[Grundlehren der mathematischen Wissenschaften] Finite Groups III Volume 243 ||

Grundlehren der mathematischen Wissenschaften 243 ASeries ofComprehensive Studies in Mathematics

Editors M. Artin S. S. ehern J. L. Doob A. Grothendieek E. Heinz F. Hirzebrueh L. Hrmander S. MaeLane W Magnus C. C. Moore J. K. Moser M. Nagata W Sehmidt D. S. Seott J. Tits B.L. van der Waerden

Managing Editors B. Eckmann S. R. S. Varadhan

B.Huppert N.Blackbum

Finite Groups 111

Springer-Verlag Berlin Heidelberg N ew York 1982

Bertram Huppert Mathematisches Institut der Universitt Saarstrae 21 D-6500 Mainz

Norman Blackbum Department ofMathematics The University GB-Manchester MI3 9 PL

Library ofCongress Cataloging in Publication Data. Huppert, Bertram, 1927-. Finite groups III. (Grundlehren der mathematischen Wissenschaften; 243). Bibliography: p. Inc1udes index. I. Finite groups. I. Blackbum, N. (Norman). 11. Tide. III. Series. QA 171.B 578.512'.22.81-2288.

ISBN-13: 978-3-642-67999-5 e-ISBN-13: 978-3-642-67997-1 DOI: 10.1007/ 978-3-642-67997-1

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Preface

Und dann erst kommt der "Ab - ge - sa.ng\' da./3 der nidlt kurz und nicht zu la.ng,

From "Die Meistersinger von Nrnberg", Richard Wagner

This final volume is concerned with some of the developments of the subject in the 1960's. In attempting to determine the simple groups, the first step was to settle the conjecture of Burnside that groups of odd order are soluble. The proof that this conjecture was correct is much too long and complicated for presentation in this text, but a number of ideas in the early stages of it led to a local theory of finite groups, so me aspects of which are discussed in Chapter X. Much of this discussion is a con-tinuation of the theory of the transfer (see Chapter IV), but we also introduce the generalized Fitting subgroup, which played a basic role in characterization theorems, that is, in descriptions of specific groups in terms of group-theoretical properties alone. One of the earliest and most important such characterizations was given for Zassenhaus groups; this is presented in Chapter XI. Characterizations in terms of the centralizer of an involution are of particular importance in view of the theorem of Brauer and Fowler. In Chapter XII, one such theorem is given, in which the Mathieu group 9J'l1l and PSL(3, 3) are characterized. This last chapter is mainly concerned with some aspects of multiply transitive permutation groups loosely connected with the Mathieu groups or with sharp n-fold transitivity, and several results from Chapter XI are used in it. The two last chapters are, however, independent of Chapter X.

Again we wish to acknowledge our indebtedness to the many colleagues who have assisted us with this work. In addition to those named in the preface to Volume II, thanks are due to George Glauber-man, who read an earlier version of Chapter X. The contributions of all have done a great deal to improve this volume, and it is with the greatest pleasure that we express our gratitude to them.

January, 1982 Bertram Huppert, Mainz Norman Blackburn, Manchester

Contents

Chapter X. Local Finite Group Theory ....................... . 1. Elementary Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Groups of Order Divisible by at Most Two Primes. . . . . . . .. 11 3. The J-Subgroup ...................................... 19 4. Conjugate p-Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28 5. Characteristic p-Functors.. . .. . . . . .. . . . .. . .. .. .. . . . . . . .. 35 6. Transfer Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 7. Maximal p-Factor Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52 8. Glauberman's K-Subgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57 9. Further Properties of J, ZJ and K . . . . . . . . . . . . . . . . . . . . . .. 69 10. The Product Theorem for J. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76 11. Fixed Point Free Automorphism Groups . . . . . . . . . . . . . . . .. 91 12. Local Methods and Cohomology ........................ 106 13. The Generalized Fitting Subgroup . . . . . . . . . . . . . . . . . . . . . .. 123 14. The Generalized p'-Core ............................... 131 15. Applications of the Generalized Fitting Subgroup .......... 142 16. Signalizer Functors and a Transitivity Theorem. . . . . . . . . . .. 148 Notes on Chapter X ....................................... 158

Chapter XI. Zassenhaus Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160 1. Elementary Theory of Zassenhaus Groups . . . . . . . . . . . . . . .. 161 2. Sharply Triply Transitive Permutation Groups ............ 172 3. The Suzuki Groups .................................... 182 4. Exceptional Characters ................................ 195 5. Characters of Zassenhaus Groups ....................... 205 6. Feit's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 219 7. Non-Regular Normal Subgroups of Multiply Transitive

Permutation Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227 8. Real Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 234 9. Zassenhaus Groups of Even Degree . . . . . . . . . . . . . . . . . . . . .. 246

VIII Contents

10. Zassenhaus Groups of Odd Degree and a Characterization of PGL(2, 2f ) ............................... . . . . . . . . . . .. 256

11. The Characterization of the Suzuki Groups ............... 264 12. Order Formulae ...................................... 286 13. Survey of Ree Groups .................................. 291 Notes on Chapter XI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295

Chapter XII. Multiply Transitive Permutation Groups ........... 296 1. The Mathieu Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 297 2. Transitive Extensions of Groups of Suzuki Type ........... 314 3. Sharply Multiply Transitive Permutation Groups . . . . . . . . .. 325 4. On the Existence of 6- and 7-Fold Transitive Permutation

Groups .............................................. 339 5. A Characterization of 9)111 and PSL(3, 3) .. . . . . . . . . . . . . . .. 341 6. Multiply Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . .. 366 7. Doubly Transitive Soluble Permutation Groups . . . . . . . . . .. 378 8. A Characterization of SL(2, 5) . . . . . . . . . . . . . . . . . . . . . . . . . .. 387 9. Sharply Doubly Transitive Permutation Groups ........... 413 10. Permutation Groups of Prime Degree .................... 425 Notes on Chapter XII ...................................... 438

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 439

Index of Names ........................................... 449

Index . ................................................... 451

Index of Symbols

JA'l3) 11 J('l3) 24 ZJ('l3) 24 K('l3), K('l3) 59 Rp(fj) 77 j('l3) 91

T~ 107 R(fj, f)) 107 C(f), (fj) 107 if>,,

Chapter X

Local Finite Group Theory

The word local is used in finite group-theory in relation to a fixed prime p; thus properties of p-subgroups or their normalisers, for example, are regarded as local. In the case of a soluble group, then, everything is local, but an insoluble group also has global aspects. Now the local behaviour influences the global, that is, there are theorems in which the hypothesis involves only p-subgroups and their normalisers, but the conclusion involves the whole group. This chapter is an introduction to theorems of this sort.

Some such theorems are already known from Chapter IV; for example, Burnside's transfer theorem, which asserts that if the centre of the normaliser of a Sylow p-subgroup 6 contains 6, then the whole group is p-nilpotent. This is proved by showing that the transfer into 6 is an epimorphism. An essential lemma (IV, 2.5) states that two 6-invariant subsets of 6 are conjugate in (f) if and only if they are conjugate in N(jj(6). This has many other applications, being a link between the global and local properties. More generally, the situation in which two subsets A, B of 6 are conjugate in (f) frequently arises; such sets A, Bare often described as fused, particularly when they are not conjugate in 6. In general, fusion can be reduced not to one but to a sequence of local transformations. This is the subject matter of 4, where the precise way in which A can be transformed into B is inves-tigated. It is shown that if Ag = B, then g = glg2 ... gn where gi nor-malises so me subgroup 'l3i of 6 and Ag,'" 9 ,_, ~ 'l3i' Moreover there are certain sets fF of subgroups of 6 for which the additional con-dition that 'l3i E fF may be imposed. These sets fF are called conjugation families.

Another theorem with a local hypothesis but a global conclusion is the theorem of Thompson (IV, 6.2) that, for p odd, if C(jj(Z(6)) and N(jj(Jo(6)) are p-nilpotent, so is (f). Here J o(6) denotes a characteristic subgroup of 6. Certain similarly defined characteristic subgroups are very useful for establishing non-simplicity criteria; this is shown in 2, where a character-free proof of the solubility of groups of order paqb is

2 X. Local Finite Group Theory

given. In 3, it is shown that there is such a eharaeteristie group ZJ(6) whieh is always normal in (fj whenever Op(fj) ~ Ccr.(Op(fj)) and (fj is p-stable. This ean be used to give another proof of the above theorem of Thompson; it was also used by BENDER [3J to simplify greatly a seetion of the proof of the solubility of groups of odd order. For all such applieations, eriteria for p-stability are of course required, such as those given in Chapter IX.

Now J(~) is defined for any p-group ~ by eertain rules. To analyse these, we eonsider first, in 5, complete1y general rules, supposing only that there is defined in each p-group ~ a subgroup W(~) and that whenever r:x is an isomorphism of ~ onto ~, W(~)r:x = W(~). Such a W is ealled a characteristic pjunctor. In order to study fusion, a eon-jugation family is defined in 5 corresponding to any characteristie p-funetor W. This enables us to prove, for example, that (fj and Ncr.(W(6)) have isomorphie maximal p-faetor groups if and only if the same is true in the normaliser of any non-identity p-subgroup (Theorem 7.3). By eombining this with some results about the transfer developed in 6, a eommutator eondition is obtained which implies that (fj and Ncr.(W(6)) have isomorphie maximal p-factor groups. In 8, two characteristie p-funetors K, Kare defined, and, completely within the eontext of p-groups, a complementary commutator condition is established (Theorem 8.10). Putting the two together, a theorem of Glauberman, which states that for p ~ 5, (fj, Ncr.(K(6)) and Ncr.(K.(6)) all have isomorphie maximal p-faetor groups, follows. Grn's second theorem (IV, 3.7) makes a similar assertion, but requires that (fj be p-normal. Glauberman's theorem, however, has no such hypothesis. Among its consequences are the fact that if (fj is not a p-group, there exists a Sylow subgroup 6 of (fj for which N(!i(6) > 6. In 9, it is shown that K, K could be used in place of Jo in the theorem of Thompson, and before this, it is shown that when every section of (fj is p-stable, K, K and, for p odd, ZJ have a property which is described as strongly controlling fusion: whenever A, Ag are contained in 6, there exists hE Ncr.(W(6)) such that ag = ah for all a E A.

In 10, we consider another property of J. If (f) is p-soluble, Op,(f) = 1 and 6 E Sp(f), then the equation

holds under many eireumstanees; eertainly if p > 3. This kind of factorization is of considerable importance and made its first appearance (implicitly) in Thompson's theorem. Conditions for its validity when p is 2 or 3 are found in 10, and in 11 these are applied to prove a

1. Elementary Lemmas 3

theorem on fixed point free automorphism groups. It is conjectured that if21 is a fixed point free group ofautomorphisms offfi and (1211,1 ffil) = 1, then ffi is soluble; in 11 this is proved when 21 is elementary Abelian.

Since ffi/ffi' is in duality with Hom(ffi, C X ) = H1 (ffi, C X ), where C X is regarded as a trivial Zffi-module, the transfer of ffi into a subgroup ~ gives rise to a homo mo rphi sm of Hl(~, CX) into H1(ffi, CX ). This is a special case of the corestriction homomorphism of Hn(~, M) into Hn(ffi, M) described in I, 16.17. It is shown in 12.8 that Hn(ffi, M) and Hn(N(fj(W(6)), M) have isomorphic Sylow p-subgroups if M is a trivial ffi-module and W is a characteristic p-functor which strongly controls fusion in ffi-Grn's second theorem is a special case of this. This is applied to the Schur multiplier of ffi in 12.17; if 6 E S p(ffi) and the dass of 6 is at most 1P, the Sylow p-subgroups of the Schur multipliers of ffi and of N~(6) are isomorphic.

In addition to the transfer, a number of results, which have be-come very familiar in finite group theory, are frequently used in proving these theorems; these indude the properties of the centralizers of the Fitting subgroup and 0p',p(ffi) and a number of other facts which are collected in 1. In 13 and 14, some ofthese results are generalized in such a way that solubility hypotheses are removed. In doing this, the role of the nilpotent group is taken by the quasinilpotent group (13.2) and that of the p'-group by the p*-group (14.2). It is shown, for example, that every group ffi has a unique maximal normal quasinil-potent subgroup F*(ffi) and that C(fj(F*(ffi)) ~ F(ffi); again, every group ffi has a generalized p'-core Op.(ffi), and if ~ is a p-subgroup of ffi, Op.(C(fj(~)) ~ Op.(ffi). Finally, in 16, another aspect oflocal properties is briefly considered; this involves the relationship between the various soluble p'-subgroups of a group ffi which are normalised by a fixed Abelian p-subgroup of G>.

1. Elementary Lemmas

In this chapter a number of elementary results will be used frequently. Some of these have already appeared in the previous chapter; the remainder are collected together in this section.

First, we establish a lemma for characteristic subgroups of p-groups analogous to the theorem (111, 7.3) that a maximal normal Abelian subgroup of a p-group is its own centralizer.


1.1 Lemma. Let ffi be a p-group and let 21 be a characteristic Abelian subgroup of ffi. Then there exists a characteristic subgroup ~ of ffi such that

(i) ~ ~ C(!j(~) = Z(~) ~ 21, and (ii) ~jZ(~) is an elementary Abelian subgroup of Z(ffijZ(~)).

In particular, the dass of ~ is at most 2.

Proof Let f!( be the set of characteristic subgroups X of ffi such that Z(X) ~ 21 and XjZ(X) is an elementary Abe1ian subgroup of Z(ffijZ(X)). Thus 21 E f!(. Let ~ be a maximal element of f!(. If ~ ~ C(!j(~), then ~ has all the required properties. Suppose then that ~ "* C(!j(~), that is,

Z(~) < C(!j(~). Let !ljZ(~) be the set of elements of order at most p in (C(!j(~)jZ(~)) n Z(ffijZ(~)). Thus !l ~ C(!j(~), !lP ~ Z(~) and [!l, ffi] ~ Z(~). By 111, 7.2, !l > Z(~), so !l $, ~. But !l~ E f!(, since

[!l~, ffi] = [!l, ffi] [~, ffi] ~ Z(~) and (!l~)P = !lP~P ~ Z(~). This contradicts the maximality of ~. q.e.d.

1.2 Lemma. Suppose that ffi is a p-group and that r:t.. is an automorphism of ffi of order prime to p. If there exists a subgroup ~ of ffi for which

~C(!j(~) ~ C(!j(r:t..), then r:t.. is the identity automorphism.

Proof This is proved by induction on I ffi I. There is nothing to prove if ~C(!j(~) = ffi. Otherwise, there exists a maximal r:t..-invariant proper subgroup rol of ffi such that rol ~ ~C(!j(~). Since N(!j(rol) is r:t..-invariant, N(!j(rol) = ffi, by 111, 2.3. Thus rol


Proof If a E 21, let p(a) denote the automorphism of m induced by a. Then p is a homomorphism of 21 into the group of automorphisms of m. Let St = ker p, 210 = im p, ~o = p(~). Then there is an isomorphism between 21/St and 210 in which ~/St and ~o correspond. Now

~o = {I E 210 , (xmi) = xm i for all XE mi - 1 (i = 1, ... , k)}. By IX, 7.3, ~o is anormal p-subgroup of 210 . Hence ~


.0 ~ ~ and C5 (.o) = {(u, v)lu E Z(~), V E C!I(~)}' Since [Sl, C!I(~)] = 1, 'oCI)('o) ~ C5(ct). By 1.2, ct is the identity automorphism. Thus X E C(!j(~) for every element x of Sl of order prime to p. Since Sl = OP(Sl), it follows that ~ centralizes R q.e.d.

1.6 Lemma. If ~ is a p-subgroup of the p-constrained group (fj,

Op,(N(!j(~)) ~ Op,(fj).

Proof Let (fj = (fjjOp,(fj), ~ = ~Op,(fj)jOp,(fj), 91 = N16(~)' Sl = Op,(91L Then Sl = OP(Sl) since ft is aJ'-group, and [~, Sl] = 1 since ~, ft are normal subgroups of 91 of coprime orders. Also

~~ ~ N16(Op(fj)) and

[Sl, CO,(16)(~)] ~ Op(fj) (\ [ft, 91] ~ Op(fj) (\ Sl = 1.

Hence by 1.5, [ft, Op(fjU = 1. But~p,(fj) = 1 and (fj is p-constrained (VII, 13.3), so ft ~ 0i(fj). Since Sl is a p'-group, ft = 1. Now by IX, 6.11, 91 = N(!j(~)Op,(fj)/Op,(fj), so 91 ~ N(!j(~)j9Jl, where 9Jl = N(!j(~) (\ Op,(fj) is anormal p'-subgroup of N(!j(~)' Since Op,(91) = ft = 1, it follows that Op,(N(!j(~)) = 9Jl ~ Op,(fj). q.e.d.

1.7 CoroUary. Suppose that ~ is a p-subgroup ofthe p-constrained group (fj. If~ contains every p-element ofC(!j(~), then

Proof By hypothesis, Z(~) is the only Sylow p-subgroup of CQ;(~)' Since Z(~) ~ Z(CQ;(~, CQ;(~) = Z(~) x 91 for some 91, by IV, 2.6. Thus 91 = 0p,(C(!j(~ ~ Op,(N(!j(~)) ~ Op,(fj) by 1.6, and 91 =

CQ;(~) (\ Op,(fj). q.e.d.

We now turn to a generalization of part ofIX, 6.11.

1.8 Lemma. Suppose that m: is a group of operators on a group (fj and that either m: or (fj is soluble.

a) If 91 is anormal m:-invariant subgroup of (fj and (1911, Im:i) = 1, then

CQ;/91(m:) = C(!j (m:) 91j91.

b) If(i[(fj, m:]I, Im:!) = 1, then (fj = C(!j(m:)[(fj, m:J.


Proof a) Obviously Co;(~)91/91 ~ Cijj/IJl(~)' If x91 E Cijj/IJl{~)' then by I, 18.6, there exists y E Cijj(~) such that y91 = x91. Hence X E Cijj(~)91.

b) By III, 1.6b), Wl = [m, ~J is a normal ~-invariant subgroup of m. Clearly CijjJ!(~) = m/Wl. But by a),

Cijjtl1Jl(~) = Cijj(~)WlIWl.

Thus m = Cijj(~)Wl = Cijj(~)[m, ~J. q.e.d.

Note that on account of the solubility of groups of odd order, the hypothesis that either ~ or (fj be soluble is unnecessary.

1.9 Lemma. Suppose that m is a p'-group and ~ is an Abelian p-group of operators on m. Then

If also ~ is not cyclic, (fj = (Cijj(x)lx E ~, x =1= 1).

Proof The first assertion is proved by induction on 1 (fj I. If q is any prime other than p, (fj possesses an ~-invariant Sylow q-subgroup ,0, by IX, 1.11, and (fj is generated by all such Sylow subgroups. Thus it suffices to prove that ,0 =


1.10 Lemma. a) 1f 6 E Sp(f) and m: is a maximal normal Abelian sub-group of 6, C(lj(m:) = m: x '!l for some p' -subgroup '!l.

b) 1f p is odd, 6 E Sp(f) and m: is a maximal normal elementary Abelian subgroup of 6, every element of order p in C(lj(m:) lies in m:.

e) Suppose that m: is a p-subgroup of (f) and every element of order p in C(lj(m:) lies in m:. 1f Sl ::;; (f), m: ::;; N(lj(Sl) and Sl n m: = 1, Sl is a p'-group.

Proof a) See IX, 5.9. b) Sinee 6 E Sp(N(lj(m: and C(lj(m:)


p'-subgroup !l. Thus !l is a characteristic subgroup of Co;(21) and Co;(21)


satisfies the conditions of the theorem. Since ffi is a minimal counter-example, it follows that Sl91/91 ~ Op,(ffi/91) = 1 and Sl ~ 91 = Op,(ffi).

Let [I' be the set of subgroups X of Op(ffi) such that ~m ~ N(D(X) but Sl $ C(D(X). Since Op,(ffi) = 1, ffi is p-constrained and Sl is a non-identity p'-group, Op(ffi) E [1'. Thus [I' is non-empty. Let ~ be a minimal element of [1'. Thus

b) ~ is a subgroup of Op(ffi), m:Sl ~ N(D(~), Sl $ C(D(~)' We prove next that c) the dass of ~ is at most 2 (cf. 111, 13.5). Indeed, by minimality of~, Sl ~ C(D(~'). Thus [~',~, Sl] = [Sl, ~',~]

= 1. By 111, 1.10, [~, Sl, ~'] = 1. Now since Sl $ C~~), there exists Y E Sl such that y induces a non-trivial automorphism 1'/ on ~. In fact 1'/ is not of order apower of p, since Sl is a p'-group. But Sl ~ N(D([~, Sl]), so 1'/ leaves [~, Sl] fixed and induces the identity automorphism on

N[~, SlJ. It follows from 1,4.4 that 1'/ induces a non-identity auto-morphism on [~, SlJ. Thus Sl $ C(D([~, Sl]). But m:Sl ~ N(D([~, Sl]), so

[~, Sl] E [1'. By minimality of~, [~, Sl] = ~. Thus

[5, ~'] = [~, Sl, ~'] = 1 and the dass of ~ is at most 2.

Since I~I is odd, each element of ~ has a unique square root. Thus by c) and VIII, 9.16, there exists an addition on 5 with respect to which

~ is an Abelian group and m:Sl is a group of operators on . By 111, 13.4b),

here [f), SlJ is understood in the sense of the additive structure of f) and C~(Sl) = Cl,i(Sl). Thus [, Sl] is an m:-invariant subgroup of , and by b), [, Sl] # O. Since [, Sl] and 21 are p-groups, there exists an element U of order p in [, Sl] such that U E C~(m:) = Cl,i(m:). But then, by hypo thesis, U E 21. Thus if g E Sl and the commutator [u, g] is now understood in the ordinary sense,

Thus U E Cl,i(Sl) = CS(Sl) and

U E C~(Sl) n [, Sl] = 0,

a contradiction. q.e.d.

2. Groups of Order Divisible by at Most Two Primes 11

2. Groups of Order Divisible by at Most Two Primes

The main aim of this section is a proof of Burnside's theorem on the solubility of groups (f) for which I (f) I is divisible by at most two primes. This was proved in V, 7.3, following Burnside's original method (1904), which uses the theory of group characters. In 1961, Thompson gave a proof in the case when I (f) I is odd, which made no use of character-theory. About 10 years later a number of improvements were made by various authors, and the proof given here is a culmination of these.

We begin with a rather technicallemma.

2.1 Lemma. Suppose that (f) is a p-soluble group of odd order, Op,(f)) = 1 and the Sylow p-subgroups of(f)/Op,p,(f)) are cyclic. Suppose also that if 3 = .Ql(Z(Op(ffi))), CIfj(3) = Op(ffi). Let 6 be a Sylow p-subgroup of ffi and let ~ be an elementary Abelian subgroup of 6 of order as large as possible. Then ~ ~ Op(f)).

Proof Let ~ = Op(f)),9l = Op,p,(ffi). Since 69l/9l is cyclic and ~9l/9l is elementary Abelian, 1~9l/9l1 ~ p. Since ~ n 9l = ~ n ~,it follows that I~: ~ n ~I ~ p. But (~ n ~)3 is an elementary Abelian subgroup of 6, since 3 ~ Z(~). Hence I(~ n ~)31 ~ I~I and

13: ~ n 31 ~ I~: ~ n ~I ~ p. But ~ n 3 ~ Z(~3), so 3/(~ n 3) is anormal subgroup of ~3/(~ n 3) of order at most p. Thus [3, ~] ~ ~ n 3 and [3, ~, ~] = 1. But by IX, 7.4, ffi is p-stable, since Iffil is odd; (since ISA(2, p)1 is even, this also follows from IX, 7.10). Since CIfj(3) = ~, Op(NIfj(3)/CIfj(3)) = Op(ffi/Op(ffi)) = 1, so it follows that ~ ~ ~. q.e.d.

To use this, we make the following definition.

2.2 Defmition. For any p-group 1, let Je(l) be thesubgroup of 1 gen-erated by all elementary Abelian subgroups ~ which are of order as large as possible. Thus Je(l) is a characteristic subgroup of 1.

2.3 Lemma. Suppose that ffi is a p-soluble group of odd order, 0p,(ffi) = 1 and the Sylow p-subgroups of (f)/Op,p,(f)) are cyclic. Suppose also that if 3 = .Ql(Z(Op(ffi))), CIfj(3) = Op(f)). Then if 6 E Sp(f)), JA6)


Proof Let ~ = Op(fj). It follows from 2.1 that JA6) ::; ~. Hence the elementary Abelian subgroups of ~ of greatest possible order are the same as those of 6. Thus Je(6) = JA~). Hence Je(6) is a characteristic subgroup of ~ and hence of (fj. q.e.d.

The significance of the conclusion of 2.3 lies in the fact that anormal subgroup of (fj is constructed from 6 alone. In applications, it leads to a situation in which the following version of I, 8.8 can be applied.

2.4 Lemma. Suppose that ~ is a p-subgroup of a group (fj, U is a charac-teristic subgroup of~ and ~ E Sp(N(D(U)). Then ~ E Sp(fj).

Proof If ~ < 6 E Sp(fj), then ~o = N6(~) > ~,by 1,8.8. But ~

2. Groups of Order Divisible by at Most Two Primes

For (fj = mm: and mo


Thus there exists e; E Sp(m) such that Wl n Z(.o) ::::; N(fj(3), where

3 = q. Then p does not divide I Aut 0iWl) I , by 1,4.6. Thus if 'l3 E Sp(Wl), 'l3 centralizes Oq(Wl). Since 0iWl) ::::; 'l3 and tY = Op(Wl) x Oq(Wl), it follows that Z('l3) ::::; C!\JlOp(Wl)Oq(Wl)) = C!\JltY). By III, 4.2, ~(m ::::; tY. Thus Z('l3) ::::; tY and Z('l3) ::::; Op(Wl). Since Op(Wl) is cyclic, Z('l3) is a characteristic subgroup of Op(Wl) and hence of Wl. By 2.9a), it follows that Wl = N(fj(Z('l3)), since Wl is a maximal subgroup of m. By 2.4, 'l3 E Sp(m). By 2.9c), Oq(Wl) = 1. Thus tY is a p-group, contrary to hypothesis.


Now suppose that 02(IDl) is non-Abelian, and let p be the odd prime divisor of Iffil. Suppose.o E S2(IDl). Then .o/Co(Op(IDl is isomorphie to a group of automorphisms of the eyelie group Op(IDl) and is therefore Abelian. Henee Co(Op(IDl ~ .0'. Thus Z(.o) (') .0' ~ C!\JI(Op(IDl)02(IDl = C~ty) ~ ty, as in the previous ease, and Z(.o) (') .0' ~ 02(IDl). But sinee .0 ~ 02(IDl), .0' =F 1. Thus Z(.o) (') .0' =F 1. Let U be a subgroup of Z(.o) (') .0' of order 2. Since 02(IDl) has only one subgroup of order 2, U is a eharaeteristie subgroup both of Z(.o) (') .0' and of 02(IDl). Henee U is a eharaeteristie subgroup of .0 and of IDl. Thus IDl = NQj(U) and by 2.4, .0 E S2(ffi). By 2.ge), Op(IDl) = 1 and ty is a 2-group, eontrary to hypothesis.

b) Suppose that IDl is a maximal subgroup of ffi and 3 = Z(F(IDl)). Suppose that F(IDl) is not of prime-power order. If i! is a maximal sub-group of ffi and i! eontains 3, then F(i!) is not of prime-power order and F(i!) ~ F(IDl).

Denote the prime divisors of Iffil by p, q. Then 3 = 3p x 3q , where 3 p E Sp(3), 3 q E Si3). By hypothesis, 3 p =F 1,3q =F 1. Henee, sinee IDl is a maximal subgroup of ffi, IDl = NQj(3p) = NQj(3q). Thus N1\(3q) ~ IDl. But3p


Nm('l3) :::;; 9Jl and, by 2.4, 'l3 E Sp(


normalised by m:, .st/ is non-empty. Choose an element (m:, in) of .st/ for which ! C!Il(m:)! is maximal.

Since in ~ Ql(Z(Oq(C(9(in)))), it follows from the maximality of in that in = Ql(Z(OiC(9(in)))). By 2.12c), C(9(in) is a q-group. Thus C!Il(m:) < in. But by 1.9,

Thus there exists XE m: such that x i: 1 and C!Il(x) > C!Il(m:). Put U = C!Il(x) and m: = p. In the former case, the Frattini argument gives


and 9Jl/Op,p,(9Jl) ~ N!IJI(O)/~ for some ~, so the assertion is c1ear. Suppose then that q > p. In this case, 2.13 shows that the Sylow q-subgroups of NfD (Op(9Jl)) = 9Jl are cyc1ic. Hence 9t = Op,p,(9Jl)/Op(9Jl) is a cyc1ic q-group. Since q is odd, it follows that Aut 9t is cyc1ic. But by IX, 1.4, 9Jl/Op,p,(9Jl) is isomorphie to a subgroup of Aut 9\, so the assertion is c1ear in this ca se also.

Put ~ = p(9Jl), 3 = Q 1 (Z(~)). Since ~

3. The J-Subgroup 19

2.17 Remark. Burnside also proved that if Iffil = paqb, where p, q are primes and pa > qb, then Op(ffi) #- 1 except possibly when a) p = 2 and q is a Fermat prime, or b) q = 2 and p is a Mersenne prime.

GLAUBERMAN [8] proved the following analogous result. Suppose that Iffil = paqb, where p, q are primes. If ffi has a p-subgroup of dass at most 2 of order greater than that of any q-subgroup of dass at most 2, Op(ffi) #-1.

3. The J-Subgroup

A crucial point in the proof in 2 was the use of the subgroup Je(l). A similarly defined subgroup was used in IV, 6; it was denoted there by J(l). In this chapter it will be denoted by Jo(l), and we shall use J(l) for yet another subgroup defined in a similar way. The properties of it depend on areplacement theorem of Thompson (3.3). We begin with the following lemma.

3.1 Lemma. Suppose that ~ is a p-group and 21: is an Abelian subgroup of ~. Let x be an element of ~ for which IDl = [x, 21:] is Abelian, and put

a) 23 is an Abelian subgroup of~, and 123 n 911 ;?: 121: n 911 for any 91


If 1(91 n m)c


It will be noted that there does not necessarily exist anormal Abelian subgroup 21 of ~ such that no Abelian subgroup of ~ is of order greater than 1211 (lU, Aufg. 31).

3.3 Theorem (Replacement theorem). Suppose that ~ is a p-group, .0

22 x. Local Finite Group Theory

since [x, b] E U and [x, b, a] E Z(U). Hence x2 commutes with [x, a, b J. Since p is odd, it follows that x commutes with [x, a, b J. Therefore

[x,a,b] = [x,a,b]X = [x, a, bX ] = [x, a, bEb, x]] = [[x, a], [b, x]] [x, a, bJlb.xI = [[x,a],[b,x]][x,a,bJ.

It follows that [x, a] and [x, b] commute. Let ~ = IDlC9I(IDl). Then 21 n 2lx ~ 21 n Z(U) ~ Ct((IDl) ~ ~. But

since 21 n .0 ~ .0', x E N~ (21 n .0) and

Hence 21 n .0 ~ ~ n .0. Since x ~ 91, x ~ N~(21), so [21, x] f;, 21. Thus IDl f;, 21 n .Q, although IDl ~ ~ n .0. Hence 21 n .0 < ~ n .0, and a) is proved. b) follows from 3.1. Since IDl ~ U, ~ ~ U ~ N~(21). Since IDl ~ .0 and Ct((IDl) ~ 21 n ~, ~ ~ .0 (21 n ~). If p is odd and 21 is elementary Abelian, then U is of exponent p, since U is of dass at most 2 and is therefore regular (111, 10.2). Since ~ ~ U, it follows that ~ is elementary Abelian. q.e.d.

We prove the following related theorem.

3.4 Theorem (THOMPSON [7]). Suppose that pr > 2, ~ is a p-group and I~: ~,~p'l > 1.0: .o'.op'l Jor every proper subgroup .0 < ~. Then the class of ~ is at most 2. (We remark that I~: ~'~PI = I~: 1.0: .Q'.QP'j. Since.QP'.Q' ~ ~', this gives a contradiction.

Let IDl = [~', ~J. Since ~' f, Z(~), IDl =F 1. Hence there exists a normal subgroup IDlo of ~ such that I IDlo I = p and IDlo ~ IDl. Since

~/IDlo satisfies the hypotheses of the theorem but is not a counterexample, [~'/IDlo, ~/IDlo] = 1; thus IDl = [~', ~] ~ IDlo, IDl = IDlo and IIDlI = p. Hence IDl ~ Z(~), and by 111, 2.12, ~" ~ [~', ~, ~] = 1. Thus ~' is Abelian. Let [ = C~(~'); then ~' ~ [ and [


~/[ is elementary Abelian. For if x E ~ and y E ~', we have [x, y] E 9R ~ Z(~), so by III, 1.3, [xP, y] = [x, y] P = 1 and xP E c,,(~') = [. Similarly, if x E ~ and y E ~', [x, yP] = [x, y]P = 1, so ~'P ~ Z(~). Hence if 3 = Z(~) n ~', ~'/3 is elementary Abelian.

By VIII, 6.1, there exists a mapping y of (~/[) x (~'/3) into 9R, such that

(x[, Y3)Y = [x, y] (x E~, Y E ~');

also y is bilinear over Z. Since 9R, ~/[ and ~'/3 are elementary Abe1ian, 9R, ~/[ and ~'/3 can be regarded as vector spaces over GF(p), and y is bilinear over GF(p). y is non-singular, on account of the definitions of [and 3. It follows that I~/[I = 1~'/31. Hence I~: ~'~P'I = I~: ~'I = 1[:31

Now by III, 1.10b),

[[I, ~] = [[, [, ~] ~ [[, ~,[] ~ [~', c,,(~')] = 1. Hence [' ~ Z(~). We show also that [P' ~ Z(~). For suppose that x E [ and y E ~. Since [x, y] and x commute,

(x[x, y])P' = xP'[x, yy.

Thus

But also yP' E ~P' ~ ~', so [x, yP'] = 1. Hence

Since [x, y, y-l] E 9R ~ Z(~), it follows from III, 1.3b) that ([x, y]y-l)P' = [x, y]P'y-p'[y-l, [x, y]](f).

Since pr > 2 and 9R is of exponent p, it follows that

and [x, y]P' = 1. Hence [xP', y] = 1 and [P' ~ Z(~). Thus ['[P' ~ Z(~) n ~' = 3. Thus


eontrary to hypothesis. q.e.d.

3.5 CoroUary. Suppose that p is odd and that 21 is a maximal element of the set of normal elementary Abelian subgroups of the p-group ~. If 1211 = pli, ~ can be generated by!n(n + 1) elements.

Proof Let [ = C~(21). We prove by induetion on II that if :s;; [, then I: cP() 1 :s;; pli. First of all, ifthe dass of is at most 2, then is regular, sinee p is odd (111, 10.2). Henee the elements of of order at most p form a subgroup l of order I: PI (111,10.5 and 10.7). Sinee :s;; [ = C~(21), l :s;; 21 by 111, 12.1 sinee p is odd. Thus

as required. Seeondly, if the dass of is greater than 2, then by 3.4, there exists a subgroup 2 < sueh that

Sinee I2: cP(2)1 :s;; pli by the inductive hypothesis, we again have I: cP() 1 :s;; pli.

Taking = [, 1[: cP([)1 :s;; pli, so [ean be generated by n elements. By I, 4.3, ~/[ is isomorphie to a group of automorphisms of 21. Sinee the order of the group of all automorphisms of 21 is p11l(II-l)q for some integer q with (q, p) = 1, ~/[ ean be generated by!n(n - 1) elements. Thus ~ ean be generated by!n(n - 1) + n = !n(n + 1) elements.

q.e.d.

3.6 Notation. a) For any p-group ~, we define d(~) to be the set of Abelian subgroups 21 of ~ of maximal order.

It is dear that if 21 E d(~), 21 is a maximal Abelian subgroup of~; that is, C~(21) = 21. In particular 21 ~ Z(~).

The automorphisms of ~ perrnute the elements of d(~). b) For any p-group ~, write

J(~) =


So far the theorems in this section have all been about p-groups. We now use them to prove an important theorem about p-stable groups.

3.7 Lemma. Suppose that p is odd and that (1) is a p-stable group. If 6 E Sp((1)), ZJ(6) n Op((1))


(6) By (2), 0' ::; 3 ::; ID:, and by (5),0 f. Ns(ID:). Hence by 3.3, there exists an Abelian subgroup ~ of 6 such that I~I ~ I~I, [~, ID:, ~] = 1 and ~ n 0 > ~ n O. Since ~ is Abelian and I~I ~ IID:I, ~ E d(6). Since ~ n 0 > ID: n 0, ~ ~ d. Thus ~ ::; 91 Hence d(6 n 91) ~ d(6). Also 0 n 3 centralizes every element of d(6) and hence every element of d(6 n 91). Since 0 n 3 ::; ~ ::; 6 n 91, it follows that o n 3 ::; Z(,3) = 1. Now if g E ffi, there exist a E 91 and b E N(fj(l) such that g = ab, by (3); thus (0 n 3Y = (0 n 3)ab = (0 n 3t, since 91 ::; N(fj(O n 3). Hence (0 n 3)g ::; 1b = 1 for all g E ffi. By (1), o ::; 1. Since ~ E d(6 n 91), 1 ::; ~. Hence 0 ::; ~. Since [~, ~, ~] = 1, [0, m:, ID:] = 1. This contradicts (5). q.e.d.

3.8 Theorem (GLAUBERMAN [3J). Suppose that p is an odd prime and that ffi is a group which is p-stable and p-constrained. If 6 E Sp(ffi),

ZJ(6)Op,(ffi)


Proof By IX, 1.4, (f) is p-constrained, and by IX, 7.4 or 7.10, (f) is p-stable. The assertions thus follow from 3.8. q.e.d.

3.10 Remark. As an example of the use of 3.8, we show how the proof of 2.8 can be simplified by using it. Suppose that we have reached 2.12a) in this proof; we may then proceed as follows.

(1) Every maximal subgroup Wl of (f) contains a Sylow subgroup of (f).

By 2.10, Op'(Wl) = 1 for some p. If ~ E Sp(Wl), ZJ(~)

28 X, Local Finite Group Theory

4. Conjugate p-Subgroups

Ir A s;;; '131 ~ 6 E Sp(ij) and gl E N(fj('131)' then Ag, s;;; '13~, = '131 ~ 6. Repeating this, if Ag, s;;; '132 ~ 6 and g2 E N(fj('132), then Ag,g2 S;;; 6. This may be repeated further, and we shall see that all Ag which are contained in 6 may be obtained in this way. Moreover, so me restrictions may be placed on '131' '132' . , .. We therefore make the following definition.

4.1 Definition. Suppose that 6 E Sp(ij). A conjugation family for 6 is a set $' of subgroups of 6 such that whenever A is a non-empty sub set of 6, g is an element of ij) and Ag s;;; 6, there exist subgroups '131' ... , '13n in $' and elements gi E N(fj('13i) (i = 1, ... , n) such that g = gl ... gn and

Ag,"'g,-, c m, (I' = 1 n) - +" , ... , .

Since gi E N(fj('13i)' it follows at once that Ag,'''g, c m (I' - 1 n)' 1_ +'i -, ... , ,

in particular, A s;;; '131 and Ag S;;; '13n. Note that for such a family, ij) =

4. Conjugate p-Subgroups 29

Thus ~ is an equivalence relation on S(~), and, if ~ < 6, each equiv-alence ~class is a union of left co sets of N6i( 6). 4.3 Lemma. Suppose that 6 E Sp((fj) and ~ ~ 6.

a) If ~g ~ 6, S(~g) = g-lS(~), and for a, b in S(~g), a ~ b if and ~.

only if ga ~ gb. b) Giv!n a E S(~), there exists b E S(~) and c E N6i(~) such that NS(~b) E Sp(N6i(~b)) and a ~ b ~ c. Thus each equivalence dass of ~ contains an element of N(jj(~).

c) IfS is the equivalence dass of ~ containing 1, thenfor any a E N6i(~)' the equivalence dass containing a is aSo

d) If a E S(~), there exist c E N6i(~) and d E S(~) such that d ~ 1 and a = cd.

Proof a) This is obvious. b) If ~ = 6, take b = c = a. Suppose that ~ < 6. Then since ~ ~ 6 a- ' , the p-subgroup Ns.- (~) is contained in a Sylow p-subgroup 6 1 of N6i(~)' and 6 1 ~ 6 b- ' for some bE (fj. Then Ns'-' (~) is a p-subgroup ofN6i(~) containing the Sylow p-subgroup 6 1 and

Since ~ ~ 6 1 ~ 6 b- ' , b E S(~). Since ~ < 6, we have ~ < 6 a- ' and by I, 8.8, ~ < Ns'-'(~)' Then

so a ~ b. By Sylow's theorem, Ns(~) ~ 6; for some c E N6i(~)' Thus 6 b- ' ?!.. 6 c- ' ~ N\$(~)'-' > ~C-l = ~,so b ~ c.

c) By a), x ~ 1 if and only ifax ~ a.~Thus y ~ a if and only if -1 S ~ ~ ~

a YE . d) By b), there exists c E N6i(~) such that a ~ c. By c), a = cd for

some d ~ 1. q.e.d.

4.4 Theorem (DOLAN). Suppose that (fj :f. 1 and 6 E Sp(f)), and let ~ be a set of subgroups of 6. Then ~ is a conjugation family if and only if given any subgroup ~ of 6 for which j has more than one equivalence dass, there exists a E S(~) such that ~a E ~.

Proof First suppose that ~ is not a conjugation family. Then we may choose a subset A of 6 with lAI maximal for which there exists 9 E (fj such that Ag ~ 6 and


(1) whenever 9 = g1 ... gn with gi E Nli(~i) and ~i E $', then Agl ." g'-I ~ ~i for some i (1 ~ i ~ n).

Let ~ = 6 n 6 g - ' Then A ~ ~, ~9 ~ 6 and (1) holds with A replaced by ~.It follows from the maximality of lAI that A = ~.

First suppose that ~ = 6. By (1), 6 ~ $'. But since (fj =1= 1, ~ has more than one equivalence dass, so the condition of the theorem does not hold for 6.

Suppose then that ~ < 6. We prove the following. (2) If x ~ y, there exist ~r E $' (r = 1, ... , m) and gr E Nli(~r) such

that x-1y = g1 ... gm and ~Xgl' "gj-I ~ ~j (j = 1, ... , m). To see this, observe that since x ~ y, there exist elements a1, ... , an

of S(~) such that x = a1, Y = an and

~a~1 ~a~1 m (' 2 ) I;;> ,-I n 1;;>' > 't' I = , ... , n .

Let.Qi = 6 a,la'_1 n 6. Then.Qi > ~ai-l, so l.Qd > IA I. Since .Qf;:\a, ~ 6, it follows from the maximality of lAI that there exist ~ij E $' (j = 1, ... , mi) and bij E NID(~i) such that ai-\ ai = bil ... bim, and

.ob "'b m (' 1 ) iil i,j-l:::; +'ij } = , ... , mi

Thus

- b b b b - 212m2 n1 nm.'

and

Thus (2) is proved. It folIo ws from (1) and (2) that g, 1 are not equivalent under ~. Thus ~

'll 'll has more than one equivalence dass. Suppose that a E S(~) and ~a E $'. By 4.3d), a = cd, where c E NID(~) and d ~ 1. Thus ~a = ~d E $'. By 4.3b), 9 ,f for some fE N(lj (~), and by 4.3c), f ~ fd. Thus 9 ~ fd. Since 1 i d, it follows from (2) that there exist ~i E $' (i = 1, ... , m) and gi E No;('l3;) such that d = g1 ... gm and ~gl '''9,-1 ~ ~i (i = 1, ... , m). Put ~m+1 = ~d and gm+1 = fd; thus ~m+1 E $' and gm+1 E N(lj(~m+1)' Since fd ~ g, there exist ~j E $' (j = m + 2, ... , n) and gj E NID(~j) such that (fdt1g = gm+2 ... gn and ~fd9"+2 '''gJ-I ~ ~j (j = m + 2, ... , n). Thenfd = dfd = g1 ... gmgm+1 and 9 = g1 ... gn'

4. Conjugate p-Subgroups 31

Also

ffi9," 'gl-1 < ffi. (I' = 1 n) +' - +" , ... , .

This contradicts (1). Hence '13a ~ $' for every a E S('13). Conversely, suppose that $' is a conjugation family, '13 ~ 6 and

i has more than one equivalence dass. Thus some element 9 of S('13) is not equivalent to 1 in i' Since '139 ~ 6 and $' is a conjugation family, there exist subgroups '131' ... , '13n in $' and elements gi E Nt('13i) (i = 1, ... , n) such that 9 = 9 1 ... gn and

ffi9,"'91-1 < ffi. (I' = 1 n) +' - 1-', , ... , .

IfI'13il> 1'13I,then6n69,' ~ '13i > '139""91-1,sog1 "'gi-1 ~g1 "'gi' Thus if 1 '13 i 1 > 1'131 for all i,

contrary to the definition of g. Hence 1 '13 i 1 = 1'131 for some i, and '139,' "9,-, = '13i E $'. q.e.d.

4.5 Definition. Suppose that 6 E Sp(fj). The subgroup '13 of 6 is said to be extremal in 6 if Ns('13) E Sp(Nt('13)).

For example, any normal subgroup of 6 is extremal in 6.

4.6 CoroUary. Suppose that 6 E Sp(fj). The set $' oJ extremal subgroups oJ 6 is a conjugation Jamily Jor 6.

Proof Suppose that '13 ~ 6. By 4.3b), there exists bE S('13) such that NS ('13 b) E Sp(Nt('13b)), or '13b E $'. Hence by 4.4, $' is a conjugation family for 6. q.e.d.

The existence of conjugation families is thus established. The follow-ing is useful in applications.

4.7 Theorem (ALPERIN). Suppose that 6 E Sp(fj) and that $' is a conjuga-tion Jamily Jor 6 such that every element oJ $' is extremal in 6. 1J A is a non-empty subset oJ 6, gE (fj and A9 S;;; 6, there exist '13i E $' (i = 1, ... , n) and gi E Nt('13i) such that 9 = 9 1 ... gn' A9, ... 9,-, s;;; '13 i and, Jor each i = 1, ... , n, either '13i ~ C6 ('13i) or gi E Ct('13J


Proof This is proved by induction on 16:

4. Conjugate p-Subgroups

where gj = h;., and }

Agl" 'gj-I = Ah," 'hiJ- 1 S .Qi.' }

33

q.e.d.

To find smaller conjugation families, we investigate the condition in 4.4 involving ~.

4.9 Lemma. Suppose that 6 E Sp(fj), ~ ~ 6 and ~ has more than one equivalence dass. Then there exist elements a, b ofS(~) such that b ~ 1, 6 a - ' n 6 b- ' = ~ and ~ is extremal in both 6 a- ' and 6 b- '

Proof Let g be an element of S(~) such that g, 1 are not equivalent in ~. By 4.3b), there exist elements a, b in S(~) such that g l a, 1 l band ~a, ~b are both extremal in 6. Thus a, b are not equivalent in l' so 6 a- ' n 6 b - ' = ~. Clearly ~ is extremal in 6 a- ' and in 6 b- ' q.e.d.

4.10 Definition. Suppose that 6, 6 are Sylow p-subgroups of (fj. We put 6 '" 6 if there exist Sylow p-subgroups 6 1, ... , 6 n of (fj such that 6 = 6 1 , 6 = 6 n and 6 i- 1 n 6 i > 1 (i = 2, ... ,n). Then '" is an equivalence relation on Sp(fj). We say that (fj is p-isolated if '" has more than one equivalence dass.

4.11 Theorem. a) (fj operates transitivelyon the set of equivalence dasses defined by the relation given in 4.10, and if f) is the stabiliser of the dass C(/, C(/ = Sp(f)).

b) Suppose that p divides l(fjl. Then (fj is p-isolated if and only if (fj has a proper subgroup f) such that p divides 1f)1 and f) n f)g is a p' -group whenever g E (fj - ~.

c) Suppose that 6 E Sp(fj) and ~ < 6. If ~ has more than one equivalence dass, N(fj(~)/~ is p-isolated. 'll Proof a) If 6 1 '" 6 2, then 6~ '" 6~ for any gE (fj; thus (fj operates on the set of equivalence dasses defined by "'. By Sylow's theorem, (fj operates transitively.

Let f) be the stabiliser of the dass C(/. If 6 E C(/ and x E 6, then C(/x is the dass containing 6 x = 6. Thus C(/X = C(/ and x E f). Hence 6 E Sp(f)), and C(/ s; Sp(f)) s; Sp(fj). Suppose, conversely, that ~ E Sp(f)). Choose any ~o E C(/; then ~ = ~~ for some h E f) and ~ E C(/h = C(/. Thus C(/ = Sp(f)).

b) Suppose that (fj is p-isolated. Let C(/ be an equivalence dass and let f) be the stabiliser of C(/. Then f) < (fj and p divides 1f)1. Suppose that


~ () ~g is not a pi -group and that ~ E Sp(~ () ~g). Then ~ #- 1, and ~ ~ 6 1 E Sp(~), ~ ~ 6 2 E Sp(~g). By a), 6 1 E ~ and 6 2 E ~g. But 6 1 () 6 2 ~ ~ > 1, so 6 1 '" 6 2 , Hence ~ = ~g and g E~. Hence ~ () ~g is a pi -group whenever g E (f) - ~.

Conversely, suppose that (f) has a proper subgroup ~ such that p divides I~I and ~ () ~g is a p'_group whenever gE (f) - ~. Suppose that ~ E Sp(~); thus ~ #- 1. If g E N(D(~), ~ ~ ~ () ~g, so g E ~. Thus

N(D(~) ~ ~,and by 2.4, ~ E Sp(f)). Thus Sp(~) is a non-empty subset of Sp(f)). Since ~ cannot contain anormal subgroup of (f) of order divisible by p, Sp(~) #- Sp(f)). We observe that if 6, 6 1 are Sylow p-subgroups of(f), 6 () 6 1 #- 1 and 6 ~ ~,then 6 1 ~ ~. For if 6 1 = 6 g, ~ () ~g

~ 6 () 6 1 #- 1, so g E ~ and 6 1 ~ ~. It follows that if 6,6 1 are Sylow p-subgroups of (f), 6 '" 6 1 and 6 ~ ~, then 6 1 ~ ~. Thus Sp(~) contains an equivalence dass and (f) is p-isolated.

c) By 4.3b), there exists a E N(D(~) such that a, 1 are not equivalent in ~. Suppose N6(~) ~ ~o E Sp(N(D(~))' Thus ~-I E Sp(N(D(~))' Sup-pose that ~o/~ '" ~r'/~. Then there exist Sylow p-subgroups ~1' ... ,

~n ofN(D(~) such that ~i-1 () ~i > ~ (i = 1, ... , n) and ~n = ~f.1f ~fi ~ 6 (Xi E (f)), Xi- l ~ Xi' Thus Xo ~ Xn But 6 x ' () 6 ~ N6(~) >

, -I I \jl -I -I ~, so 1 ~ xo, and 6 x () 6 a- ~ (~o () 6t ~ N6(~)a >~, so Xn ~ a. Thus 1 ~ a, a contradiction. Hence N(D(~)/~ is p-isolated.

q.e.d.

We now obtain the following generalization of 4.6.

4.12 Theorem (ALPERIN, GOLDSCHMIDT [1]). Suppose that 6 E Sp(f)) and that !F is the set 0/ all subgroups ~ 0/ 6 such that

(i) there exists 6 0 E Sp(f)) such that ~ = 6 () 6 0 and ~ is extremal in both 6 and 6 0 , and

(ii) either ~ = 6 or N(D(~)/~ is p-isolated. Then is a conjugationfamily for 6.

Proof Suppose that .0 ~ 6 and that % has more than one equivalence dass. By 4.9, there exist elements a, b ofS(,Q) such that b ~ 1, 6 a - 1 () 6 b- 1 = .0 and .0 is extremal in both 6 a- 1 and 6 b- l Let ~ = .ob = 6 a- 1 b () 6. Then ~ is extremal in both 6 a - Ib and 6. By 4.11c), either .0 = 6 or NI\i(,Q)/,Q is p-isolated. Hence either ~ = 6 or N(D(~)/~ is p-isolated. Thus ~ E . By 4.4, is a conjugation family. q.e.d.

4.13 CoroUary (ALPERIN). Suppose that 6, 6* are Sylow p-subgroups of Q). Then there exist Sylow p-subgroups 6 1"", 6 n of Q) and gi E N(D(6 () 6 i) (i = 1, ... , n) such that

5. Characteristic p-Functors 35

a) 6*91 .g, = 6, b) 6 n 6 i is extremal in both 6 and 6 i , c) (6 n 6*)91 9,-1 ~ 6 n 6 i (i = 1, ... , n).

Proof Let ff' be the conjugation family of 4.12. By Sylow's theorem, 6*9 = 6 for some gE m. Hence (6* n 6)9 ~ 6. Hence there exist

~l' ... , ~n in ff' and gi E ND(~i) (i = 1, ... , n) such that g = gl ... gn and (6* n 6)91 9,-1 ~ ~i. But ~i = 6 n 6 i for some 6 i E Sp(m) and

~i is extremal in both 6 and 6 i q.e.d.

4.14 Remarks. BENDER [5] has proved that if m is a 2-isolated group, either (i) the Sylow 2-subgroups of mare either cyclic or generalized quaternion, or (ii) m has aseries of normal subgroups 1 ~ 9Jl < E ~ m such that 9Jl and m/E are of odd order and Ej9Jl is isomorphie to one of the simple groups PSL(2, q), Sz(q) or PSU(3, q), where q is apower of2 and q ~ 4.

According to 4.11 b), such groups are characterized by the existence of a subgroup ~ of even order such that I~ n ~91 is odd for all gE m - ~. Such a subgroup ~ is called strongly embedded.

5. Characteristic p-Functors

5.1 Definition. Let p be a prime. A characteristic p1unctor is a function W, defined on the class of all finite p-groups, such that

a) for any finite p-group ~, W(~) is a subgroup of~, and b) if ~l' ~2 are finite p-groUPS and r:J. is an isomorphism of ~l onto ~2' then W(~l)r:J. = W(~2).

Typical of the characteristic p-functors which will be considered are J, ZJ (3.6).

5.2 Lemma. Let W be a characteristic p-functor. a) 1f~ is a p-group, W(~) is a characteristic subgroup of~. b) 1f m is a finite group, ~ is a p-subgroup of m and gE m, then W(~)9 = W(~9).

c) 1f 91 is anormal p'-subgroup of the finite group m and ~ is a p-subgroup of m, then W(~91/91) = W(~)91/91, and NQ;/9l(W(~91/91)) =

NQ;(W(~))91/91.

Proof a) By 5.1b), W(~)r:J. = W(~) for every automorphism r:J. of~, so W(~) is a characteristic subgroup of~.


b) Since the mapping x -+ x 9 (x E ~) is an isomorphism of ~ onto ~9, W(~)9 = W(~9) by 5.1b).

c) The mapping x -+ x91 (x E~) is an isomorphism 0( of ~ onto ~91/91. Thus W(~91/91) = W(~O() = W(~)O( = W(~)91/91 By IX, 6.11, NQj/91(W(~91/91)) = NQj/91(W(~)91/91) = NQj(W(~))91/91. q.e.d.

5.3 Defmitions. a) Suppose that W is a characteristic p-functor and that 1 is a finite p-group. For each subgroup ~ of 1 and each integer n ~ 0, we define a subgroup wn(~) = wn.z{~) by induction on n. For n = 0, define wo(~) = NI(~)' and for n > 0,

For n > 0, W(Wn-l(~)) is a characteristic subgroup of Wn-l(~)' by 5.2a). Hence W(Wn-l(~)) 0 define .on to be a Sylow p-subgroup of NQj(W(.on-l containing .on-I. Thus

Let ~ = Un~O.on. Thus ~ is a p-subgroup of ffi. By Sylow's theorem, there exists g E ffi such that ~9:S; e. Let ~n = .o~ (n ~ 0). Then

~n :s; 1$ and

5. Characteristic p-Functors 37

Further, ~oESp(Nfj(Og))and ~nESp(N(!)(W(~n_I)))Hence~o = Ns(Og) and, for n > 0, ~n = NS(W(~n-I))' since, for example, N6 (Og) is a p-subgroup of N(!)(Og) containing the Sylow p-subgroup ~o' Thus og and each W(~n) is extremal in 6. We prove by induction on n that

~n = wn(og); this is clear for n = and, for n > 0,

Thus og and each W(wn(og)) is extremal in 6 and og is W-extremal. q.e.d.

5.5 Theorem. Let W be a characteristic p1unctor. 1J 6 E Sp(fj), the set oJ W -extremal subgroups oJ 6 is a conjugation Jamily Jor 6.

Proof This follows at once from 4.4 and 5.4. q.e.d.

We shall study the fusion of sequences of elements of a subgroup.

5.6 Defmition. Suppose that 1 is a subgroup of a finite group (fj. a) Denote by Y(l) the set of all finite sequences (Xl' ... ,Xm) of

elements Xi of 1. b) Ifx = (Xl' ... , Xm) E Y(l) and g E (fj, write xg = (xi, ... , x!). c) Two elements x, y of Y(l) are said to be conjugate in (fj if there

exists g E (fj such that xg = y. Conjugacy in (fj is an equivalence relation on Y(l).

5.7 Definition. Suppose that (fj is a finite group, ~ ~ (fj and 6 E Sp(fj). Let '" be an equivalence relation on Y(6). We say that '" contains fusion in if x '" y whenever x, y are elements of Y(6 (l ) conjugate in .

It is clear that if '" contains fusion in ~, then '" contains fusion in any subgroup of ~.

5.8 Definition. The characteristic p-functor W is called positive if W(~) =f. 1 for every p-group ~ =f. 1.

For example, Z, J, ZJ are all positive.

Roughly speaking, the following theorem shows that if '" contains fusion in N(!)(W(6)) but not in (fj, then the same is true in the normaliser of so me p-subgroup.

5.9 Theorem. Suppose that W is a positive characteristic p1unctor, (fj is a finite group and 6 E Sp(fj). Suppose that '"" is an equivalence relation


on Y(6), that '" containsfusion in N(i(W(6)) and that '" does not contain fusion in (fj. Then there exists a subgroup ~ of 6 having the following properties.

a) ~ :# 1 and ~ is W-extremal in 6. b) '" does not containfusion in N(i(~). c) '" contains fusion in N(i(W(N$(~))) and hence in N(i(W(N6(~))) n N(i(~)'

d) Either (i) there exists a subgroup .0 of 6 such that ~ = W(.Q), .0 > Opfj)

and .0 ~ C6 (.Q), or (ii) ~ ~ Opfj) and there exist elements x, y of Y(~) such that

x f y but x, y are conjugate in N(i(~). (Note that the condition (ii) of d) is only a slight strengthening of b)).

Proof Since '" does not contain fusion in (fj, there exist XE Y(6) and gE (fj such that x9 E Y(6) but x9 f X. If x = (Xl' ... , xm), put 9l = Op(ffi). For in case (i), ~ Op(ffi). And in case (ii),

~ :::; ~o < 6, so Op(ffi) :::; ~ < N6(~) = ~o. Note also that

It follows that W(~o) satisfies d) (i) (with ~o in the place of .0). We observe that W(~o) satisfies a). Since ~o =1= 1 and W is positive, W(~o) :# 1. Since wo(~) = ~o and ~ is W-extremal,

6. Transfer Theorems

It is easy to deduce by induction on n that if n ~ 0,

for ifn > 0,

Hence since ~ is W -extremal,

N=(W(wn(W(~O)))) = Wn+1(W(~O = N.r;(W(Wn+l (~))) E Sp(NQ;(W(Wn+l (~)))) = Sp(NQ; (W (wn(W (~o))).

Thus W(~o) is W-extremal.

39

Since W(~o) is a characteristic subgroup of ~o, N=(W(~o ~ N6(~O)' Since ~o < 6, N6(~O) > ~o; thus

It follows from the definition of~ that W(~o) ~ ". Since W(~o) satisfies a) and d), it follows that W(~o) does not satisfy b). Thus '" contains fusion in NQ;(W(~o; that is, ~ satisfies c). q.e.d.


To apply the results of 5 to particular characteristic p-functors, we shall need some results proved by using the transfer (Chapter IV). The following is equivalent to IV, 2.2.

6.1 Lemma. Suppose that ~ is anormal p-subgroup of (D and 6 E Sp(D). Then ~ n (D' = [~, (D](~ n 6').

Proof. Let 3/[~, (D] = Z(D/[~, (D]). Obviously ~ :::;; 3. By IV, 2.2 applied to (D/[~, (D],

(6/[~, (D]) n (D'/[~, (D]) n (3/[~, (D}) :::;; (6/[~, (D])'.


Thus

6 n (fj' n 3 ::; [~, (fj] 6'.

Since ~ ::; 3, it follows that

~ n (fj' ::; ~ n [~, (fj] 6' = [~, (fj] (~ n 6'),

whence the assertion. q.e.d.

Next we prove the important "focal subgroup" theorem.

6.2 Theorem (D. G. HIGMAN[l]). 1/6 E Sp(fj),

6 n (fj' =

6. Transfer Theorems 41

6.4 Lemma. Suppose that e E Sp(f)), m


Proof (Blessenohl). Since ~ is subnormal in m (see VII, 16.1), there exists aseries

such that ~i-l 0, then by the inductive hypothesis, there exists a subset Tl of m such that every element of m is uniquely expressible in the form yt1 with Y E 21~1' t1 E Tl' Now 21~1 is a union of double cosets 21u~, and there exists a subset U of ~1 such that every element of21~llies in 21u~ for a unique u EU. We show that T = UT1 has the required property.

Since ~


6.9 Lemma. Suppose that (f) is a group of operators on the finite Abelian p-group 58 and that e; E Spf)). Let m be a weakly closed subgroup in e; with respect to (f) and suppose that Nm(m) ~ 9Jl. Let d be a set of sub-groups which generates m. 1f C!!l(9Jl) > C!!lf)), there exist 21 E d, g E (f) - 9Jl and v E C!!l(21 n e;g) such that 21 n e;g = 21 n 9Jl9 < 21 and

n (vr) ~ C!!lf)) 'ER

for any transversal R of 21 n e;g in 21. Proof. Choose u E C!!l(9Jl) - Cilf)). By 6.5,

where 0 is any transversal of 9Jl in (f). But the set of right co sets of 9Jl in (f) is the disjoint union of the sets of those lying in the double cosets 9Jltm and 9Jl = 9Jllm is one of these double cosets. Since u ~ Cilf)), it follows that there exists t ~ 9Jl such that

(1) ut = n uh ~ Cilf)), hEQ,

where Ot is a complete set of representatives of the right co sets of 9Jl in 9Jltm. Note that since u E Cij(9Jl), ut is independent of the choice of Ot.

By 6.4, there exists g E 9Jlt such that mo < m, where

mo = m n e;9 = m n 9Jl9.

Since m is generated by d, there exists 21 E d such that 21 1, mo. Thus

Let R be any transversal of21 n mo in 21. By 6.7, there exists a subset S of m such that every element of m is uniquely expressible in the form xs with x E mo 21, SES. Thus each coset of mo in m is in mo 21s for a unique SES and is therefore, by 6.8b), of the form mors for a unique rE R. By 6.8a), each coset of 9Jl in 9Jlgm is of the form 9Jlgrs for unique elements r E R, SES. Since mgm = 9Jltm, it follows from (1) that

n n ugrs ~ Cilf)) SES 'ER


Hence

n ugr ~ C~(m). rER

But

so the stated condition holds with v = ug. q.e.d.

Our next aim is to remove the condition in 6.9 that m is Abelian. For this we need a lemma similar to 1.8.

6.10 Lemma. Suppose that m is a group of operators on a group m. Let U be an Abelian m-invariant subgroup of m and suppose that m has a subgroup 9R such that (Im: 9R1, IUi) = 1. 1f v E C~(9R) and vU is m-invariant, then vU contains am-invariant element.

Proof For each g E m, there exists f(g) E U such that vg = vf(g), since vU is m-invariant. If also h E m,

vf(gh) = (vg)h = (vf(g))h = (vh)(f(g)h) = (vf(h))(f(g)h),

so

(2) f(gh) = f(h)(f(g)h) (g E m, h E m). Since v E C~(9R), f(x) = 1 for all x E 9R. Thus by (2),

(3) f(xg) = f(g) (x E 9R, g E m).

Let m = Im: 9R1 and let T be a transversal of 9R in m. By hypothesis, ml == 1 (lUi) for some positive integer 1. Put

u = (n f(t))l. lET

(The order of the factors in this product need not be specified, since U is Abelian). Then u E U, and by (3), u is independent of the choice of the transversal T. But if g E m, T g is a transversal of 9R in m, so

u = (n f(tg))l. lET


By (2), it follows that

u = (n f(g) (f(t)g))' = f(gr' n (f(t)'g) = f(g)(ug). lET lET

Thus

(vu)g = (vg)(ug) = vf(g)(ug) = vu

for all g E (fj. q.e.d.

6.11 Lemma. Suppose that 58 is afinite p-group, (fj is a group of operators on 58 and 9Jl is a subgroup of (fj of index prime to p. Suppose that Cu(9Jl) = Cu(fj)for any (fj-compositionfactor U (see I, 11.5) of58. Then Ctl (9Jl) = Ctl(fj) Proof This is proved by induction on 158 I. If 58 = 1, there is nothing to prove. Suppose that 58 # 1 and that 3 is a minimal (fj-invariant sub-group of the centre of 58. Thus 3


If [m, IDl] '" 1, then since [m, IDl]


L fr- 1 E wt + wt = (W1 n W3)J., reR

0= w L fr- 1 = (L wr)f reR reR

for all fE wt. Thus LreR wr E W2 and LreR ur = 0. Hence W satisfies the conditions of a).

By a), Cw(Wl) = Cw(. Using VII, 8.3g),

6.14 Theorem. Suppose that e; E Sp( and ~ is anormal p-subgroup of >. Let !B be a weakly closed subgroup in e; with respect to > and suppose that N(fj(!B) :s:; Wl. Suppose also that !B is generated by a set A having the property that whenever a E A, 1/~ is a chief factor of > and 1 :s:; ~, then [1, a, ... , a] :s:; ~. Then Wl ~ e; and

p-l

Z(Wl) n ~ = Z( n~, [~, Wl] = [~, >], ~ n Wl' = ~ n >'. Proof Since !B is weakly closed, !B , regarded as a group of operators on ~. Let d = {-invariant and are normal subgroups of >; also

1/~ is a chief factor of >. Let R be a transversal of 210 in 21. The condi-tions a) and b) of 6.13 will be satisfied if it is shown that for any x E 1,

rr x r E ~, rr x r - I E ~. reR reR

Since 21 is Abelian, R -1 is also a transversal of 210 in 21; thus it is sufficient to prove that

rr xr E ~ reR

for all XE 1. Suppose that 21 =


for all X E 1.

pk-l

TI x al E I!), i=O

We regard I/I!) as a vector space over GF(p) and denote by rx the linear transformation tl!) --+ tal!) of I/I!). By IX, 1.8,

(tl!))(rx - l)i = [t, a, .... , a]l!). J

for any tEl. Since I/I!) is a chief factor of (f) and a E A, [1, a, ... , a] p-l

~ I!) by hypothesis. Thus (rx - l)p-l = 0. Since the characteristic ofthe ground-field is p, (rx - 1)P = rxP - 1. Hence rxP = 1 and xl!) = (xl!))rxP = xaPI!). Hence x aP == x mod I!). Also

1 + rx + rx2 + ... + rxP-l = (rx - W- l = 0,

so

or x = xx a XaP-I E I!). It follows that

Thus the conditions of 6.13 are satisfied. By 6.13, C,l9Jl) = C\l(f)), or ~ n Z(9Jl) = ~ n Z(f)). Also [~, 9Jl]

= [~, (f)]. It fOllOWS from 6.1 that ~ n 9Jl' = ~ n (f)'. q.e.d.

We shall also require a generalization of IV, 4.7. First, we mention a few elementary facts about OP(f)).

6.15 Lemma. Suppose that 6 E Sp(f)) and 6 ~ f) ~ (f). a) (f)/OP(f)) ~ f)/(f) n OP(f))) and f) n OP(f)) ~ OP(f). b) The following assertions are equivalent.

(i) (f)/OP(f)) ~ f)/OP(f). (ii) !(f)/OP(f))! = !f)/OP(f)!.

(iii) f) n OP(f)) = OP(f). c) (f)/OP(f))(f)' ~ f)/(f) n OP(f))(f)') and f) n OP(f))(f)' ~ OP(f)f)'.

Also (f)/(f)' = (6(f)' /(f)') x (OP(f))(f)' /(f)').


d) The following assertions are equivalent. (i) (fjjOP(fj)(fj' ~ NOP(m~'.

(ii) I (fjjOP(fj)(fj' I = l~jOP(~)~'I (iii) ~ n OP(fj)(fj' = OP(~)~'. (iv) 6 n (fj' = 6 n ~'. (v) 6(fj'j(fj' ~ 6~'j~'.

49

Proof. a) Since (fjjOP(fj) is a p-group, 60P(fj) = (fj. Since 6 ~ ~, ~OP(fj) = (fj. Thus (fjjOP(fj) = ~OP(fj)jOP(fj) ~ N(~ n OP(fj)). Thus ~j(~ n OP(fj)) is a p-group and ~ n OP(fj) ~ OP(~).

b) (i) implies (ii) trivially. If (ii) holds, then I~ n OP(fj) I = IOP(~)1 by a). Since ~ n OP(fj) ~ OP(~), (iii) holds. Finally (iii) implies (i), bya).

c) As in a), (fj = ~OP(fj), so (fj = NOP(fj)(fj'). Thus (fjjOP(fj)(fj' ~ ~j(~ n OP(fj)(fj'), ~j(~ n OP(fj)(fj') is an Abelian p-group and ~ n OP(fj)(fj' ~ OP(~)~'.

Since 6(fj' j(fj' E Sp(fjj(fj') and (fjj(fj' is Abe1ian, (fjj(fj' = 6(fj' j(fj' X l:j(fj' for some l:


~ = {(g, :!h)lg E m, h E ~, 91g = 91:!h = 91h}

(cf. I, 9.11). Since m = 91~, there is an epimorphism 1t 1 of ~ onto m given by

(g, :!h)1t i = g. Let9Jl = ker1t i ;thus

9Jl = {(1,:!h)lhE.Q}

and 9Jl ~ .Q/:!. Let ~* be the inverse image of ~ under 1t i , so that I~: ~*I = Im: ~I and

~* = {(h', :!h)lh E~, h' E .Qh}. And if

!l = {(h,:!h)lhE~},

then ~* = 9Jl!l and 9Jl !l !l = 1. If 6 E Sp(~*), then 6 E Sp(~), since I ~ : ~* I = Im: ~ I is prime to p. Also 9Jl is an Abelian p-group and 6 splits over 9Jl, so by I, 17.4, there exists a complement m for 9Jl in ~. We have

~ I ~. I I I I I I Q I I I

~ , , , , , , , ,

l:


and the restrietion 1t: of 1t: 1 to (f) is an isomorphism of (f) onto (f). Let 1t:2 be the projeetion of ~ onto f)j':! given by

(g, ':!h)1t:2 = ':!h,

b) Let 91* be the inverse image of 91 under 1t:1 . Thus

91* = {(x, ':!h)lx E 91, h E.o}

and 911t:-1 = 91* 11 (f). Thus 914> ~ .oj':!. But .oj':! is a p-group and OP(91) = 91,so914> = 1. Henee lim4>1 divides I (f)j911 = IN.oI.Butsinee f)j':! = (.oj':!) (im 4,

.0 _ lim 4>1 IN I - I(.oj':!) 11 (im 41

This implies that(.oj':!) 11 (im 4 = 1. q.e.d.

6.17 Corollary. 1/ 6 E Sp(f) and .0 = 6 11 OP(f), 6j.o' splits over .oj.o'.

Proof In 6.16, take f) = 6,91 = OP(f) and ':! = .0'. q.e.d.

6.18 Theorem (TATE). Suppose that f) ~ (f) and I(f): f)1 is prime to p. Then 6>jOP(6)) ~ ~jOP(~) if and only if6>jOP(6))6>' ~ ~jOP(~)~'.

Proof If 6>jOP(6)) ~ NOP(~),

(6)jOP(6>))j(6>jOP(6>))' ~ (~jOP(f)j(NOP(f)',

whieh implies that 6>jOP(f)6>' ~ f)jOP(ff)'. If (f)jOP(f) and NOP(f are not isomorphie, then .0 = f) 11 OP(f)

> OP(f, by 6.15b). Now .0


l! = ~ or l! = ~. Hence ~jOP(~) is a maximal subgroup of the p-group NOP(~). Thus ~ l:, 0 $. ~, so ~ n OP(fj)(fj' $. R Thus ~ n OP(fj)(fj' ::j:. OP(~)~', and by 6.15, (fjjOP(fj)(fj' is not isomorphie to ~jOP(~)~'.

q.e.d.

7. Maximalp-Factor Groups

We now combine the results of the last two sections.

7.1 Definition. Suppose that W is a characteristic p-functor and (fj is a finite group. We say that W controls transfer in (fj if (fjjOP(fj) ~ 91jOP(91), where 91 = Nlfj(W(6for some 6 E Sp(fj)(and hence for any 6 E Sp(fj)).

Suppose that 6 E Sp(fj) and 91 = Nlfj(W(6)). By 6.15 and 6.18, the following conditions are equivalent.

(i) W controls transfer in (fj. (ii) (fjjOP(fj) ~ 91jOP(91).

(iii) 1 (fjjOP(fj) 1 = 1 91jOP(91)I (iv) 91 n OP(fj) = OP(91). (v) (fjjOP(fj)(fj' ~ 91jOP(91) 91'.

(vi) 1 (fjjOP(fj)(fj'1 = 1 91jOP(91) 91' I. (vii) 91 n OP(fj)(fj' = OP(91)91'.

(viii) 6 n (fj' = 6 n 91'. (ix) 6(fj'j(fj' ~ 691'j91'.

It will be noted that in this terminology, Grn's second theorem (IV, 3. 7) states that if (fj is p-normal, Z controls transfer in (fj.

In order to prove the first reduction theorem on controlling transfer, we restate 6.2 in the terminology of 5.7.

7.2 Theorem. Suppose that 6 E Sp(fj), ~ =:;; (fj and 6 n ~ E Si~). Suppose that ~ =:;; (fj and the relation", on 9'(6) is defined as follows: x '" y if either (i) x and y are both sequences of length greater than 1, or (ii) x = (x), y = (y) and xy-l E ~. Then '" is an equivalence relation, and '" contains fusion in ~ if and only if 6 n ~' =:;; ~.

Proof Obviously, '" is an equivalence relation on 9'(6). First suppose that 6 n ~' =:;;~. It is to be proved that if

XE 9' (6 n ~), y E 9' (6 n ~) and y = xg for some 9 E ~, then either (i) x and y are both of length greater than 1, or (ii) x = (x), y = (y) and

7. Maximal p-Factor Groups 53

xy-l E ~. If X is of length greater than 1, so is y = xg and (i) holds. If X = (x), where x E 6 n Sl, then y = (xg) and xg E 6, so xx-g = [x-i, g] E 6 n Sl'; since 6 n Sl' ~ ~,(ii) holds.

Conversely, suppose that '" contains fusion in R Thus, if XE 6 n Sl, gE Sl and xg E 6 n Sl, then (x- 1) '" (x- g) and x-1xg E~. Hence, by 6.2 (applied to the Sylow p-subgroup 6 n Sl of Sl), 6 n Sl' ~ ~. q.e.d.

7.3 Theorem (ALPERIN and GORENSTEIN [2]). Suppose that W is a positive characteristic p-functor and that (fj is afinite group. IfW controls transfer in Nr;(~) for every non-identity p-subgroup ~ of (fj, then W controls transfer in (fj.

Proof Suppose that 6 E Sp(fj) and 91 = Nr;(W(6)). Define the relation '" on 9"(6) as folIows: X '" Y if either (i) x, y are both sequences of length greater than 1, or (ii) X = (x), y = (y) and xy-l E 91'. It fOllows from 7.2 that if Sl ~ (fj and 6 n Sl E Sp(Sl), then '" contains fusion in Sl if and only if 6 n Sl' ~ 91'.

Suppose that W does not control transfer in (fj. Then 6 n (fj' f;, 91', so '" does not contain fusion in (fj. However, '" contains fusion in 91, so by 5.9, there exists a non-identity W-extremal subgroup ~ of 6 such that

a) '" does not contain fusion in N(lj(~), and b) '" contains fusion in N(fj(W(N;;(~))).

Now since ~ is W-extremal, 6 n N(lj(~) E Sp(N(lj(~)) and 6 n N(fj(W(N;;(~))) is a Sylow p-subgroup of N(fj(W(N6(~)))' Hence a), b) are respectively equivalent to the following.

c) 6 n N(fj(~)' $, 91'. d) 6 n N(lj(W(N6(~)))' ~ 91'.

Thus 6 n N(fj(~)' t N(fj(W(N;;(~)))'. Since N6(~) E Sp(N(fj(~, this shows that W does not control transfer in N(f;(~), by 7.1 (viii). q.e.d.

7.4 Lemma. Suppose that ~, Sl are subgroups of the finite group (fj such that (fj = ~Sl and ~ n Sl ~ 6 E Sp(fj).

a) 1f A s;;; 6, gE (fj and Ag s;;; 6, there exist u E ~, V E Sl such that g = uv and AU s;;; 6.

b) 1f '" is an equivalence relation on 9"(6) and '" does not contain fusion in (fj, then either '" does not contain fusion in ~ or '" does not contain fusion in R

Proof a) Since (fj = ~Sl, there exist x E~, Y E Sl such that g = xy. It folIo ws from Ag s;;; 6 that AX s;;; 6 y - 1 Thus AX s;;; ~x n Sly-l = ~ n R Since A S;;; 6,


some Sylow p-subgroup 6 1 of ~ n.R Since 6 ::5: ~ n Sl, we have 6 1 = 6: for some Z E ~ n .R Thus AX s; 6 1 = 6: and Ax:-' s; 6. If u = xz- 1 and V = zy, then uv = xy = g, U E ~, V E Sl and AU s; 6.

b) Since '" does not contain fusion in m, there exist elements x, y of 9'(6) such that y = xg for some gE m but x f y. If Ais the set of elements of 6 in x, then A s; 6 and Ag s; 6. By a), there exist u E ~, V E Sl such that g = uv and AU s; 6. Thus XU E 9'(6) and XUV = xg = Y E 9'(6). Since x f xUv, it follows from the transitivity of '" that either x f XU or XU f XUV In the former case, '" does not contain fusion in ~, and in the latter, '" does not contain fusion in .R q.e.d.

7.5 Lemma. a) Suppose that n is a set oJ primes and ~ = O,,(m). IJ ~C(D(~)/~ is a n'-group, then C(D(~) = O",(m) x Z(~) and O,,(m) ~ c"((O,,(m)), where m = m/o",(m).

b) Suppose that 6 E Sp(m) and that m cannot be expressed as the product of two proper subgroups of m each oJ which contains 6. Then either m = 6C(D(Oim)), or Op(m) ~ C"((Oim)), where m = m/op,(m).

Proof a) Since ~C(D(~)!~ is a n'-group, it follows from IX, 1.2 that

where 2 is a n'-subgroup of Cm(~) and 2

7. Maximal p-Factor Groups 55

satisfies 0lO) ~ Co(OlO)), then 0/0) n .0' = Op(.o) n No(W(60))' for 60 E Sp(.o). Then W controls transfer in (.

Proof Suppose that this is false and that ( is a counterexample of minimal order. Let 6 E Sp((), 91 = N(W(6)). By 7.1, 6 n (' #-6 n 91', since W does not control transfer in (. We obtain a contradic-tion in a number of steps.

(1) W controls transfer in every proper section of (. For every proper section of ( satisfies the hypothesis of the theorem

and is of smaller order than (. (2) ~ #- 1, where ~ = Op((). W does not control transfer in (. By 7.3, there exists a non-identity

p-subgroup l: of ( such that W does not control transfer in N(l:). By (1), N(l:) = (, or l:


6 1\ ffi' ~ 6 1\ '-9Jl' = '-( 6 1\ 9Jl') ~ '-( 6 1\ 91'),

and

6 1\ ffi' = ('- 1\ ffi')( 6 1\ 91').

Since 6 1\ 91' =F 6 1\ ffi', it follows that '- 1\ ffi' $. 91'. (6) If ffi = N~, where 6 ~ ~ ~ ffi and 6 ~ Sl ~ ffi, then either ~ = ffior"Sl = ffi.

Let '" be the equivalence relation defined on Y(6) by putting x '" y if either (i) x and y are both sequences of length greater than 1, or (ii) x = (x), y = (y) and xy-l E 91'. It follows from 7.2 that if 6 ~ E ~ ffi, then '" contains fusion in E if and only if 6 1\ E' ~ 91'. Hence if '" does not contain fusion in E, then 6 1\ E' $. (91 1\ E)" so 6 1\ E' =F 6 1\ (911\ E)' and W does not control transfer in E.

Since 6 1\ ffi' $. 91', '" does not contain fusion in ffi. By 7.4, either '" does not contain fusion in ~ or '" does not contain fusion in R From above, it follows that W does not control transfer in ~ or W does not control transfer in R By (1), either ~ = ffi or Sl = ffi.

(7) ffi =F 6C(f;('-). For suppose ffi = 6C(f;('-). Then by III, 1.10,

By 6.1, '- 1\ ffi' = '- 1\ 6'. But 6 ~ 91, so '- 1\ ffi' ~ 91', contrary to (5).

(8) By (4), (6), (7) and 7.5b), Op(ffi) ~ C(f;(Op(ffi)). By the hypothesis of the theorem, '- 1\ (' = '- 1\ 91', contrary to (5). q.e.d.

7.7 Theorem. Suppose W is a positive characteristic p1unctor and that ffi is afinite group. Suppose thatfor every section.Q offfi satisfying Ca(Op(.Q))

~ Op(.Q), and for every 6 0 E Sp(.Q), either W(60) -

8. Glauberman's K-Subgroups 57

if No(W(60)) = .0, so we may suppose that W(60) is not normal in .0. Let A be the set of aB elements a of 6 0 which have the property that [X, a, ... ,a] ::;; '!> for every chief factor XI'!> of.o satisfying X ::;; Op(.o).

p-l

By hypo thesis A % Op(.o). If a E A and 9 E .0, [X, ag, ... ,ag] ::;; '!> for any chief factor XI'!> of.o satisfying X ::;; Op(.o). Hence if a E A, 9 E .0 and ag E 6 0 , then ag E A. Thus if 9 E .0 and Ag ~ 6 0, Ag = A.1t follows that if'!3 = of.Q with

X ::;; 0i.Q) such that [X, g, ... ,gJ $ ,!>. p-l

Condition (1) immediately implies that W(60 ) i= W(Op(.Q)). We therefore seek a characteristic p-functor K with the property that when-ever ~ is a p-group, 91 is a minimal

p-l

characteristic factor of 91. The problem is thus reduced to one on p-groups. By 1.1, there is a characteristic subgroup 5l of 91 such that it is sufficient to consider the operation of 9 on 5l, so K has to be defined in such a way that K(~) i= K(91) implies the existence of a subgroup 9i such that 9i $, 91 and [5l, 9i, ... , 9i] = 1 for some k. Glauberman's

k

definition of such a characteristic p-functor K is quite complicated.


8.1 Defmitions. Let ~ be a p-group and let ~ be the set of subgroups of ~ of class at most 2. We define subsets ~(~) of ~ and subgroups KII(~) of ~ for n = -1, 0, 1, ... by induction on n.

(1) For n = -1, put .)(-1 (~) = ~ and K_1 (~) = ~. (2) For n even, define ~(~) to be the set of subgroups ~ E ~ which

satisfy the following conditions.

(2a)

(2b) If ~ E ~, ~ ~ K II - 1 (~), ~ ~ N~(~), ~ ~ C~(~/) and [Z(~), ~, ~] = 1, then [~, ~] ~ Z(~).

(3) For n odd and n > 0, define ~(~) to be the set of subgroups ~ E ~ for which [~, KII-1(~)] ~ Z(~).

(4) Forn ~ O,defineKII(~) =


[Z(~), m, m] = 1. Then m :$; Kn(,Q), so by 2b), [~, m] :$; Z(~). Hence by (2), ~ E X"m+l (9l).

Next, suppose that m, n are even and ~ E .x:. + 1 (.0). Again ~ :$; Kn+1(,Q) :$; 9l. Also [~, Km(9l)] :$; [~, Kn(,Q)] :$; Z(~) by (3); thus

~ E X"m+l (9l). b) By two applications of a) (with .0 = 9l = '.p),

for all 1 ~ -1. Sinee X"-l('.p) = fJI :2 X"l('.p), the assertion folIo ws at onee.

e) is an immediate eonsequence of b). q.e.d.

8.4 Definition. For any p-group '.p, define

K('.p) = n K2n- 1 ('.p), K('.p) = U K2n('.p) n~O n~O

By 8.2e) and 8.3e), K and Kare eharaeteristic p-funetors. By 8.2, K('.p), K('.p) both eontain Z2('.p) and all normal Abelian subgroups of '.p, so K, K are positive eharaeteristie p-funetors.

8.5 Lemma. If'.p is any p-group,

Proof If ~ is a maximal normal Abelian subgroup of '.p, ~ :$; K('.p), so C~(~) ~ C'll(K('.p)). But by III, 7.3, C'll(~) = ~, so C'll(K('.p)) :$; ~ :$;

K(~). Similarly C'll(K(~)) ::;; K(~). q.e.d.

8.6 Lemma. Suppose that '.p is a p-group and .0 ::;; ~. a) If K('.p) ::;; .0, then K('.p) :$; K(,Q) and K('.p) ~ K(,Q). b) If K('.p) :$; .0, then K('.p) :$; K(,Q) and K('.p) ~ K(,Q). e) If K('.p) K('.p) ::;; .0, then K('.p) = K(,Q) and K('.p) = K(,Q).

Proof a) By hypothesis, K2n('.p) :$; .0 for all n; and of course K2n+ 1 (.0) ::;; '.p. Henee by 8.3a),

X"2n-l (.0) ~ X"2n-l ('.p) ~ X"2n('.p) ~ X"2n(,Q) ~ X"2n+1 (.0) ~ X"2n+l ('.p).


%2n-1 (.0) ~ %2n-1 (~) for all n ~ O. Hence K2n(~):$ K2~(.Q) and K(~) :$ K(.Q); also K2n- 1 (.0) :$ K2n- 1 (~) and K(.Q) :$ K(~).

b) Since ~ is finite, there is an integer r such that %2n+1 (~) = %2r+1 (~) for all n ~ r, by 8.3b). Then K2n+1 (~) = K(~) for all n ~ r. It follows from 8.3a) that if n ~ r,

%2n+1 (~) ~ %2n-2r-1 (.0) => %2n-2r(.Q) ~ %2n+2(~) =>%2n+3(~) ~ %2n-2r+1(.Q),

since K2n+3(~) = K(~) :$ .0. But %2r+1 (~) ~ %-1(.0) since K(~) :$ .0; thus %2n+1 (~) ~ %2n-2r-1 (.0) and %2n-2r(.Q) ~ %2n+2(~) for all n ~ r. Hence K2n+1(~) :$ K2n-2r-1(.Q) and K(~) :$ K(.Q); also K2n- 2r(.Q) :$ K2n+2(~) and K(~) ~ K(.Q).

c) This follows from a) and b). q.e.d. We shall now obtain the properties of .K. K mentioned in the

introduction to this section. This is thus the last explicit use that we make of Definition 8.1.

8.7 Lemma. Suppose that ~ is a p-group, that 91


Since [ft,91] ::; Z(ft), it follows from the last of these conditions that ~ $. 91. Also

Hence we may take in = ~, the condition d) being satisfied with 3 = 1, .0 = Z(ft).

Next suppose that ft E .ff,,-l (~). Since Kn(~) 1. 91, there exists [ E .ff,,(~) such that [ 1. 91. By 8.2d), Kn- 1 (~) ::; N'll([)' Since ft E

.ff,,-l(~)' ft ::; N'll([)' Thus [ft, Z([)] ::; Z([) and [ft, Z([), Z([)] = 1. If Z([) $ 91, choose in = Z([); d) is satisfied with 3 = 1, .0 = Z(ft). Suppose, then, that Z([) ::; 91; we show that we may then take in = [. This is c1ear if [ft, [, [] = 1, for then d) is again satisfied with 3 = 1, .0 = Z(ft). Suppose then that [ft, [, [] =F 1. Since ft::; Kn- 1 (~), [Kn- 1 (~), [, [] =F 1 and [Kn- 1 (~), [] $ Z([). But [ E %n(~), so by 8.1(3), n is even. Let 3 = Z(ft) (') Z([), .0/3 = Z(ft/3). Then

and

[Z([), .0, .0] ::; [91, ft, ft] ::; [Z(ft), ft] = 1.

Hence since [ E .ff,,(~), 8.1(2) gives [[,.0] ::; Z([). Thus [.0, [, [] = 1. And since ft ::; N'll([)'

[ft, [, [, ft] ::; ft' (') [' ::; Z(R) (') Z([) = 3. Thus [R, [, [] ::; .0 and d) is satisfied. q.e.d.

8.8 Lemma. Let U, V be z'(fj-modules. Ifu EU, V E V,

for all g E (fj and r ~ O.

Proof This is a straight-forward calculation, using induction on r. It is trivial for r = O. If r > 0, we have

u v(g - 1) = (ug- 1 v)(g - 1) - ug-1(g - 1) v.


Replacing u by ug-(,-l)(g - 1r1- j,

ug-(,-l)(g - 1r1- j v(g - 1) = (ug-'(g - 1r1- j v)(g - 1) - ug-'(g - 1rj v.

But by the inductive hypothesis,

u v(g - 1)(g - 1)'-1

= 'f (_1r1-j(r-:-1)(ug-(,-1)(g - 1r1- j v(g - 1))(g - 1)i. j=O ]

Hence

u v(g - 1)' = 'f (_1r1- j (r ~ 1)((ug_,(g - 1r1- j v)(g - 1) j=O \ ]

- ug-'(g - 1rj v)(g - 1)i f .(r - 1) . .

= L. (-1r J (ug-'(g - l)'-J v)(g - 1)1 j=l ] - 1

+ 'f (_1rj (r ~ 1) (ug-'(g - 1rj v)(g - 1)i j=O ]

= t (-1rj(~)(ug-'(g - 1),-j v)(g - 1)i. q.e.d. j=O ]

8.9 Lemma. Let X, U be normal subgroups of the group GJ. Let IX be an automorphism of G) for which X, U are IX-invariant and

EU, IX, ... , IXJ = [EU, XJ, IX, ... , IXJ = 1. m 11

Then if x E X, U E U,

[u, [x, 1X~+~':IIX]] E EU, I, G)] [G), U, IJ [I, G), U].

Proof. Let f> = (IX) and let U, V denote the lf>-modules U/[U, G)], 1/[1, G)] respectively, written additively. By VIII, 6.1, there is a l-bilinear mapping y of U x V into W = EU, 1]/91, where

91 = EU, I, G)J [G), U, IJ [I, G), U],


given by

(y[U, ffi], x[l, ffi])y = [y, x]91 (y E U, X EI).

This gives rise to a Z-linear mapping of U V into W. Now W is also a Zf}-module and is a Zf}-homomorphism, for

(((y[U, ffi]) (x[l, ffi]IX) = ((YIX[U, ffi]), (xlX[l, ffi]y = [YIX, XIX] 91 = ([y, X] 91) IX = (((y[U, ffi]) (x[l, ffi])IX.

By hypothesis, CU, IX, ... , IX] = [W, IX, ... , IX] = O. It follows from 8.8 that if u E U, V E V, thenm n

u V(IX - 1r+n- 1 m+n-l (m + n - 1)

= L (-lr+n- j +1 . (ulX-(m+n-1)(IX-l)m+n-1-j V)(IX-1)i j=O ]

= t(1X - 1)"

for some tE U V. Thus

(U V(IX - l)m+n-l) = (t)(1X - l)n = 0,

since t E W. Hence if y E U, X E 1,

(y[U, Gi], [x, Cl, ... , oc][;E,


Proo! By 1.1, there exists a characteristic subgroup .R of 91 such that .R ~ C91(.R)and.R/Z(.R)isanelementary AbeliansubgroupofZ(91/Z(.R)). By 8.7, there is a subgroup 91 of dass at most 2 such that .R s N'll(91), 91 1, 91 and [.0, 91, 91J = 1, where .0 is a subgroup of .R containing Z(.R). Thus

[Z(~), 91, 91J s [.0, 91, 91J = 1

and

[.R, 91, 91, 91J s [91, 91, 91J = 1.

Choose c to be an element of 91 not lying in 91. Since Z(91) S C91(~) s Z(.R),

[Z(91), c, c J s [Z(.R), 91, 91J = 1.

Thus b) is proved. We prove a) by induction on lXI. Let { = Aut 91, and consider (

as a group of operators on 91. Thus X/I!) is a (-chief factor of 91. If X s~,

[X, c, c, c, c J s [~, 91, 91, 91, 91] = 1,

as required. Suppose that X ~ R Then [~, XJ :f. 1 since .R ~ C91(.R). Let U be a minimal characteristic subgroup of 91 such that U s .R and m = CU, XJ :f. 1. Since [91, UJ < u, it folIo ws that [91, U, XJ = 1. Hence by HalI's three subgroup theorem (III, 1.10),

CU, X, 91J [91, U, X] [X, 91, U] s CU, X, 91] [91, U, X] = [m, 91]. Let rl be the automorphism of 91 induced by c. Then

CU, rl; 3J = CU, c, c, c] S [~, 91, 9\, 9\J = 1.

Also m = CU, XJ s [~, 91] s Z(~), so

[m, IX; 2J = [m, c, c J s [Z(~), 91, 91J = 1. By 8.9,

[u, [x, rl, rl, rl, rl JJ E [m, 91J


for any U E U, X E I. Let [/[IJ3, 91] = Cl /[m,91] (U). Thus [x, IJ., IJ., IJ., IJ.] E [ for all x E I, and the assertion follows at once if [ =:; 'D. Suppose then that [ f;, 'D. Thus 'D < 'D[ =:; I. But [ is a characteristic subgroup of 91 and there is no characteristic subgroup properly between land 'D. Hence 'D[ = land I/'D is G>-isomorphic to [/([ 11 'D). But [ < I, since otherwise,

m = CU, I] $; [m, 91], and this is impossible since lJ3 # 1. Hence by the inductive hypothesis, [[, c, c, c, c] =:; [ 11 'D. Since c induces the automorphism IJ., [[, IJ., IJ., IJ., IJ.] =:; [ 11 'D and

[I,c,c,c,c] = [I,IJ.,IJ.,IJ.,IJ.] =:; 'D. q.e.d.

8.11 Theorem (GLAUBERMAN [5]). If P ~ 5, K and K control transfer in any group G>.

Proof Write K for K or K. Let ,0 be a section of G> for which Op(,Q) ~ Co(Op(,Q)) and K(60 ) ~ ,0, where 6 0 E Sp(,Q). Since K(Op(,Q)) is a characteristic subgroup of Op(,Q), K(Op(,Q)) 'D and there is no characteristic subgroup properly between land 'D. Since a characteristic series of Op(,Q) may be refined to part of a chief series of ,0, [I, g, g, g, g] $; 'D for every chief factor I/'D of,Q with I =:; Op(,Q). Since p ~ 5, it follows that [I, g, ... ,g] =:; 'D

p-l for every such chief factor. Thus K and K control transfer in G>, by 7.7. q.e.d.

8.12 Remarks. For p = 2, no positive characteristic 2-functor controls transfer in every finite group. For there exist simple groups G> in which the Sylow 2-subgroup 6 is a maximal subgroup; (for example, G> = PSL(2, 17) (II,8.27)). Then N(6(W(6)) = 6 for any positive charac-teristic 2-functor W, and 6 11 G>' = 6 > 6' = 611 N(6(W(6))'.

For p > 5, J controls transfer in any group (GLAUBERMAN [2]).

As an application of Theorem 8.11, we have the following.

8.13 Theorem (THOMPSON). Suppose that p ~ 5, 6 E Sp(G and 6 # 1. IfN(6(6)/C(6(6) is a p-group, then OP(G < G>.


Proof Suppose that this is false and that (fj is a counterexample of minimal order. Thus (fj = OPfj). By 8.11, 9l = OP(9l), where 9l = N(!;(K(6)).

Since (fj = OPfj) and 6 =f. 1, (fj is not p-nilpotent. Hence by Burn-side's transfer theorem (IV, 2.6), C(!;(6) < N(!;(6). But by hypothesis, N(!;(6)!C(!;(6) is a p-group, so C(!;(6) ~ OP(N(!;(6)). Hence N(!;(6) =f. OP(N(!;(6)). Since 9l = OP(9l), it follows that N(!;(6) =f. 9l. Thus 6 =f. K(6), so 6jK(6) is a non-identity Sylow p-subgroup of9ljK(6). Hence 9ljK(6) satisfies the hypothesis ofthe theorem. Since 6 =f. 1, K(6) =f. 1 and 19ljK(6)1 < l(fjl. Since (fj is a counterexample of minimal order, it follows that 9ljK(6) > OP(9ljK(6)). But this implies that 9l > OP(9l), a contradiction. q.e.d.

8.14 Remark. s. D. SMITH and A. P. TYRER [1] have studied groups in which IN(!;(6): 6C(!;(6)I = 2. For p odd and OPfj) = (fj, they have shown that if 6 is non-cyc1ic but of c1ass at most 2, then (fj is p-soluble of p-Iength 1. The proof is mainly by modular representation theory.

The following is the proof of a long-standing conjecture of Zassenhaus.

8.15 Theorem. If (fj is a finite group and N(!;(6) = 6 for every non-identity Sylow subgroup 6 of(fj, l(fjl is apower of a prime.

Proof Suppose that (fj =f. 1; we show that OPfj) < (fj for some prime p. This is c1ear if (fj is soluble. But otherwise, I (fj I is divisible by at least one prime p ~ 5, by 2.8. Let 'l3 E Si(fj). By hypothesis, N(!;('l3)jC(!;('l3) is a p-group, so by 8.13, OPfj) < (fj.

Now if G> is not a p-group, OP(G contains a non-identity Sylow subgroup .Q of G>. By the Frattini argument,

But by hypothesis, N(!;(.Q) = .Q ~ OPfj), so (fj = OPfj), a contradic-tion. Thus (fj is a p-group. q.e.d.

The results for K and K corresponding to the ZJ-theorem (3.9) hold also for p = 2. To prove them, we need the following lemma.

8.16 Lemma. Suppose that 1 < 9l ~ 9Jl ~ (fj, where 9l is a minimal normal subgroup of (fj and 9Jl is a sub normal subgroup of (fj. Then 9l is the direct product ofminimal normal subgroups of9Jl.


Proof We use induction on Im: 9Jl1. If 9Jl = m, the assertion is trivial. If9Jl < m, there exists anormal subgroup i! of m such that 9Jl ~ i! < m. Let r be the greatest integer for which there exist minimal normal sub-groups m1, ... , mr of i! with mi ~ m and

Since m > 1, r ~ 1. If m =


are both normal. Hence either K(6) # K(~) or K(6) # K(~). By 8.7, ~ has a subgroup 9l with the stated properties.

(2) If G: = C6i(Z(21)) and IDl = 9lG:, then 21 ::;;; G:

9. Further Properties of J, ZJ and K 69

9. Further Properties of J, ZJ and K

9.1 Defmition. The characteristic p-functor W is said to control fusion strongly in (f) if whenever 6 E Sp(f)), ~ :::;; 6, g E (f) and ~g :::;; 6, there exists h E N(jj(W(6)) such that gh- 1 E C(jj(~)'

We shall not need the corresponding notion of controlling fusion, (which means that ~g :::;; 6 implies that gh- 1 E N(jj(~) for some hE N(jj(W(6))).

9.2 Lemma. Suppose that 6 E Sp(f)) and that W is a characteristic p-functor.

a) W strongly controls fusion in (f) if and only if, whenever x, y are elements of 9'(6) (see 5.6) conjugate in (f), then x, y are conjugate in N(jj(W(6)).

b) Suppose that W strongly controls fusion in (f). Then (i) W controls transfer in (f), and (ii) if(f) is p-constrained, 0p,(f))W(6)


Proof Suppose that 6 E Sp(f)) and define the relation", on 9'(6) by putting x '" y ifx, y are conjugate in N(D(W(6)). Clearly, '" is an equiv-alence relation which contains fusion in N(D(W(6)). By 9.2a), W strongly controls fusion in (f) if and only if '" contains fusion in (f). Suppose that this is not the case. By 5.9, there exists a non-identity W-extremal sub-group ~ of 6 such that '" does not contain fusion in N(D(~) but '" con-tains fusion in N(D(W(Ns(~)))' Since Ns(~) E Sp(N(D(~)), it follows that there exist x, y in 9'(N(D(~)) such that y = xg for some g E N(D(~), but x + y. Since '" contains fusion in N(D(W(N($(~))) and 6 fI N(D(W(N(D(~)))

~ N($(~)' it follows that x, y are not conjugate in N(D(W(N($(~)))' Thus W does not strongly control fusion in N(D(~), contrary to hypothesis.

q.e.d.

9.4 Lemma. Suppose that W is a positive characteristic plunctor. Let ~ = Op(f)). Suppose that ~ =1= 1 and (f)/C(D(~) is a p-group. IfW strongly controls fusion in every proper section of (f), then W strongly controls fusion in (f).

Proof Suppose that 6 E Sp(f)). If 6 = ~, N(D(W(6)) = (f) and there is nothing to prove. Suppose then that 6 > ~. Since W is positive,

W(6/~) =1= 1. Thus if U/~ = W(6/~), U > ~. Since ~ = 0p(f)), it follows that N(D(U) < (f). Clearly 6 ~ N(D(U),

Suppose that ~ ~ 6, g E (f) and ~g ~ 6. Let .0 = ~R Thus ,Qg ~ 6. By 4.3b), there exists t E (f) such that 9l = ,Qt is extremal in 6. By hypo thesis, ~ =1= 1 and W strongly controls fusion in (f)/~, so there exists a' E N(D(U) such that 9l = ,Qa'. And since N(D(U) < (f), W strongly controls fusion in N(D(U), so there exists a E N(D(W(6)) such that 9l = ,Qa. Thus 9la- 1 9 ~ 6. Thus, again, there exists c' E N(!;(U) such that x a- 1 9 == xc' mod ~ fr all XE 9l, and there exists c E N(fj(W(6)) SUC;l that xc' = XC fr all x E 9l. Thus x a- I gc- I == x mod ~,and a-1gc-1 E 1), where

1)/~ = C(!;/'ll(9l/~). Now

so 1)

9. Further Properties of J, ZJ and K 71

Thus a-1ge- 1 = bd for some bE C(!;(9t), d E 1) (") 6. Thus xa-Igc-I = X bd = xd for all x E 9t. Since 9t = .oa, it follows that ygC- 1 = yad for all y E .0; hence g(adefl E C(!;(Sl). But ade E N(!;(W(6)). Thus W strongly cont

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