=
C-rC2rXCr
c-rc2r-lxcr
=
=
Cr(XCrX)X
Cr-l(XCrX)X
=
x,
=c
- 1x
= y.
Since the set G C\ is invariant under the inner automorphisms of G it must consist of at most two classes of conjugate elements. It will consist of a single conjugacy class if and only if the involutions x and y are conjugate. An arbitrary conjugate of x is XXCs
=
XCs
=
c-2s X
Thus the involutions x and y are conjugate in G if and only if there exists a positive integer s such that cZs = c, that is, if and only if the order ICI of C is finite and odd. 1 Clearly the finite dihedral group whose cyclic normal subgroup C has order n can be defined by
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47
D, = ( x , y ; x2 = y 2 = (xY)” = 1). The group G is called IocaIIy dihedraI if every finite subset of G is contained in some dihedral subgroup of G. From 1.1.1 one obtains rather precise information about this class of groups. We only note the following result. 1.1.2 Proposition. The IocaIIy dihedral group G has a IocaIIy cyclic normaI subgroup of index 2, and every etement of G \ C is an invoIution that inverts every element of C . In view of 1.1.1 this needs proof only if the group G is not finitely generated.
Proof. If the group G has order larger than four, then there are non-involutory elements in G. Let C be the subgroup of G generated by these elements. If X is any finite subset of C , then there exists a finite set Y of noninvolutory elements of G such that X E ( Y ) . By our assumption on the order of G the locally dihedral group G contains at least two non-commuting involutions i andj, say. Let g be any element of G. Then there exists a dihedral subgroup D of G containing the subset { i , j , g } v Y. Denote the unique cyclic normal subgroup of D of index 2 by C,. By 1.1.1 one clearly has X E ( Y ) E ( d E D ; d 2 # 1) = C D E C. Since the subgroup C, is cyclic, the subgroup C is a locally cyclic normal subgroup of G. If i E C,, then ij = j i , which we had assumed not to be the case. Thus for the element g one has g E D = C D V ~ C D EC v i C .
Since g is an arbitrary element of G this implies that G = C u iC. The group G is non-abelian; thus IG : CI = 2. Also if x E X , then x E C,, and so xi = x-‘. Therefore i inverts every element of C , and hence every element of the coset iC = G\C inverts every element of the abelian subgroup C . Finally, if c E C, (ic)’ = (ici). = 1. Thus every element of iC is an involution. I Up to isomorphism there exists only one infinite locally cyclic 2-group namely the Prufer 2-group C Z m .Hence, up to isomorphism, there exists only one infinite locally dihedral 2-group, namely,
D,,
=
.
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$1
1.1.3 Lemma. The group DzoDcontains a single conjugacy class of non-central involutions.
Proof. In the above notation, D,,\C element d E C such that dZ = c. Then
=
iC. If c E C, then there exists an
id = j(id-'j)d = id2 = jc.
Thus all the elements of D2,\C are conjugate in D z m . The subgroup C being locally cyclic, contains only one involution, z say, and z is central in D Z msince z' = z-l = z. 1 The (generalized) quaternion group Q,,, is closely related to the dihedral group D,,.It is defined by
Qzn= (x, y ; x2
=
y 2 = (xy)'" = z, where z 2
=
1).
The group Q2 is the ordinary quaternion group of order eight. Clearly, the element z of the (generalized) quaternion group QZn is in the centre of Q2.. Thus one visibly has the isomorphism D,, N Q2./(z>.One easily shows that the element z is the only involution of the (generalized) quaternion Q,.. As in 1.1.1 one shows that the (generalized) quaternion group Q,, contains a normal cyclic subgroup of index 2, which is characteristic if n > I, and every element of Q",C \ is of order four and inverts every element of C. The group G is called locally quaternion if every finite subset of G is contained in some (generalized) quaternion subgroup of G. Clearly, for locally quaternion groups one has an analogue to 1.1.2. Since a locally quaternion group is, by definition, a 2-group, one finds that, up to isomorphism, there exists only one infinite locally quaternion group, namely
QZm= .
In Gorenstein [l] p. 193 a survey is given of all finitep-groups containing a cyclic normal subgroup of index p. For p = 2 besides the dihedral and (generalized) quaternion groups two more types occur. We shall comment only briefly on one of these types, namely the quasi-dihedral groups, since these groups play a special role in characterizations of certain finite and locally finite simple groups (see 4.20 and 4.22 or Alperin, Brauer and Gorenstein [I]). The quasi-dihedral group Q-D,n is defined by Q-D,, = ( x , y ; x2" = y 2 = 1, x y = x-'+'"-'
) for n 2 3.
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In Theorem 5.4.3 of Gorenstein [ I ] p. 191, some properties of the groups Q-D2. are given. In particular, the maximal subgroups of Q-D,,, are cyclic, (generalized) quaternion, or dihedral. Since the (generalized) quaternion and dihedral groups do not contain quasi-dihedral subgroups, one sees that nothing is gained by introducing the class of locally quasi-dihedral groups, since these groups must be quasi-dihedral and hence finite. Very sharp information has been obtained in Gorenstein and Walter [2] on the structure of finite groups with dihedral Sylow 2-subgroups. These results extend to locally finite groups (see 4.19). Brauer and Suzuki [I] show that a finite group cannot be non-abelian and simple if its Sylow 2-subgroups are (generalized) quaternion. This result has since been considerably extended by Glauberman [l]. Both these results extend to locally finite groups. Let us first show this for the more general result of Glauberman [l]. 1.1.4 Theorem. If the locally jinite group G contains an involution i such that every maximal 2-subgroup of G contains at most one conjugate of i, then the centre of the factor group G / O G contains the involution iOG, and thus is non-trivial. Proof. Since every 2-subgroup of G is contained in some maximal 2subgroup of G, the hypothesis of 1.1.4 implies (and is equivalent to) the fact that every 2-subgroup of G contains at most one conjugate of the involution i. If Z, denotes the local system of all finite subgroups of G containing the involution i, then the hypothesis of 1.1.4 holds in each SEZ. But then Glauberman’s Theorem yields that [S, i ] E 0s. By 1.B.9 and comments there is a subgroup T E C containing S such that S n OT = S n O G . Thus one obtains [S, i ] E S n [T, i] E S n OT G OG.
But this implies that [G, i ] G O G , as contended. I The extension of the theorem of Brauer and Suzuki [I] is an immediate corollary, for in a locally cyclic or in a locally quaternion group there is at most one involution. 1.1.5 Corollary. If every 2-subgroup of the locallyjinite group G is either locally cyclic or locally quaternion, then either G = OG or the centre of the factor group G / O G contains precisely one involution.
Remark. Since one does not know very much about the set of all maximal 2-subgroups of a locally finite group G (see Section D), it seems difficult to
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check the conditions of 1.1.4 in any given group. To lessen that task it would seem desirable to obtain, if possible, a similar result under weaker assumptions involving only certain maximal 2-subgroups. Section J. Frobenius groups
A Frobenius group (after G. Frobenius 1849-1917) is a group G that contains a subgroup H with G 2 H 2 (1) such that H n H g = (1) for every element g E G \ H . Such a subgroup H of the Frobenius group G is usually called a Frobenius complement of G , although in general the subgroup H need not complement any subgroup of G. If in the Frobenius group G with Frobenius complement H the subset K = (G\ U g E G H 9u) (1) is a subgroup, then the subgroup K is called the Frobenius kernel of G (with respect to H). If the Frobenius kernel K exists in the Frobenius group G (with respect to the Frobenius complement H ) , then one often has G = HK. It is this situation from which the names derive. The classical theorem of Frobenius asserts that for a finite Frobenius group G the set of Frobenius complements in G consists of a single conjugacy class of subgroups and that the Frobenius kernel of G (with respect to any of these subgroups) exists and is a characteristic subgroup of G. It is our aim in this section to extend these and some further properties of finite Frobenius groups to locally finite Frobenius groups. We thus obtain a rather satisfactory survey of the structural properties of locally finite Frobenius groups. These results essentially appear in Kegel [1] and Busarkin and Starostin [l]. For a discussion of finite Frobenius groups see Huppert [I], 6 V.8 or Gorenstein [I], 5 2.7, 0 4.5, and 5 10.3. The first result holds for Frobenius groups in general; it just gives a description of Frobenius groups in the language of permutation groups. It is in this context that one often encounters Frobenius groups. 1.J.l Proposition. The group G is a Frobenius group i f and only if it has a faithful, transitive permutation representation on a set 52 with 1521 > 1 such that every element g # 1 of G satisfies wg = w f o r at most one w E SZ, and i f furthermore there is a non-trivial element of G leaving an element of 52 Jixed. Proof. If G is a Frobenius group and H a Frobenius complement in G, then take as 52 the set of all conjugates of H in G with G operating via conjugation. From the definition of a Frobenius complement it is clear that G operates in the desired way on the set SZ. If on the other hand, the group G has such a permutation representation on a set 52, then choose any element
CH.
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FROBENIUS GROUPS
51
w E 52 and put H = G, = ( g E G ; og= o}.Since G acts transitively on Q one has H # ( l ) , and since no non-identity element of G has more than one fixed-point in 0,H has trivial intersection with every conjugate. Since 1521 # 1, the subgroup H i s not the whole of G. Thus G is a Frobenius group with Frobenius complement H. I If G is the free group on two generators x and y , and if H = ( [ x , y ] ) , then it is easy to check that for every g E G \ H the subgroup ( H , H g ) is not cyclic. Hence the group ( H , H g ) is a non-abelian free subgroup of G and its centre is trivial. But then, since the intersection H n Hg lies in the centre of ( H , H g ) , one has H n H g = ( 1 ) for every g E a H , and the group G is a Frobenius group with Frobenius complement H. Suppose K is any normal subgroup of G with HK = G . Since H is abelian, we have G' -c K. But H c G', and so K = G. Hence the subgroup H of G does not have any proper normal supplement in G. Further it is easy to check that every element of R = (UgeGHg)\(l) has length at least four, so x-'y-' and x y are elements of K = C\R whose product visibly does not lie in K. Thus H does not determine a Frobenius kernel of G either. For locally finite groups the situation is very much better. 1.5.2 Theorem. Let H be a Frobenius complement of the locally finite Frobenius group G. Then G contains a unique Frobenius kernel K , and K is a normal complement of H in G. The subgroup K is nilpotent, and there exists a set rc of primes such that G is rc-closed and K = 0, G. Any Frobenius complement of G is conjugate to H under an element of K and every abelian subgroup of H is locally cyclic. If S is any non-trivial subgroup of G such that K n S = (l), then ZS # (1) and S is conjugate to a subgroup of H.
Proof. If G is a finite group then the theorem is well known to be true. The uninitiated reader should consult Gorenstein [I], Q 10.3 or Huppert [ I 1, 0 V.8. The full theorem is deduced from the finite case. Put K = (G\UgEG H 9 ) u { 1) and let Z denote the set of finite subgroups F of G satisfying ( 1 ) F n H 2 F. This set is clearly a local system of G such that each of the subgroups F E Z is a finite Frobenius group with Frobenius complement F n H. If g is any element of G such that (1) F n H g $ F where F E Z, then F n H g is also a Frobenius complement of F. A finite Frobenius group has a unique Frobenius kernel and this kernel is a normal complement of every Frobenius complement of the group. Hence for each F E Z, it follows that F n K is the Frobenius kernel of F and that F = ( F n H)(F n K). Therefore K is a Frobenius kernel of G, K is unique, and G = HK.
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For each F E Z the kernel F n K is a nilpotent Hall subgroup of F. Hence K is locally nilpotent and K = 0 , G for some set x of primes such that G is n-closed. Further, since every non-trivial element of H induces on K an automorphism of finite order that leaves only the identity element fixed, a theorem of G. Higman [l ] yields that K is even nilpotent (of class bounded by a function of the smallest prime dividing the order of an element of H). By a result of Burnside for each F E Z any subgroup of F n H of order p q , where p and q are (possibly equal) primes, is cyclic. In particular the abelian subgroups of H are locally cyclic and the p-subgroups of H are locally cyclic or (if p = 2) locally quaternion. Suppose that H contains an involution i. Then i acts fixed-point freely on K since H n H k = (1) for all k E K, and so k' = k - for each k E K. But this automorphism lies in the centre of the automorphism group of K. Hence i lies in the centre of H and is the unique involution of H. Let S be any non-trivial subgroup of G such that K n S = (1). The subgroup S is isomorphic to a subgroup of H. Hence if S contains an involution, this is unique and is contained in the centre ZS of S. Suppose that S does not contain any involutions. Then every primary subgroup of S is locally cyclic, and S is metabelian, S' and SJS' are locally cyclic and the elements of S' and SJS' have coprime orders (Huppert IV.2.11). If S' # (1) let s and t be elements of prime orders p and q respectively, with s E S' and t E S\S'. Now (s) is clearly a normal subgroup of S, so the subgroup (s, t ) has order p q and thus is abelian. Hence t centralizes s. In view of the structure of the automorphism groups of the cyclic and Pruferp-groups (Scott [ l ] 5.7.12; KuroS [I] Vol. 1, p. 155) the element t centralizes the maximal p-subgroup of S'. This holds for each primary component of S'. Hence the element t centralizes S', and so ( t ) is a characteristic subgroup of the abelian normal subgroup (S', t>of S. As such ( t ) is normal in S. Further ( t ) n S' = (I}, so t is central in S. We have now shown that S always contains a non-trivial central element, z say. It follows from HK = G and the definition of K that for some element k E K we have z E Hk. Thus if x E S, then z E H k n Hkx,and so S c NGH k = H k , from the definition of Frobenius complement. Finally if H, is a second Frobenius complement of G then the uniqueness of K implies that G = H , K and H , n K = (1). But then H I G Hkfor some element k E K and so H k = H,(Hk n K ) = H,. I The reader has probably guessed by now that the following characterization of locally finite Frobenius groups holds.
CH.
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FROBENIUS GROUPS
53
1.5.3 Proposition. If G is a locallyfinite group then the following three statements are equivalent. 1. G is a Frobenius group. 2. G has aproper normal subgroup N such that for every non-trivial element n o f N the centralizer CGnis contained in N. 3. G has a local system Z consisting offinite Frobenius groups. Proof. If G is a locally finite Frobenius group with Frobenius complement H let N be the Frobenius kernel of G. For any n EN\( 1) and any g E G, we have H gn C G n E H g n Hg" = (1). Hence CGn E N, and thus 1 implies 2. Suppose 2 holds and let C be the set of finite subgroups F of G such that (1) # F n N # F. It follows at once from Scott [I], 12.6.1 that C satisfies condition 3. Now assume that C is a local system of G consisting of finite Frobenius groups. Let S E C, and denote by C any (fixed) Frobenius complement of S. If T E C is such that S E T, then there exists a unique Frobenius complement C , of T containing C, for the Frobenius kernel of T is nilpotent, S is not nilpotent, and any Frobenius complement of T intersecting S properly is a Frobenius complement of S and thus is conjugate to C. Put
H = . For any x in G, if ik" has even order, then k" E S. Hence the elements x of G for which ik" has even order lie in fewer than N right cosets of CGk.Since i and k are conjugate the set of all
u,
70
CENTRALIZERS IN LOCALLY FINITE GROUPS
[CH.
2
these x E G has cardinal less than N and if M = {x E T : ik" has odd order}, then IMI 2 N. Let c E M and put s = ik'. Since s has odd order, there exists an element s1 of (s) such that i = (kc)sl.But then i = k"' = k'", so h = csl a; E CGk . Now j = ci is an involution of G and
'
since c and s1 lie in T. Therefore, for every c E M there exists an element h E C G ksuch that the involutionj = ci conjugates ha, onto its inverse. Since k is conjugate to i we have that lCGkl < N. The regularity of N now implies that there exists an element r of C , k and a subset N of M of cardinal at least N such that ci conjugates y = ra, onto its inverse for every c E N. Clearly cd-' E C G yfor all c, d E N , and SO lCGyl 2 N. If C G y = G , then r E C G k implies that k = k"' = i, a contradiction which shows that y is not central in G . Let c E N . Since the element c has odd order, there exists an element c1 E "= i. Put g = y"'. Then g is a non-central element of G such that ICGgl 2 N and gi = g - ' . Finally, the centralizer of an involution of C G ihas cardinal less than N and IcGgl 2 N, so g must have odd order. b. By a there exists a non-trivial element g 1 E G such that g: = g;' and IC,gll 2 N. Clearly the group is an infinite hypercyclic group and g,gt
.
,g 2 , . .,g,-,>
E I.
The group (i, H ) is also hypercyclic, and therefore by 1.A.7 the group ( i , H ) contains a self-centralizing abelian normal subgroup A , and A will also be infinite. A is a direct product of a 2-group A , and a 2'-group A,,, at least one of which is infinite. Clearly i acts fixed-point freely on A,, and ai = a- for every a E A , , . Thus we may assume that A , is infinite. The map a H d-' is an endomorphism of A , with kernel CA2iand image [ A z , il, so at least one of the subgroups CA2iand [ A , , i] is infinite. Finally (ai-l)i = which implies that ai = a - 1 for every a E [ A , , i ] , and the result follows. I
'
CH.
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71
2.2 Corollary. Let G be a periodic group and x a regular cardinal such that IG( 2 X . I f G contains an involution, then G contains a non-trivial element g such that lcGgl 2 K.
2.2 is an immediate consequence of 2.la. Our next major objective is to obtain the conclusion of 2.2 for an arbitrary locally finite group. A major part of the argument is isolated into the following lemma.
2.3 Lemma. Let G be a group, p a prime and R an elementary abelian psubgroup of G or orderp2. Suppose that A?isa set ofjnite subgroups of G each containing R such that 1-41 = x is a regular cardinal. Ifevery M E & contains a non-trivial abelian normal subgroup N M ,then there exists an element g # 1 of some M E -4 such that (C,gl 2 N. Proof. Suppose that lcGgl < N for every g E M\(1> and every M E A. Put C = UgaR\ xo (as is also the case in the more special situation of the exercise below). Exercise. If N is a regular cardinal and G is a periodic group containing an involution i such that IC,zl < N for every involution z of G centralizing i, prove that all the involutions of G are conjugate.
In 2.8 we have seen that it is possible that no maximal abelian subgroup of the infinite, locally finite group G has the same cardinal as G . Hence the condition that in the infinite, locally finite group there is a large abelian subgroup is a restriction on the structure of G. Unfortunately, however, this restriction does not seem to lend itself to further investigation. Let us tighten up this situation. Question 11.3. What can be said about an infinite locally finite group G such that for every infinite factor S of G and for every (regular) cardinal N5I S1 every maximal abelian subgroup M of S satisfies K S IMI? In particular, can such a group be simple? In the context of 2.10 and 3.33 we see that the existence of an involution i in the infinite, locally finite group G such that the centralizer C,i is finite, forces the existence of a proper normal subgroup of finite index in G. Thus if the infinite, locally finite group G is, in particular simple, then every involution has infinite centralizer. Question 11.4. Is it true that in every infinite, locally jinite simple group G one has lC,gl = [GI for every element g in G?
CH.
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79
Throughout this chapter our methods, that is the pigeon hole principle, have forced us to restrict our attention to regular cardinals. Does this restriction depend on the method or does it reflect some of the inherent difficulties of the situations considered? Question 11.5. Are any of the results of this chapter involving a regular cardinal still true for arbitrary injinite cardinals?
Note. The existence of infinite abelian subgroups in infinite, locally finite groups was shown (independently) in Kargapolov [6] and P. Hall and Kulatilaka [l]. Further proofs were given in Sunkov [2] and [6], see also Strunkov [3]. All these proofs make use of the Feit-Thompson Theorem stating that a finite group of odd order is soluble. However, special cases of this result have been known for some time, for example the locally soluble case is dealt with in Cernikov [17]. The 2-group case of 2.5 appears in Held [3 1, Sunkov [6] and Strunkov [ l ] and [3]; see also Kegel 151. We have taken 2.1 from Sunkov [8] (for the special case N = N ~ ) a, preliminary version appearing in Sunkov [6]. Also 2.9 and 2.10 can be extracted from Sunkov [8]. The idea of generalizing these various results by considering regular cardinals instead of N~ first appears in print, it seems, in Strunkov [2] and [3], although these papers are more concerned with properties of the class of 2-finite groups. The connection between Smidt’s Problem and the other topics of this chapter had long been realized, of course, and thus many of the papers referred to above derive the corresponding case of 2.6 from their respective results. Specifically this connection is mentioned in Kargapolov [6], Strunkov [l], [2], and [3], and in Sunkov [2], [5], and [6]. The locally finite case of 2.7 appears in Sunkov [5].
CHAPTER 3
Locally Finite Groups with min-p
A group satisfies min-p for the prime p if each of its p-subgroups satisfies min. This is the case in a number of important situations. Thus certain properties of locally finite groups with min-p play an important role during reduction arguments leading to the characterizations of the groups PSL(2, F ) over certain locally finite fields F (see Chapter 5). These results in turn provide the key to the solution of the structure problem for locally finite groups satisfying min (see Question 1.5 and comments). The object of the present chapter is to study the relatively general notion of min-p is some depth. Although Sylow’s Theorem does not extend to locally finite groups satisfying min-p, such groups do contain p-subgroups whose behaviour reflects various important properties of the Sylow p-subgroups of a finite group. In general the maximal p-subgroups of a locally finite group G with min-p will not be conjugate, nor even isomorphic. But among the maximalp-subgroups of G there are some which contain an isonlorphic copy of all the others, and these p-subgroups of G are isomorphic. We call these subgroups the Sylow p-subgroups of G. They are well behaved under joins and intersections with normal subgroups of G, but they need not be conjugate. Thus it would seem that the Frattini argument applied to the Sylow p-subgroups of a normal subgroup is not available. However, something can be salvaged since every finite p-subgroup of G is conjugate to some subgroup of any Sylow p-subgroup of G, and is contained in some finite p-subgroup of G, whose conjugates are permuted transitively by the automorphisms of G. This substitute for a Sylow theory is expounded in Section A . We put it to use in Section B. In that section we obtain the structure of a locally finite group G satisfying min-p if G has many normal subgroups; in this case Opt,G has finite index in G. Of course some restriction of the normal structure of G is needed, since PSL(n, F ) satisfies min-p for every prime p distinct from the characteristic of F. A suitable such condition can be phrased in terms of Sunkov’s local notion of the range of an element (see 3.29). The section concludes with some applications of these results (promised at the end of Chapter 2), which show 80
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that the centralizer of an involution in an infinite locally finite group is very often infinite. For example a special case of 3.33 is that in an infinite, simple, locally finite group the centralizer of any involution is infinite. Section A. A Sylow theory for locally finite groups with min-p As shown in the Introduction maximal p-subgroups of a locally finite group need not all be conjugate (or isomorphic), even if they are Cernikov groups. For this situation we discuss an example in some detail. If all the psubgroups of the locally finite group G are Cernikov groups then their sizes are bounded, and those of largest size are isomorphic. Since these maximal p-subgroups enjoy many of the properties of the Sylow p-subgroups of a finite group, we shall call them Sylowp-subgroups of G. In this section we shall study some of the properties of these Sylow p-subgroups. The main result will be that these subgroups, or rather their size, lend themselves to treatment by induction, and so in the applications in Section B induction on the size of the Sylow p-subgroups will appear, generally in the guise of “least criminal” arguments.
3.1 Lemma. The locally finite group G satisJes min-p if and only if some maximal elementary abelian p-subgroup E of G is finite. Proof. If the group G satisfies min-p, then every elementary abelian p subgroup of G is finite. Let E be a finite maximal elementary abelianp-subgroup of G and suppose that Q is a p-subgroup of G. By 1.E.3 the group Q satisfies rnin if and only if all its countable subgroups satisfy min. Let R be an arbitrary countable subgroup of Q, and consider the subgroup H = ( E , R ) of G. By 1.A.9 the countable, locally finite group H has a local system Z consisting of finite subgroups that is linearly ordered by inclusion, and we may assume that the finite subgroup E of H is contained in the smallest member of Z.By 1.D.3 there exists a p-subgroup P E Max, H which reduces into Z and contains E. Now 1.6.6 applied to P yields that P is a Cernikov group, and so there is an abelian normal subgroup A of finite rank r and finite index IP : A [ in P. Clearly, every finite subgroup of P has an abelian normal subgroup of rank at most r and of index at most IP : A [ . Since every finite p-subgroup of H is conjugate to a subgroup of P,this last statement also holds for all finite subgroups of R. But then 1.K.2. yields that R, too, has an abelian normal subgroup of rank at most r and of index at most IP :A ] . Thus R is a Cernikov group, and Q satisfies rnin. I This result may be slightly rephrased.
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min-p
[CH.
3, 9
A
3.2 Corollary. If the locally finite group G does contain elements of order p, then G satisfies min-p if and only if there is an element of order p in G such that the centralizer CGgsatisJies min-p. Proof. The necessity of this condition is obvious. Its sufficiency follows from 3.1, since every maximal elementary abelian p-subgroup of G containing the element g is contained in the centralizer CGg and so is finite. I Exercise. Let p be a prime, H a locally finite p'-group, and K a locally finite group of automorphisms of H such that the set { C H X ;X E K}, partially ordered by inclusion, satisfies the descending chain condition. Prove that Ksatisfies min-p. (Effectively a special case of this result is that a linear group over a field of positive characteristic other than p satisfies min-p; see 1.L.3.) For a proof, consult the proof of 4.6 in B. Hartley [4].
Even if the locally finite group G does satisfy min-p the elements of Max,G need not be isomorphic, let alone conjugate in G. This can be seen by choosing the factors suitably in the universal counter example given in the Introduction. We shall now look in detail at a very similar example.
3.3 Example. For any pair (p, q) of distinct primes there exists a countable metabelian {p, 4)-group G satidying min-p such that G contains non-isomorphic maximal p-subgroups. Proof. Let A = (ai; a; = a i - l , a,, = 1, i E N) be a Pruferp-group, and B = (b) a cyclic group of order p. For the prime q # p let X = (x) be a cyclic group of order q. Denote by M the set of all mappings of the direct product A x B into X which take a value different from 1 on finitely many elements of A x B only, and consider M a s a right A x B-module in the usual way. Put G = X ? ( A x B), the split extension of M by A x B. Clearly M is an elementary abelian q-group, and G is a countable metabelian {p, q)-group satisfying min-p. Put H = MA. For each natural number n define the mapping p, E M by
For every eIement g E A x B one thus has g>Pn = (g>P?-'* Hence as elements of G the elements p, and a,- commute. Put c, = a : ' ' * . P n and C = (c,; n E N). The order of the element c, is p* and (g)Pu, =
cl =
(an-1
(az)B1P2...Bn
=
,
(a"Pn
)PI 1
...Pn-1 -
CH.
3,
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SYLOW THEORY
83
Thus C is a p-subgroup of G. Clearly, one has H = MC, and C is isomorphic to A. We prove that C is a maximal p-subgroup of G. Since A x B is trivially a maximal p-subgroup of G, and since A x B and C clearly are not isomorphic, this will complete the proof. Cis a maximalp-subgroup of H , the index IG : HI = p , and thep-subgroups of G are abelian. Thus if C is not a maximal p-subgroup of G , then G = M C , C . In particular, there exists a mapping p E M such that the element pb-' centralizes C. Thus, for each natural number n one has a;~...~n~b-'
= a;'"'"".
Since the element b centralizes A, the mapping?, = p1 . . . pnp((pl . . .p n ) - l ) b , qua element of G, centralizes the element a, and thus also the element an-l = a,". But ( l ) Y , = (1)p * x-,, and (1)yF-l = ( a , - , ) p x-'. Hence one has (a,,-l)p . X"-' = (I)p for all n E N. Consequently for infinitely many a E A the value (a)p # 1,which contradicts the definition of M. Therefore no such mapping p can exist, and hence C is its own centralizer in G. That is, C is a maximal p-subgroup of G. 1 Using the above method it is possible to construct for each cardinal ct = 1, 2, . . ., No a countable metabelian {p, 9)-group satisfying min-p whose maximal p-subgroups fall into exactly ct isomorphism classes. Further, if p is any infinite cardinal or a finite cardinal prime top, then there exists a metabelian {p, q}-group G satisfying min-p such that the set of maximal p-subgroups of G consists of one isomorphism class but precisely p conjugacy classes; see Wehrfritz [7], Theorems 3, 4, and 5. Under relatively mild additional assumptions on the locally finite group G satisfying min-p, the maximal p-subgroups of G are all conjugate. This general and vague statement is illustrated by the following result, taken from Zalesskii [I] and Wehrfritz [I] and [7]. 3.4 Theorem. If in the Iocally Jinite group G satisfying min-p for euery
properly ascending chain {A,}6N of abelian Jinite p-subgroups the descending chain of centralizers {C, An}neNbecomes stationary after finitely many steps, then the maximal p-subgroups of G are conjugate. Notice that if a group satisfies the hypothesis of the theorem, then so does each of its subgroups.
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mfn-p
[CH.
3,
A
Proof. Suppose the theorem is false and let H be a counter example. If for every subgroup X of H that is also a counter example to 3.4 there is a finite non-central abelian p-subgroup A , such that the centralizer C,A, is also a counter example to 3.4, then we shall derive a contradiction. Put X , = H , and choose inductively I,,,, = CxnAxn.If we put A, = (Ax,; j = 1, 2,.
. ., n>,
,
then A,, is a finite abelian p-subgroup of H and X,,, = C , A,,. The sequence {A,,}noNof finite, abelian p-subgroups is properly ascending, and - by construction - the sequence {X,,},,eN of centralizers is properly descending, which contradicts our general assumption. Hence the group H contains a subgroup G that is a counter example to 3.4 and such that for every finite, non-central abelian p-subgroup A of G the maximal p-subgroups of C,A are conjugate in CGA. Let P be any maximal p-subgroup of G and A its minimal subgroup of finite index; let B be any maximal radicable, abelian p-subgroup of G and Q any maximalp-subgroup of G containing B. By 1.E.6 the group A is radicable and abelian, and B is the minimal subgroup of finite index in Q. To obtain a contradiction - and hence the non-existence of H - it suffices to prove that the subgroups P and Q are conjugate. Put A,, = { a E A ; uPn= l}. By hypothesis one has CGA = C , A,, = C , A j for some natural number j . Hence A contains a finite subgroup A* (= A j + , ) such that C,A = CG(A;). In the same way B contains a finite subgroup B* such that C , B = CG(B,P).By Sylow’s Theorem there exists an element g in the finite sub-group (A,&) of G such that the subgroup S = ( A * , BZ) is a finite p-group. Suppose that S contains an element z such that z E Z, SIZ, S and z pE Z, S. The map x w [x, z ] is a homomorphism of S into Z S with kernel C,z. Moreover, since z p E Z, S, the group [S, z ] has exponent p and S p c C,Z. Consequently
nnsN
,
Z
E C,(A$) n C,((BS,)’) =
CGA n C,(B9).
Hence A u B9 c C,z. But z is not central in S, hence not in G, and so the maximal p-subgroups of C G z are all conjugate. The maximality of B and min-p ensure that some conjugate of B contains A . If S is abelian but not central in G, then S
G
CG(A$) n C,((BB,)P) = CGA n cG(Bg).
Hence A u Bg c C , S. Since the maximal p-subgroups of C , S are conjugate,
CH.
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85
we again get that some conjugate of B contains A . Finally if S is central in G , then C G A = G = CG(Bg),and thus A 5 3. We have now shown that there always exists an element h in G such that A c Bh. The factor group P / A is a finite maximal p-subgroup of N,A/A, and every maximal p-subgroup of N , A contains A . Thus by 1.D.12 the maximal p-subgroups of N , A are all conjugate. Since Bh c NG A , some conjugate of Bh lies in P . Hence A and B have the same rank, and A = Bh. But then Qhc NG A , and so P and Qhare conjugate, which is the desired contradiction. I As an immediate consequence of 3.4 we obtain a result of Baer [ 5 ] . It should be observed, however, that this result is obvious, once one knows that the answer to Question 1.5 is positive, at least for locally finite groups, see 5.8.
3.5 Corollary. If the locally$nite group G satisfies min, then for each prime p the maximal p-subgroups of G are all conjugate. Many theorems on the conjugacy of the maximal p-subgroups of a locally finite groupare known; see Chapter 1, Section D (in particular the Appendix) for comments on this kind of result. In spite of Example 3.3 and the counterexamples of Wehrfritz [7] mentioned above, there does exist a “Sylow theory” which is just satisfactory for the class of locally finite groups satisfying min-p. Definition. Let p be a prime and G any group. Then the subgroup P is a Sylowp-subgroup of G if P is a maximal p-subgroup of G and contains an isomorphic copy of every p-subgrofip of G .
Clearly, the set Syl, G of all Sylow p-subgroups of the group G is a subset of Max, G which remains invariant under the action of every automorphism of G. If the maximal p-subgroups of G are all isomorphic or even conjugate, then trivially they are all Sylow p-subgroups of G. Thus the Sylow p-subgroups of a finite group are just its maximal p-subgroups, and our terminology is consistent with the ordinary terminology for finite groups. Unfortunately, however, a locally finite group need not contain Sylow p-subgroups for a given prime p . The reader should convince himself of this fact in the following exercise. Exercise. Let A be an uncountable abelian p-subgroup and B an uncountable locally finitep-subgroup such that every abelian subgroup of B is countable (that such a group B exists was seen in 2.8). Then the locally finite group G = ( A * B ) / K K 4 does not contain any Sylow p-subgroups. Here K denotes the kernel of the projection of the free product A*B onto the direct
86
LOCALLY FINITE GROUPS WITH
n7in-p
[CH.
3, 0 A
product A x B, and q is an arbitrary prime # p . (The same construction, repeatedly applied for different primes, allows one to produce a locally finite group not containing a Sylow p-subgroup for any p . ) In view of this fact, it seems natural to ask: What conditions on the locally finite group G assure the existence of Sylow p-subgroups? We shall see that min-p is such a condition. The main tool for this result is the following lemma. 3.6 Lemma. If P and Q are Cernikov groups such that Q contains an isomorphic copy of every finite subgroup of P, then Q contains a subgroup isomorphic to P. Proof. Let A (respectively B ) be the minimal subgroup of finite index in
P (respectively Q ) . For every natural number i put A,
=
,the factor group G/B cannot be simple if is a local system of G/B. Since G/B is simple the set Z3 is not a local system of G, and thus Z4 = Z2\Z3 is a a local system of G. Let H EZ4.If Q is any maximal normal subgroup of H that does not contain the element 9, then B n H E Q and Q / ( B n H ) is a p‘-group. If R is any other maximal normal subgroup of H, then by 3.23 there exists an element x E N,H such that g I$ R”. Hence B n H c R” and R“/(Bn H ) is d p’-group. Since x normalizes B and H, the intersection B n H lies inside R, and R / ( B n H ) is a p’group. Now R # Q , and thus H = RQ. Therefore H / ( B n H ) is a p’-group. But g E H B \ is ap-element. This contradiction shows that no such maximal normal subgroup R can exist in H, and thus Q is the unique maximal normal subgroup of H . By the definition of the range k of g in G, one has IH : Ql 5 k. Since this holds for all subgroups H E Z4,1.K.2 applies and yields that in G there is a proper normal subgroup of index bounded by k , which does not contain the element 9. This again contradicts the assumption that G/B is infinite. This final contradiction shows that our initial assumptions are untenable and thus proves the result. I Since Question III.1 remains open, there is definite interest in Sunkov’s generalization of 3.17, were it only to illustrate further the power of the notion of the range of an element.
z3
3.29 Theorem. If the locally $nite group G satisfies min-p,then every pelement of G hasjinite range in G if and only i f the index IG : Op,p GI is Jinite. Proof. By 3.26 everyp-element of G has finite range in G if and only if every p-element of G/O,G has finite range in G/O,,G. I f the index ( G : O,,, GI is finite, then every composition series of G/O,, G has finite factors, and thus - by 3.27 - every element of C / O ,G has finite range in C/O,, G. In the opposite direction, we assume that the result is false and choose among the hypothetical counter examples a group G with an abelian-byfinite normal subgroup B such that the Sylowp-subgroups of G/Bare as small as possible. By 3.26 and 3.16 we may assume that 0,. G = (1) and OP’G=
CH.
3,
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APPLICATIONS OF SYLOW THEORY
G . Therefore B is a finite extension of a p-group and so has min. If PIB is an infinite normalp-subgroup of G/B, then P is abelian-by-finite by 3.16 and the Sylowp-subgroups of GIP are of smaller size than those of GIB. This contradiction of the choice of G and B shows that O,(G/B) is finite. Suppose that GIB is simple. By 3.28 G contains a proper normal subgroup N not contained in B. Then G = BN. Since B satisfies min we can find a supplement M of B in G such that B n M is minimal. M / ( B n M ) N GIB, so by 3.28 again M contains a proper normal subgroup M , not contained in B n M . Then B M , = G and B n M , c B n M , a contradiction to the minimality of B n M . Therefore G / B is not simple. There exists a normal subgroup B, of G such that B c B, c G. Since OP’G= GI the group G/B, is not a p’-group. Therefore the Sylow p-subgroups of B J B are smaller than those of GIB and consequently IB, :O,.,B1l is finite, But 0,.G = (1) and O,(G/B) is finite. Thus BJB is finite and nontrivial. We can repeat the above argument with B, in the place of B. Continuing in this way we can define an ascending series B = Bo c B, c B2 c . . of normal subgroups of G such that B,/B,_ is finite and non-trivial for each i. If all but a finite number of the Bi/Bi- are p‘-groups, then D = Bi contains a non-trivial normal p‘subgroup by 3.16. Therefore DIB contains infinite p-subgroups. Let Q be the subgroup generated by the divisible abelian p-subgroups of DIB. The group Q is infinite and contains no proper subgroup of finite index. But clearly IQ : C,(B,/B)I is finite, so Q lies in the centre of D . Consequently Q is an infinite p-group, contradicting the finiteness of O,(G/B). The theorem is now proved. I In view of the result of Feit and Thompson [l] that every finite group of odd order is soluble, we obtain the following result as an immediate consequence of 3.29.
.
,
uz
3.30 Corollary. The locally finite group G satisfying min-2 is an extension of a local/y soluble group by afinite group if and only if each of its 2-elements has finite range in G . There also is a variation on 3.18. 3.31 Corollary. Let G be a locally finite group satisfying min-p for every prime p. If every 2-element of G hasJinite range in G, then there is a radicable abelian normal subgroup A in G such that there factor group GIA is residually finite, and every Sylow subgroup of GIA isfinite.
Proof. From 3.30 and 1.B.4 it is obvious that no infinite simple group can be involved in G, and thus the conditions of 3.18 are met. I
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[CH.
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5B
Observe that for results like 3.29 or 3.30 to hold some condition with bearing on the normal structure of G is necessary, since the simple groups PSL(n, F ) over infinite, locally finite fields F satisfy min-p for every prime p except the characteristic of F. In view of several of the preceding results we ask Question 111.3. Does the locallyfinite group G have a locally soluble subgroup offinite index if it satisfies min-p for every prime p. The following lemma reduces a number of problems to the study of certain infinite simple groups. It allows numerous variations, for example 1.F.4 could have been incorporated into it.
3.32 Lemma. Let G be a Iocallyfinite group satisfying min-2 such that no locally soluble subgroup of G has finite index in G. Then G contains subgroups H a n d N such that H is perfect, N is normal in H , and the factor group HIN is infinite and simple. Also every subgroup of H of 2-size smaller than H (in particular N ) is (locally soluble)-by-finite. Further, if G satisfies any one of the following four conditions, then so does H / N . a The group satisfies min-p; b Every abelian subgroup of the group satisfies min; c The group is an XRc-group and satisfies min-p for every prime p ; d The group is an mC-group satisfying min-p for every prime p , and the centralizer of each of its non-central subgroups is abelian-by-finite. Proof. G contains a subgroup H that is not (locally soluble)-by-finite and is of minimal 2-size with respect to this property. Clearly we may assume H = 0 2 ' H .If K is any proper normal subgroup of H, then the 2-size of K is smaller than that of H, and by the minimal choice of H in G, the subgroup K is locally soluble)-by-finite. In particular, the group H is perfect. Let N denote the product of all proper normal subgroups of H . By 3.17 (or 3.30, if you prefer) the characteristic subgroup N of H is (locally soluble)by-finite, and hence H / N is an infinite simple group. By 3.13 if G satisfies min-p, then so does H/N. If AIN is any abelian subgroup of H / N and if G satisfies b, then A is (locally soluble)-by-finite and satisfies min by 3.19. Thus every abelian subgroup of H / N is a Cernikov group. Now assume that G E mCand that G satisfies min-p for every prime p . By 3.22 the normal subgroup N of H contains an abelian characteristic subgroup B = N o of finite index. Now both C H B and C,(N/B) have finite
CH.
3, 8 B]
APPLICATIONS OF SYLOW THEORY
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index in H (see 3.21) and the stability group ( C H Bn CH(N/B))/CHN is abelian ( l . C . l ) . Hence H = N C H N , H / N N C , N / Z N , and C H N is not (locally soluble)-by-finite. Thus we may assume that H = CHN, that is, N = ZH. Let a E H\N, and consider the map 6 = 6, of K = CH((a,N > N)/N) into N given by 6 : x H [x, a ] . The map is a homomorphism. Moreover, if I(a)l = r, then ( K a y = (l), and since N satisfies min-p for all primes p, the image K S is finite. Clearly we have Ker 6 = C H a ,and thus IK : CHal is finite. In particular, if every proper centralizer in G is abelian-by-finite, then every proper centralizer in H / N is abelian-by-finite. Let S1 c S , c . . . c Si c . . . be an ascending chain of subsets of H , and put K i= C H ( ( S i ,N ) / N ) . In order to prove that H / N E % , , we have to show that Ki = K j for all large i and j . Since G E E,, the collection of centralizers { C H S i ;i = 1, 2, . . .} is a finite set, and we may assume that CI,S, = C H S ifor i = 1,2, . . .. There exists a finite subset F of SI such that C H S 1= CHF, since H E %I?=. By the previous paragraph, the index ICH((a,N ) / N ) : C,al is finite for each element a E F, and thus the index JC,((F, N ) / N ) : CHFI is finite. Trivially CH((F,N ) / N ) 2 K , 3 K i3 CHSl = C H F for each i,
and the result follows. I The next result (Theorem 5 of Sunkov [2]) is of a different nature. It is closely associated with the results of Chapter 2, but we give a proof using 3.29. This theorem may in fact be used to give another proof of the result of Kargapolov [ 6 ] and Hall and Kulatilaka [I] (contained in 2.5) that every infinite, locally finite group G contains infinite, abelian subgroups, since (together with the Feit-Thompson Theorem) it provides a reduction to the 10cally soluble case, which may be dealt with by 2.3. In the proof of this result we shall need a special case of Theorem 2H of Brauer and Fowler [11, namely that if G is a finite group and if the involution i of G satisfies (CGiI = n, then there exists a normal subgroup L # G of G with IG :LI 5 [+n(n+2)]!. 3.33 Theorem. If the involution i of the IocaIlyJinite group G hasfinite centralizer C , i, then there exists a normal subgroup ofJinite index in G which does not contain i. (Hence G cannot be inJinite and simple.)
Proof. Let X be any finite subgroup of G containing i, and denote by SX the set of all normal subgroups N of X such that i # N a n d such that every composition factor of X/N has order at most m = [$lC,il(lC,il+2)]! The
106
LOCALLY FINITE GROUPS WITH
min-p
[CH.
3,
5B
set Sx is not empty, for let M be the smallest normal subgroup of X such that every composition factor of X / M has order at most m, and suppose i E M . Then by the result of Brauer and Fowler [l] quoted above, M has a normal subgroup MI with 1 < IM : NIl 5 m. Hence the intersection M , = nxsXMf is a normal subgroup of X such that every composition factor of X / M 2 has order at most m , and yet M2 < M . This contradicts the choice of M , and consequently i 4 M and M E Sx. If Y is any finite subgroup of G with i E: X c Y, then the mapping A; : N w N n X , N E S ~maps , the set Sy into the set S,. The collection {S,, A:; i e X E Y } is an inverse system of non-empty finite sets over the directed index set of all finite subgroups of G containing i. By l . K . l the inverse limit b.m Sx of this system is not empty. Let (M,) E lim Sx,where M x E S, . Then the properties of the inverse limit yield that M = u x M x is a normal subgroup of G not containing i, M x = M n X , and M X / M =! X / M x . Consequently the composition factors of every finite subgroup of G/M have order at most m, and so every element of GIM has finite range. By 3.2 the group G satisfies min-2, and so by 3.13 the group GIM satisfies min-2. By 3.29 the group G contains normal subgroups M c L c K c G such that G/K is finite, K / L is a radicable, abelian 2-group, and L / M is a 2'-group. Let P be any Sylow 2-subgroup of G containing i and A its minimal subgroup of finite index. If a E A , there exists an element b E A with b2 = a. But then a = (bb')(bb-'), bb' E C,i, and (bb-')' = ( b ! ~ - ~ ) - In ' . particular A = (CAi)[i,A ] . But A is radicable and CGi is finite. Consequently A = [i, A ] , and thus a' = aL1 for every element a E A . Now clearly 3.13 implies that A L = K , and so the involution i inverts every element of K/L. Hence if i E K, then K = L . Now L / M is a 2'-group and so i E M. Since this cannot be, we have i$ K. I 3.34 Lemma. If the element x of prime order p in the locally Jinite group G hasjnite centralizer in G, then every normal subgroup of G containing x has finite index in G.
Proof. Let N = (x'), the normal closure of x in G. By 3.2 the group N satisfies min-p, and by 3.7 and 3.10 there is a finite p-subgroup P of N containing x such that the conjugacy class of P in N is invariant under all the automorphisms of N. Hence the Frattini argument yields G = N . NGP. Since P is finite, INGP : CGPI is finite. Clearly C G P c C G x , and so the index IG : NC,xl is finite. But C,x is finite, thus IG : NI is finite. I An automorphism of a group that leaves fixed only a finite number of elements is called a nearly regular automorphism. Involutary nearly regular auto-
CH.
3, 5 B ]
APPLICATIONS OF SYLOW THEORY
107
morphisms arise in several contexts (the involution i of 3.33 induces such an automorphism on G). The reader will have no difficulty in extracting the proof of the following lemma from the last paragraph of the proof of 3.33. 3.35 Lemma. I f A is a radicable abelian group and i f i is an involutary nearly regular automorphism of A , then ai = a - l f o r every a E A . 3.36 Theorem. Let G be a locally Jinite group in which every 2-element has finite range. If G admits an automorphism a of order two such that the centralizer C G a is a C'ernikov group, then the subgroup 0 2 , z G of G has finite index in G and contains an ascending series with abelian factors consisting of subgroups invariant under G and a. If the centralizer CGais finite, then the subgroup O Z pG2is soluble. We do not know whether the condition that all the 2-elements of G have finite range can be weakened or even left out without affecting the conclusion. In fact, we are not aware of any examples of a locally finite group G which is not soluble-by-finiteand which admits an automorphism a of order two such that the centralizer CGais a Cernikov group.
Proof. Denote by K the semi-directproduct K = G X (a). By 3.2 the group K (and hence also G) satisfies min-2. Consequently, by 3.29, the subgroup 02,2 G is of finite index in G. The factor group OZr2 G/02.G is soluble, and by the result of Feit and Thompson [l] the subgroup 02' G = OG is locally soluble. Let N be any K-invariant subgroup of OG with N # OG and put H = O G / N . Observe that C,a = N C o G ~ / NWe . want to show that in the group H # ( 1 ) there exists a K-invariant abelian subgroup A # (1). Let M be maximal among the K-invariant subgroups X of H satisfying X n C,a = (1). Then clearly a acts regularly on M , and since M i s locally finite, it must be abelian. So if M # (l), we are done. Hence assume that every K-invariant subgroup # ( 1 ) of H intersects the centralizer CHm nontrivially. By 3.12 there are but a finite number of conjugacy classes of elements of prime order in C , a. Since every K-invariant subgroup # (1) of H contains at least one of these classes there exists a minimal K-invariant subgroup M # (1) of H. Now 1.B.4 asserts that M is abelian. Thus 02, G has an ascending series of K-invariant subgroups with abelian factors. If COGa is finite, then induction on the order ]COGa1 shows that the derived length of OG is bounded by 2 log 31C~GaI 1. I
+
Exercise. Let G be a locally finite group and a an involutary nearly regular automorphism of G. Prove that for every a-invariant normal subgroup
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mk-p
[CH.
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0B
N of G the automorphism induced on GIN by a is still nearly regular. (See Sunkov [12], Lemma 8.) Exercise. Let G be locally soluble group admitting an automorphism a of prime order. Show that G is hyperabelian if the centralizer C,a is a Cernikov group, and that G is soluble if the centralizer C,a is finite. (Use Thompson’s Theorem - see Huppert [l], V.8.14 - and the results of G. Higman [ l ] - see Huppert [l], V.8.8a.) 3.37 Theorem. If G is an infinite, locally finite group and V is an elementary abelian subgroup of G of order four, then for at least one involution i E V the centralizer C , i is infinite.
Proof. Suppose, if possible, that this result is false, and let G be any counter example. By 3.2 the group G satisfies min-2. Let S be any maximal 2-subgroup of G containing V, and denote by A the smallest subgroup of finite index of S. If i and j are distinct involutions of V then a’ = aj = a-l for every a E A by 3.35. But then A E CGij,which is finite. Therefore the 2-subgroups of G are all finite. We shall now assume that G is chosen to be a counter example with Sylow 2-subgroups of minimal order. By 3.33 - applied to each of the involutions of V - the group G has a normal subgroup N of finite index such that N n Y = (1). Suppose that N is a 2‘-group. For every finite subgroup F of G containing V and for every prime p there exists a Sylow p-subgroup P of F n N normalized by V. By Gorenstein [l], 5.3.16 we have
P c L = (CGi;i e V\(I)). Hence F n N G L . But then N c L , and yet L is a finite group and N is infinite group. Thus N must contain an involution. There exists an involution z in N centralizing V (the centre of a Sylow 2subgroup of G has a non-trivial intersection with N ) . If i is any involution of V, then the Sylow 2-subgroups of ( i , N ) have smaller order than those of G and hence there exists an involution j in the elementary abelian group ( i , z ) of order four such that the centralizer C < i , N )isj infinite. Put C = CGj. The infinite group C contains V,and the image of V in C/(j ) has order four. Further it contains an involution with infinite centralizer in C / (j ) , since the Sylow 2-subgroups of C/(j)are of smaller order than those of G. So let k e V\(l) be such that the centralizer K = C c ( ( j , k ) / ( j ) ) is infinite. But C c k = C&, k ) and so the index IK : C,kl is finite, which implies that the centralizer C,k is infinite. This contradicts the assumption that G is a counter example. Hence no counter example can exist and the result is proved. I
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We conclude this chapter with a further application of the characteristic conjugacy classes of subgroups constructed in 3.9 that allows many properties of the automorphism group of a Cernikov group to be read off from corresponding properties of linear groups. For example both parts of 1.F.3 follow at once from the proof below and the fact that a periodic linear group over the field of p-adic numbers is necessarily finite. Note also that since for any prime p the ring of p-adic integers is an integral domain of characteristic zero, there always exist rings R satisfying the requirements of 3.38 below. 3.38 Proposition. Let G be a C'ernikov group, A its minimal subgroup of finite index and p l , . . ., p t the primes involved in A . I f R is any integral domain that f o r i = 1, 2, . ., t contains a subring isomorphic to the ring of pi-adic integers, then the automorphism group AutG of G contains a subgroup L28 with Aut G = (Inn G)W, that is linear over R . Moreover the outer automorphism group Out G of G is linear over the quotient field of R.
.
Proof. Let d = Aut G, 4 = Inn G, K E Z ( G ) (see 3.9) and put 3 = N,(K) = {$ E d ; K# = K } . Since A is characteristic in G, C,A = {$ ~ d ; u4 = a for all a in A } is normal in d.Also d / C , A is isomorphic to asubgroup of Aut A , which in turn is a direct product over i of linear groups over the pi-adic integers. Hence d / C , A is linear over R. Since A K = G, C,A n C,K = (1). Thus C,K is linear over R. But g / C , K is a finite group as K is finite, so 53'too is linear over R. If $ E d then K@E%(G), By 3.9 there exists g in G such that KO = K g .If Ag denotes the inner automorphism x H gxg-' of G then $Ag E and so d = #a. Now 4 n C,A = {Ag; g E C G d }and so 9/($ n C,A) is finite. Hence I$*C,K: ( 4 n C C , A ) C , K J is finite. But 4 L 2 8 = d and I$: C,KI is finite, so Id : ( 4n CdA)C,Kl is finite. Now
(9n C,A)C, K 3 n C,A
N
c, K = C,K, 4 nC,A nC , K
and therefore d / ( 9n C d A ) is linear over R . Again 4/(9n C,A) is finite and thus Out G = d/9is linear over the quotient field of R (Chevally [I], p. 119). 1
Note. The concept of range (there called rank) and the fundamental theorem 3.29 is taken from Sunkov [ 5 ] , though the treatment given here follows Wehrfritz [8 1, from where the Sylow theory of locally finite groups with min-p is also taken. Of the other less trivial results of this chapter, 3.2 appears in Sunkov [3] and Kegel [3], 3.3 in Wehrfritz [7], 3.4 in Zalesskii [ 1 ]
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and Wehrfritz [l] and [7], 3.5 in Baer [ 5 ] , 3.17 in Kargarpolov [4], 3.22 in Kegel and Wehrfritz [l], 3.24 in Sunkov [ 5 ] , 3.33 in Sunkov [ 2 ] , and 3.37 in Sunkov [3]. A quite different approach to the linearity in 3.28 is given in Merzljakov [l].
CHAPTER 4
Locally finite simple groups
In the earlier chapters of this book we either did not need any assumptions about the simple groups involved in the various groups G under consideration (as, for example, in Chapter 2) or we essentially assumed that the relevant simple sections of G were all finite (see Section B of Chapter 3). Further we saw that the existence of counter examples to certain types of conjectures implies the existence of infinite, simple counter examples (1.F.4, and 3.32). Such counter examples must be studied, of course, in close connection with the particular conjectures they contradict, but there are a few general considerations about infinite, locally finite, simple groups that may be of help in any such closer scrutiny. For example, we show that any infinite simple group has a local system consisting of countable simple subgroups. Moreover a countable, locally finite, simple group is a limit (of sorts) of a sequence of finite simple groups, and such a sequence contains a good deal of information about any locally finite, simple group that may be its limit. Section A below studies such sequences and contains some results that prove quite strong whenever we know infinitely many terms of the sequence rather well. For example, if all the finite simple groups in such a sequence are linear of degree at most n (over possibly different fields) then any limit of the sequence is linear of degree n, and then 4.6 yields that we may assume that the terms of the sequence are subgroups of the limit (and not just sections). It is essentially this situation that we consider in Section B. We give an axiomatic (and hence considerably more general) treatment of the theorem that a group G with a local system of finite subgroups, all isomorphic to PSL(2, q ) for various prime powers q is isomorphic to PSL(2, F ) for some locally finite field F. In this axiomatic approach, the symbol PSL(2, 4) is considered essentially as a functor from tKe category of locally finite fields to that of locally finite (simple) groups. The simplicity of the groups PSL(2, ) is nowhere required in the axioms, indeed we give examples of functors that satisfy our axioms and yet map into the category of soluble groups. 111
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However, we exploit the results of this axiomatic treatment exclusively in order to study certain types of infinite, locally finite, simple groups and the structure of groups involving sections of this kind. In Section C these results are used to extend several rather deep structural theorems about finite groups to locally finite groups. Of course, the principal labour in these results resides in the finite case, which we usually admit without question. With the ever expanding supply of extendable theorems on finite (simple) groups the subject of this section has vast potential for growth. We give only a few salient examples of such extendability, our choice being greatly influenced by the tools that we require for Chapter 5, where some of the results of Section C will assume a critical importance. Section A. Infinite and finite simple groups, an approximation principle
Every infinite, simple group has a local system of countably infinite, simple subgroups. If the simple group G is locally finite, one might ask whether there is already a local system consisting of finite simple subgroups. For linear groups this question will be answered affirmatively. But in general, we can only show that a countably infinite, locally finite, simple group G can be approximated by a sequence of finite simple groups that appear as sections of G, and is linearly ordered by involvement. We study, superficially, the influence that such a sequence has on the structure of G. This study will also show that, should several “natural” conjectures on locally finite, simple groups be wrong, then there would be infinitely many “new” finite, simple groups still to be found - an easy, but intriguing observation. For the study of infinite groups in general the following observation is sometimes quite useful. 4.1 Theorem. The group G is simple if and only if there is a local system Z of G such that for every subgroup S E C and every normal subgroup N of S with (1) # N # S there exists a subgroup T E Z containing S such that every normal subgroup M of Tsatisfies M n S # N . (If G is simple, then every local system C of G has this property.) Proof. Let the local system C of the group G have the given property. Suppose G has a non-trivial normal subgroup X . Then there is subgroup S E Z of G such that (1) # S n X # S. For every subgroup T EZ of G containing S we have
(T n X ) -=I T and ( T n X ) n S
=
S nX,
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contradicting the assumption on the local system C. Thus, such a normal subgroup X of G cannot exist, and the group G is simple. Now, let G be a simple group and C any local system of G. If for some pair S, N of subgroups of G such that S E C, N 4 S, (1) # N # S, there is no subgroup T E C containing S such that M n S # N for every normal subgroup M of T, then put C' = { T EZ; S c T ) and for T EC' put A , = { M a T; M n S = N } . In this case 2' is a local system of G , and for T EZ' the set A , is non-empty. If N7' = M , then N T is thenormal closure of N in T (hence the notation), and it follows that for T, U E C' with T E U one has N T c N U .Put X = N T ; then X is a subgroup of G (in fact, it is the normal closure of N in G) and X # (1). On the other hand we have
nMEdT
UTEf,
X nS =
(u
NT) nS
TEZ'
(NT nS ) = N # S.
= TEZ'
Hence X is a proper normal subgroup of G, contradicting the assumption that G is simple. Thus the simplicity of G forces every local system Z of G to have the desired property. 1
4.2 Corollary. Let G be a simple group with a local system C.Ifthe subgroup S E C of G has only finitely many normal subgroups, then there is a subgroup T E C of G containing S such that every normal subgroup of T meets the subgroup S of G trivially (that is, in (1) or s). Proof. By 4.1 for each non-trivial normal subgroup N of S one may choose a subgroup TNE Z containing S such that M n S # N for every normal subgroup M of T N .There exists a subgroup T E C of G containing each of the finitely many subgroups TN . For every normal subgroup M of T one has M n S = ( M n T N )n S # N for each proper normal subgroup N of S . Hence M n S is either (1) or S. I This corollary shows that such a subgroup S of the simple group G embeds naturally into some chief factor of the subgroup T of G. One would, of course, like to have some control on the position in T and on the structure of such a chief factor. This can be achieved by placing severe restrictions on the local system C.
4.3 Corollary. If the local system Z of the simple group G is such that every subgroup S E Z of G has only finitely many normal subgroups, and i f with S also all the normal subgroups of S belong to the local system Z, then for every subgroup S E Z there is a subgroup T EC containing S, such that f o r some maximal normal subgroup M of T one has M n S = (1).
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In general such a stringent condition on the local system of a simple group will have a strong influence on the structure of this group. But if Z consists of finite groups, one obtains a local system Z' of G satisfying the assumptions of 4.3 by adjoining to Z the subnormal subgroups of members of Z. Thus, in a locally finite group G, the conditions imposed on the local system Z of finite subgroups in 4.3 have no real influence on the structure of the simple group G. Proof of 4.3. Fix the subgroup S E C of G. For each subgroup X of G containing S denote by Sx the normal closure of S in X . For each pair X , Y of subgroups of G with S c X c Y one has Sx G Sy. Consequently, the set C, = (Sx; S c X E Z} of subgroups of G is a local system of the group V = UXEISX, which is clearly a normal subgroup of G. Since we assumed that the group G is simple, the set Zlis a local system of G. By our assumption on the local system Z of G, the local system Z1is a subset of Z. Applying 4.2 to the local system Zlof G, we know that there is a subgroup U E C containing S such that for every normal subgroup N of Su one has
(*)
either S G N or S n N
= (1).
Now Su E Z, so Su contains only finitely many normal subgroups. Let K be a maximal normal subgroup of Su and put L = nxeUK".Since L is normal in U and properly contained in Su we have L n S = (1). Further K has only finitely many conjugates in U and consequently Su/L is a direct product of a
finite number of (isomorphic) simple groups. Let M be a normal subgroup of Su containing L such that M n S = (1) and maximal subject to this. Then if N / M is any minimal normal subgroup of S U / Mwe have that N / M is simple and that S c N , by (*). I Before exploiting 4.3 for the study of locally finite simple groups, we present another general fact about infinite simple groups that in many instances allows us to confine our attention to countable groups. 4.4 Theorem. The injinite group G is simple if and only ifit has a local system C consisting of countably infinite simple subgroups of G.
Proof. By 4.1 the existence of a local system of G consisting of simple subgroups forces the group G to be simple. Thus we need only show that every infinite simple group has such a local system. Let G be an infinite simple group. For every pair c, d of elements of G with c # 1 # d choose a finite set S(c, d ) of elements of G such that cE
(d";x
E S(c, d ) ) ;
CH.
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since the simple group G is generated by the set of elements conjugate to the element d, such a choice is always possible. With respect to this choice we define for every countably infinite subgroup C of G another countably infinite subgroup C* by C*
=
most all of the finite simple groups Sibelong to 9. The pigeon-hole principle asserts that infinitely many of the finite simple groups Si belong to A’”, one of the families S,,, or to %?. Since Siis involved in S i + l ,it is impossible that infinitely many of the Sibelong to the set M which satisfies the maximal condition with respect to involvement. So assume infinitely many of the Sibelong to the (very) classically parametrised family S,,. The group H i s a limit of this subsequence {Sit>. Since the group H is not linear, 4.7 yields that H is large (enormous). But the assumption that G is not large (enormous) is a condition on the set 9 of isomorphism types of finite groups of prime power order involved in G. Since every isomorphism type of 9 is represented as a subgroup in H , the assumption that G is not large (enormous) entails that the group H cannot be large (enormous). This contradiction is derived from the assumption that there are infinitely many of the finite simple groups Si belonging to one of the families S,,. Thus almost all of the Si belong to %?, and hence %? is infinite. I A class 23 of groups is called a variety if every subgroup and every homomorphic image of a @-group belongs to @ and if the Cartesian product of any set of @-groups also belongs to @. The variety 8 is locallyfnite if every group G E @ is locally finite. Clearly, the groups of a locally finite variety must be of bounded exponent. It is an open question as to “how finite” the groups of a locally finite variety are. Let us specify one interpretation of this vague question as Question IV.6. Is there a locally finite variety @ containing an infnite simple group? A positive answer to Question IV.3, would answer Question IV.6 negatively, since such a simple group S would be linear of finite degree over a suitable field F, and a theorem of Platonov [2] asserts that a linear group S belonging to a variety different from the variety of all groups must be soluble by-finite. Thus the simple group S is finite. Let us give another twist to the question: “how big” (or how intricate) is an infinite, locally finite, simple group? Question IV.7. If G is an infinite, locally$nite, simple group, is the variety of all groups the only variety of groups containing G?
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An introduction to the theory of varieties of groups can be found Hanna Neumann [I]. The importance of locally finite varieties for this theory and questions related to this section are discussed in KovBcs [3]. If some locally finite, simple group G is a limit of a (very) classically parametrized family of finite simple groups, by 4.7 there are two possibilities, either G is large (enormous) - and apparently, in this case, we have no means to make any further structural statement - or G is linear, and then by 4.6 almost all the simple groups of such an approximating sequence appear indeed as subgroups of G. These subgroups have, of course, bounded rank parameter. It is essentially this situation that we shall study in the next section, although the axiomatic approach that we choose will look quite different.
Section B. An embedding theorem and concrete examples In this section we study groups G that have a local system consisting of finite subgroups all of the same type. The desired conclusion is that then the group G itself is also of this type. For example, a group having a local system consisting of subgroups isomorphic to PSL(2, q), for varying q, is necessarily isomorphic to PSL(2, F ) for some locally finite field F. In the following account locally finite fields and finite simple groups will appear only at the very end. Instead, we give an axiomatic treatment of our main results, the embedding theorem 4.10 and the identification theorem 4.13, and only then indicate how certain families of (possibly twisted) Chevalley groups fit into this pattern. However, it may be helpful for the reader to bear in mind the particular example X = PSL(2, ) and Y = PGL(2, ) in all that follows and to check the progress made in terms of this example. Simply to point out that our axiomatic set-up has a much wider range than will be exploited here, we give two examples where the groups involved are soluble (instead of simple).
Definition. An index category on the set M is a category U whose objects are subsets of M such that 1 M = u A s e A (abusing notation, we denote the set of objects of V also by U), and 2 if A , B E V, then the set %(A, B ) of morphisms from A to B is empty if A $ Band it consists only of the canonical embedding eAB:A --f B otherwise. The objects of an index category V will be called indices.
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Definition. Ajinite (group) functor on the index category $? is a functor F from V to the category of groups such that 1 the group F(A) is finite, if A is a finite object of V, and 2 the map F(e,,) is a monomorphism, if A and B are finite objects of V with A c 3. The preceding two conditions are satisfied by any group-valued functor on the index category V, if V does not have any finite objects. It is part of the usual definition of functor that if A , B and C are indices of the index category V with A s B c C, then F(e,B)F(eBc) = F(e,c).
We shall not need this coherence property in this section. However, since in our applications F really will be a functor in the usual sence, it does not seem worthwhile introducing a new term. The reader may include or exclude this axiom according to his taste. We shall say that the index category Vl on the set M , is wider than the index category V on the set M , if M G MI and V E V, . We shall say that the group-valued functor Y on V, is wider than the functor X on V, if the index category V, is wider than the index category V, if for every finite index B E V the group Y ( B ) contains a subgroup isomorphic to X(B), and if for each finite index A E V with A s B every subgroup S N X ( A ) of Y ( B ) is contained in a subgroup T N Y ( A ) of Y ( B ) . Definition. A pair of functors X , Y on the index categories V, %,, respectively, into the category of groups is coupled if Y is wider than X and 1 whenever A E V and B E V, are finite indices, then the number of conjugacy classes of subgroups of Y ( B )that are isomorphic to Y ( A ) is bounded by an integer r,, independent of B; 2 whenever A€%' and B E V ~ are finite indices, then any subgroup U of Y ( B ) that is isomorphic to X ( A ) satisfies IN,(,,U : UI 5 ,S for some integer s, , independent of B. If the functor X is coupled to X,then we shall call X a self-coupled functor. Observe that if X and Y are coupled functors on the index categories V, Vl , respectively, then the functor Ylo is self-coupled. One has only to show that for every finite index A E V there exists an integer tA such that for every finite index B E V and for every subgroup U N Y ( A ) of Y ( B ) one has INY(,)U : Ul S t A . To see this, let kA denote the number of subgroups V of U that are isomorphic to X(A). With N = N I - ( ~ ) Uthe , Frattini argument yields
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IN : UI 2 max IN, V : Vlk, V
[CH.
4, 5
B
5 max INY(B)V : Vlk, 5 s A k A . V
It seems unlikely that in general the functor X will also be self-coupled on %,' but it clearly will be in the important special cases where for all finite indices A , 3 E 9 the group Y(B) contains a unique subgroup isomorphic to X(B) and the index of this subgroup in Y(B) is bounded over all B for which Y(B) contains a subgroup isomorphic to Y(A). Notice that the functors X = PSL(n, ) and Y = PGL(n, ) meet these conditions for the index family of all locally finite fields. 4.9 Lemma. Let X and Y be coupled functors on the index categories 9
and V l , respectively, and suppose that the group H has a local system A of Jinite subgroups such that for each subgroup L E A there exists a j n i t e index F = F L E Vl with L N Y(F). If A E V and B E V are Jinite objects and if S is afixed subgroup of H isomorphic to X ( A ) then the number of subgroups of H which contain S and are isomorphic to Y(B) is bounded by a function of A and B only. Proof. Choose (if possible) subgroups S and T of H satisfying S C T, S N X ( A ) and T N Y(B). In the local system A of H there are arbitrarily large subgroups L containing T such that L N Y(F) for some finite index FE . We first prove 4.9 for L (and the pair A , B) instead of for H. Let a = u ( T ) denote the number of subgroups of L that are conjugate to T i n L and contain S , dA,Bthe number o f subgroups of T conjugate to S in L, and eA,B the number of subgroups of T isomorphic to S. Then the number of conjugates of S in L is Thus
IL : NLSI
= dA,BIL
: NL T1a-l.
U l L NLSI = d , , ~ l L : NL TI 5 eA,BlL : N, TI. Now in T the set of the eA,Bsubgroups of Twhich are isomorphic to S splits into n conjugacy classes, say, each containing at most IT : SI elements. Hence eA,B nlT : SI. Also since the functors X and Y are coupled, one has
IL NLSI = IL : SI/INLS : SI 2 sA'IL SI. Putting these inequalities together, we obtain a * sA1lL : SI
that is:
alL : NLSl 5 e,,,IL
: NLTI
- n * I T :Sl IL :NLTI I
a = a(T) 5 n
S nlL :Sly
.sA.
Again, since the functors X and Y are coupled, the number of conjugacy
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classes of subgroups of L that are isomorphic to T N Y ( B ) is bounded by rB. Hence the number of subgroups of L that contain S and are isomorphic to T is at most n'rB'sA. Note that this bound is independent of the subgroup L E A of H. Thus the number of subgroups of H that contain S and are isomorphic to T is also at most n.r,.s,. I The following embedding theorem is technically the central result of the whole section.
4.10 Theorem. Let X and Y be finite coupled functors defined on the index categories $? and Vl , respectively, and suppose that G and H a r e groups having local systems .Z and A , respectively, such that for each S E C there exists afinite index E, E %' with S ?: X(Es) and for each L E A there exists afinite index FL E %, with L N Y(F,). rffor every S in C there exists an L in A such that Es E FL, then the group G is isomorphic to some subgroup of the group H. Note that, since X and Y are finite functors, each S and each L is finite; thus the groups G and H in 4.10 are assumed to be locally finite. Further the condition Es E FL implies that some subgroup of L is isomorphic to S ; for since Y is wider than X, the group Y(Es) contains a subgroup isomorphic to X(Es) N S, and Y(eEsFr,) embeds Y ( E s ) into Y(F,) 1: L . In general, there may be many embeddings of S into H the problem is to choose such embeddings in a coherent way as S ranges over Z. Proof. Let S be any element of C. By assumption S N X(E,) for some finite index E, E V, and there exists a subgroup L E A of H such that L N Y(F,) for some finite index FL E V, , Es c F,, and some subgroup of L is isomorphic to S. Let ( T , , . . ., T i , . . .} be a maximal set of pairwise non-conjugate subgroups of H that are isomorphic to Y(E,). The cardinality t of this set is finite, since the group H has the local system A of subgroups L l N Y(FL,) and the number of conjugacy classes in L , of subgroups isomorphic to Y (Es)is at most Y E S . For each of the groups T i let ( S i , , . . ., Sim} be a maximal set of pairwise non-conjugate subgroups of Ti that are isomorphic to S (m is finite since the group Tiis finite). Denote by lijthe set of all isomorphisms of S onto S i j . By 4.9 for each S , E C with S c S , , the set of all subgroups of H containing S , j and isomorphic to y(&,) is finite. For every element a E l i j let Zija(Sl)denote the set of all monomorphisms of S , into subgroups of H isomorphic to Y(E,,) that extend a. Clearly, for any such choice of
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i, j , u and S , E Z the set Zija(Sl)is finite. Conceivably it may be empty, but fortunately the following holds. (*) There exist i', j ' and u E ZiPj. such that for every S , E C with S E S1 the set Zii.j.a(S,) is non-empty. Suppose that S , ,S , are elements of .Z with S c S , c S, . Then the restriction of S , to S , maps the set Zija(S2)into the set Zija(Sl).For if /3 E Zija(S2) then S! E. T c H for some subgroup N Y(Es,) of H . Since Y is wider than X, there exists a subgroup T of Twhich contains S , and is isomorphic to Y(E,y,).Thus BlS, E Zija(S1). If (*) is false, then for every pair i, j and every u E Zij there exists some subgroup WijaE .Z of G such that S c Wijaand the set Zija( Wija)is empty. The local system Z contains a subgroup Z of G which contains all these (finitely many) subgroups Wijaof G. Then for every choice of i, j and cz the set Zija(Z)is empty (as resgiJm: Iija(Z)-+ Zija(Wija) = 0). But by assumption there exists some embedding q of Z into a subgroup isomorphic to Y ( E J of H . Again, since Y is wider than X there exists a subgroup of H which contains s'' and is isomorphic to Y(Es). Consequently, S" is conjugate in H to some S i j and so the set Zija(Z)is non-empty for some u E Zij(S). This completes the proof of (*). Pick one such triple (i', j ' , u ) for which (*) holds and write Z for Zi,,.a. The system {Z(S,), re,: : S , , S , E Z, S E S , E S,} is readily checked to be an inverse system of finite sets and any element of the non-empty inverse limit of this system defines an embedding of G into H . I We are interested in embedding groups of type X ( A ) into groups of type Y ( B ) for infinite indices. Since virtually all our definitions up to now refer only to finite elements of V and V, , we will have to make a definition that ensures that such groups X(A) and Y(A) have local systems of finite subgroups of the same type. Then one can apply 4.10.
Definition. Let V be an index category on the set M. We shall say that $7 is a normal index category if 1 denoting { E E V; E finite and contained in C} by &c for every C E V, then a C = U E E I , Efor every C E V, and b if C E V and E l , E, E b, then there exists a finite index E E &, satisfying El L: E, c E; 2 for any ascending chain 7 of finite elements of %? the subset UTE,Tof A4 belongs to V. A functor F from the index category V on the set M into the category of groups is normal if V is a normal index category, F is a finite functor, and if
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1 whenever A , 3 are finite elements of V such that F ( A ) is isomorphic to some subgroup of F(B) then A E B; 2 for every C E V the group F(C) has a local system C of subgroups such that S E C is isomorphic to some group F(Es) where Es E gC,and the partially ordered sets C and €, are order-isomorphic via the map S H &.
4.11 Lemma. Let F be a normal functor on the index category V on the set M . Then the set M is countable, and f o r every C in V the group F(C) is a countable, locally finite group. Proof. M is the union of all the finite elements of V by the definition of normality for .'%5 Since F is also normal, the finite elements of %' are in one-to-one correspondence with non-isomorphic finite groups. There exist but countably many isomorphism types of finite groups and hence M is countable. The group F(C) has a local system Z of finite subgroups such that C is order isomorphic to gC.Since €, is countable, F(C) is a countable, locally finite group. I We can now prove the rollowing basic corollary of 4.10.
4.1 2 Corollary. Let X and Y be normal coupled functors defined, respectively, on the index categories V and V, and suppose that the group G has a local system Z such that f o r each S in Z there exists a finite index Es E V with S N X(Es). Then there exists some N in %? such that G is isomorphic to a subgroup of Y ( N ) ,and in particular G is countable. Proof. Put N = usssEs; then N is a subset of the set M upon which %? is defined. By 4.11 the subset N is countable. The set {Es; S E C} of finite subsets of N is filtered above by inclusion since C is a local system for G and X is normal. Hence {Es; SEC} contains an ascending sequence the union of which is N. Since V is a normal index category, one has N E V and since the functor Y is wider than X, the group Y ( N ) is defined. Further Y ( N )has a local system A of subgroups isomorphic to Y ( E s)as S ranges over C. Therefore by 4.10 the group G is isomorphic to some subgroup of Y ( N ) . The latter group is countable by 4.11. I In the situation of 4.12, if E is any finite subset of N then E c Ui=1 Esi for some Siin Z. There exists S i n C satisfying Ur= Sic S. By the definition of normality Esi E Es. Hence S contains a subgroup isomorphic to X(E). Thus we may assume that the map S I+ Es maps C onto 8,.Further, the map is order preserving. (But clearly it need not be one-to-one, PSL(2,q') contains many subgroups isomorphic to PSL(2, q).) Call a group G an X-group of index N E V, if G has a local system C for
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which there exists a map S H Es of Z onto Q, satisfying S N X(Es) for all S E Z. Clearly X(N) is an X-group of index N. Are all the X-groups of index N isomorphic? If not, are all those X-groups of index N for which there exists a local system C of the above type with the map S H Es a bijection isomorphic (to X(N))? If X is self-coupled the answer is yes. 4.13 Theorem. If X is a normal self-coupled functor and if G is an X-group of index N then G is isomorphic to X(N). Further every injection of X(N) into itself is an automorphism.
Proof. Let H = X(N). By 4.10 there exists an embedding 4 of G into H. By the same result, with the roles of G and H interchanged, there exists an embedding $ of H into G. Hence for both parts of the theorem it suffices to prove that any embedding 8 of an X group K of index N into itself is onto. Let y denote the first uncountable ordinal. For every ordinal a < y we define inductively X-groups K, of index Nand embeddings A,+ ;K, -+ K,, such that K, = l&,
