Conjugacy Class As a Transversal in a Finite Group

Journal of Algebra 239, 365᎐390Ž. 2001 doi:10.1006rjabr.2000.8684, available online at http:rrwww.idealibrary.com on

A Conjugacy Class as a Transversal in a Finite Group

Alexander Stein

Uni¨ersitat¨ Kiel, 24118 Kiel, Germany View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Gernot Stroth provided by Elsevier - Publisher Connector Received September 1, 2000

The object of this paper is the following

THEOREM A. Let G be a finite group and let g g G. If g G is a trans¨ersal to some H F G, then² gG : is sol¨able. Remarks.1IfŽ. g G is a transversal, the coset H contains exactly one aGg F a <<

DEFINITION. Let G be a finite group and let g g G such that G s G gCGŽ. g. Call G a CCCP-group, where CCCP stands for conjugacy class centralizer product. The idea to study these groups came from a paper by Fischerwx 10 . For details of his work and the relation to Theorem A we refer to Section 1, but we will give here a short summary: Fischer defines a so called ‘‘distributive quasigroup’’ Q and a certain finite group G s GQŽ.F Aut Ž.Q . His main statement is that G is solvable. The major interest in Q comes from the fact that Q can be defined in group theoretic terms of G itself. In fact Q can be seen as a conjugacy class of G and multiplication in Q is the conjugation action of Q on itself inside G. However, to construct a distributive quasigroup from a given group, this group has to fulfill the following two properties: Ž.1 G is a CCCP-group for some g g G in the above definition, y1 y1 Ž.2 for all a, b, c g g Gcthe following holds: a bbcs a a.

~~365 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. 366 ALEXANDER STEIN~~

This second condition reflects the left distributivity of Q, but it is hard to check. So the idea was to drop this condition: The construction of a quasigroup Q still works and out comes a so called right distributive quasigroup Q˜˜. A group G s GQŽ.can be constructed in the same way as for distributive quasigroups and an analogue theorem of Fischer’s theorem is a corollary of Theorem A. For the second approach to Theorem A we have to weaken the CCCP- s G property: If G CggGŽ. , then obviously

~~G l s ) g CgG Ž. Ä4g Ž.~~

Call this second condition the Glauberman condition, as Glauberman showed in his famous Z* by theoremwx 11 that if Ž.) holds for some involution g, then ² g G : is solvable. An analogue theorem for elements of odd prime order is still an open problem, and to give an impression how this could look like we state here the following Conjecture. Let G be a finite group and let g g G be of prime order p. G l s g r Then g CgGpŽ. Ä4g exactly if gOЈŽ. G ZG ŽOGpЈ Ž...

Note that OpЈ may be nonsolvable and that the conjecture holds if G is solvable or p s 2. However, there is another generalization of Glauber- man’s theorem which is a corollary of Theorem A: y THEOREM B. Let G be a finite group and let g g G such thatŽ g1 .GG g l s G ¨ CgGŽ. Ä41.Then² g: is sol able. The third approach to Theorem A needs another definition:

~~DEFINITION. Let G be a finite group and let ␣ g AutŽ.G such that~~

~~s wx␣ N g ␣ G Ä4g, g GCG Ž..~~

Call such a group an ␣-CCP-group, which stands for commutators centralizer product. The relation between ␣-CCP-groups and CCCP-groups is very simple: ␫ ␫ Every CCCP-group is an gg-CCP-group, where is the inner automorphism corresponding to the g g G for which G is a CCCP-group. If G is an ␣-CCP-group, then any extension G.²:␣ is a CCCP-group for ␣. Details for this can be found in Section 2. The idea behind this definition is that the authormorphism ␣ in an ␣-CCP-group is a generalized fixed point free automorphism: Indeed if G is a finite group admiting a fixed point free automorphism ␣, then G is an ␣-CCP-group as every element of G is a commutator with ␣. CONJUGACY CLASS AS A TRANSVERSAL 367

These automorphism are widely studied: Thompson showed inwx 20 that G is nilpotent if G admits a fixed point free automorphism of prime order. Rowley showed inwx 17 , using the classification of finite simple groups, that G is solvable in the general case. The proof of Theorem A is based on the following generalization of these results:

THEOREM C. Let G be an ␣-CCP-group. Thenwx G, ␣ is sol¨able. ␣ ␣ Remark. As the -CCP-property gives no restriction to CGŽ., we have wx␣ s wx␣ ␣ to restrict ourselves to G, , but nevertheless G G, CGŽ.by the ␣-CCP-property. As indicated above, Theorem A is a consequence of Theorem C and we will now give an overview of the proof of Theorem C and begin with some basic properties of ␣-CCP-groups: The ␣-CCP-property can be restated in the following way: A finite group G has the ␣-CCP-property exactly if it has the following wx␣ N g l g N g ␣ g s Property Ž.i Ä g, g G4Äh h CGŽ., g G4Ä41. From this easily follows Let G be an ␣-CCP-group. Then G has the following Ј wx␣ N g l ␣ s Property Ž.i Ä g, g G4 CGŽ. Ä41. Note that this condition is the ␣-CCP equivalent of the Glauberman condition Ž.) . The next two properties were found by studying the work of Fischer on distributive quasigroupswx 10 . The search for an analogous proof for right distributive quasigroups failed, but out came the following properties of ␣-CCP-groups: Let G be an ␣-CCP-group. Then G has the following properties: Property Ž.ii If U F G such that U␣ s U, then U is an ␣-CCP-group. Property Ž.iii If N eG such that N ␣ s N, then GrNisan␣-CCP- ␣ s ␣ r group and CG r NGŽ.C Ž.N N. These two properties show that the problem is better viewed from the group theoretical point of view than from the point of quasigroups, as it fits nicely into the existing group theory: The proof of Theorem C can make heavy use of the ‘‘minimal counterexample’’ due to these properties. This is a major difference from Fischer’s distributive quasigroups, as his proof first has to develop a theory of distributive sub- and factor quasigroups. 368 ALEXANDER STEIN

Details of these properties and some further applications can be found in Section 2. A first step toward Theorem C was to show that a minimal counterexample is a nonabelian simple group, which is quite elementary up to a special case where the odd order paperwx 9 , Thompson’s work on N-groups wx19 , and a classical result of Brauer and Suzuki wx 3 were applied. To go further, the classification of finite simple groups was used. This seems quite brutal, but that is exactly the point where the classification of finite simple groups was used by Rowley in the special case of fixed point free automorphisms inwx 17 , so there seems to be no way to avoid this. ␣ / However, the case CGŽ. 1 is quite different from the case of fixed point free automorphisms, so in the last case we use Rowley’s work. In fact ␣ / the condition CGŽ. 1 is the key idea to our proof. ⌺ s The alternating groups were treated using the fact that nnAutŽ. Alt for n ) 6 and PropertyŽ. iЈ then gave a contradiction. For the sporadic groups the information mostly came from the Atlaswx 7 and PropertyŽ. iЈ again gave a contradiction. Details for this part of the proof can be found in Section 3. A big problem was the groups of Lie type. A generalization of Rowley’s proof failed but led to interesting results: Rowley relied heavily on the fact that a fixed point free automorphism normalizes exactly one Sylow-r-subgroup for any prime r dividing <subgroup of G,ifCGŽ.contains elements of order p, where p is the characteristic of G. Another idea of Rowley was to use the building of G. In this spirit we showed the ␣ ␣ following; If CGŽ.contains elements of order p as above, either CGŽ. acts flag transitively on the building of G or the Lie rank of G is small and some more restrictions on ␣ and G hold. In the first case we can apply a result of Seitzwx 18 on flag transitive subgroups to get the contradiction s ␣ G CGŽ.. ␣ / As Rowley’s result indicates, CGŽ. 1 in the minimal counterexample, g ␣ so we can find always an element x CGŽ.of prime order r. As the centralizers of involutions in simple groups of Lie type in odd characteris- ␣ tic are all classified, we can show that CGŽ.contains elements of order p, g ␣ ␣ if it contains involutions: If i CGGŽ.is an involution, CiŽ.is -invariant but mostly contains large nonsolvable sections which are groups of Lie type in characteristic p. By minimality of G now ␣ has to centralize ␣ these sections and by PropertyŽ. iii something in CG Ž.has to cover these ␣ sections, so CGŽ.contains elements of order p. ␣ Now the last step was now to show that CGŽ.contains always involutions or elements of order p. Again Rowley’s work gave a good idea: In the case of the alternating groups he looked at cyclical Sylow-r-subgroups R CONJUGACY CLASS AS A TRANSVERSAL 369

␣ ␣ r normalized by . He showed that has to act trivially on NRGGŽ.CR Ž., forcing R to lie in the center of its normalizer, a contradiction. From this g ␣ ␣ came the observation that for x CGŽ., has to act trivially on r ␣ NxGGŽ² :.Cx Ž .. Now PropertyŽ. iii comes into play: If centralizes some ␣ ␣ -invariant section, something in CGŽ.has to cover this section. This ␣ property gave the opportunity to show that CGŽ.cannot be any subgroup ␣ F ␣ of G, as we can use PropertyŽ. iii to ‘‘blow up’’ CGGŽ.: If some U C Ž., ␣ ␣ then CGGŽ.has to cover all sections of NUŽ., which are centralized by . The major result for getting such sections is Proposition 4.1 in Section 4. Its proof makes use of the theory of algebraic groups and Frobenius morphisms to get some control over CxGŽ.for semisimple x. Another useful tool was the natural modules for the classical groups. Note that this proposition is quite independent and the entire Section 4 has no relation to the previous sections. The above mentioned ‘‘blowing up’’ strategy was used to show that ␣ CGŽ.contains always elements of order 2 or p under the assumption that G is a minimal counterexample. Now only ‘‘small’’ cases were left and a direct ‘‘blowing up’’ strategy was used to show the contradiction G s ␣ CGŽ.. This part of the proof can be found in Section 5.

Some remarks on the notations. General group theoretic terms come fromwx 1 ; the notation for the classical groups and their modules comes fromwx 13 . From w 4, 5, 19 x comes the notation for algebraic groups of Lie type, Weyl groups, and maximal tori. The notation for isomorphism types wx of groups, especially extensions, follows 7 . The symbols AqnnŽ., Bq Ž., CqnŽ., . . . always refer to the nonabelian simple groups, if these exist. Otherwise we assume that these groups are center free and have no nontrivial pЈ-factor group, where q s p f for some prime p. Under a group G of Lie type in characteristic p we understand the following:

~~s p, pЈ s r F Let S OGŽ.and S S ZSŽ.. Then CSG Ž. S and S is a direct product of simple groups of Lie type as defined above.~~

~~The author acknowledges the support of Professor Stroth and financial support by the DFG.~~

~~1. QUASIGROUPS AND CCCP-GROUPS~~

DEFINITIONS Ž.1.1 . Let Q be a finite set and let ) be a binary operation on Q. Call Q a quasigroup iff for all a, b g Q the equations a) x s b and y) a s b have a unique solution x respectively y g Q. 370 ALEXANDER STEIN

g ␭ ␳ For a Q the maps aaand are called left respectively right translations which are defined by ␭ ¬ ␭ s ) aa: Q Q, Ž.b a b ␳ ¬ ␳ s ) aa: Q Q, Ž.b b a. ␭ ␳ g Note that aa, SymŽ.Q ,asQ is a quasigroup. s ␳ N g s ␭ N g Define GQraŽ. ²a Q :and GQ la Ž. ²a Q : A quasigroup is called right distributi¨e iff the equation

~~Ž.a) b )c s Ž.Ž.a)c ) b)c~~

~~holds for all a, b, c g Q. It is called left distributi¨e iff the equation~~

~~a)Ž.Ž.Ž.b)c s a) b ) a)c~~

~~holds for all a, b, c g Q, and it is called distributi¨e iff Q is both right and left distributive. Remark Ž.1.2 . Let Q be a left distributive respectively right distributive~~

~~quasigroup. Then GQlrŽ.respectively GQ Ž.is a group of automorphisms of Q.~~

THEOREM Ž.1.3wx 10 . Let Q be a finite distributi¨e quasigroup. Then ¨ GrŽ. Q is sol able. As indicated in the preface, an analogous theorem is a consequence of Theorem A and its proof is given in the last paragraph of this article:

THEOREM Ž.1.4 . Let Q be a finite right distributi¨e quasigroup. Then ¨ GrŽ. Q is sol able. To show the relation between the right distributive quasigroups and the CCCP-groups we state some elementary lemmata:

LEMMA Ž.1.5 . Let Q be a right distributi¨e quasigroup and let a g Q. ␳ s ␳␳G rŽQ. ␳ s ␳ b Then GrGŽ. Q C ŽQ. Žaa . and a) ba. Especially G rŽ. Q is a ␳ r CCCP-group for a. Proof. From the right distributivity follows ␳␳ s ) ) ) s ) ) s ␳␳ Ž.Ž.Ž.Ž.Ž.Ž.Ž.ba) babx x b a b x a b x .

Further, let a and b be elements of Q. By the quasigroup property there is ␳ ) s ␳ c s ␳ an element c with a c b. Therefore ab. Now the set of all right translations is closed under conjugation and generates GQrŽ.. Therefore the right translations form a conjugacy class of GQrŽ.. CONJUGACY CLASS AS A TRANSVERSAL 371

For showing the above factorization we first show that C Ž.␳ s G r ŽQ. a Stab Ž.a : G r ŽQ. Ž.i Let b g Q. Then b) b s b.If b)c s b then b s c. Ž.b) b ) b s Ž.Ž.b) b ) b) b by the right distributivity. As Q is a quasigroup the left ␭ ) translation b) b is a permutation on Q. But now b and b b have the same image which forces b s b) b.Sob) b s b for each b g Q. If now ) s ) s s ) ␭ b c b then b c b b b and as b is a permutation of Q we have b s c. g s wx␳ s s Ž.ii If x AutŽ.Ž.Q , xa a, then x, a 1. If xaŽ. a then ␳ s ) s ) s ) s ␳ g xŽaa Ž..b xb Ža .xb Ž.xa Ž.xb Ž.a ŽŽ..xb for each b Q; ␳ s ␳ wx␳ s therefore aax x and x, a1. g wx␳ s s wx␳ s Ž.iii If x AutŽ.Q , x, aa1 then xaŽ. a.If x, 1 then s ) ) s ␳ s ␳ s ) xaŽ.Ž.Ž.Ž.ŽŽ..ŽŽ..Ž.Ž.xa a by i . Now xa a x aaa xa xa a.By i this forces xaŽ.s a. Now byŽ. ii and Ž iii . we have C Ž.Ž.␳ s Stab a . G rrŽQ. aGŽQ. ␥ s y1 s Let g be any element of GQrŽ., g and b gaŽ.. By the quasigroup property there is an element c g Q with a)c s b. Then a s ␥ Ž.b s ␥ ) s ␥␳ s ␳␥ s s y1␳ Ž.ŽŽ..a c cca . Setting h we have ha Ž.a and g h c. s y1 g ␳ As haŽ. a we have h CG ŽQ.Ž.aras seen above and GQŽ.is a ␳ r a-CCP-group. LEMMA Ž.1.6 . Let G be a CCCP-group for g g G. On gG define an operation ) by a) b [ ab if a, b g g G. Then this defines a right distributi¨e quasigroup. s Ghs h G h s Proof. If G CggGGŽ. then also G ŽŽ..CgŽ g. ; that is, G h GGg g cs CggGŽ . . So let a, b g . Then there is a c G with a b and s g g G c hx with x conjugate to g and h CaGŽ.. So for all a, b g there is an x g g G with a x s b and therefore the left translations are surjections. As g G is finite, the left translations are also bijections and the above solution x is unique. y1 Setting y s b a for all a, b g G there is at least this so defined y with y a s b. Therefore the right translations are surjections and even bijections as g G is finite. So the set g G is a quasigroup with the operation ). c So let a, b, c g g G.AsŽ.a) b )c s Žab.cbccbs a s a s Ž.Ž.a)c ) b)c the operation is right distributive.

~~2. CCCP-GROUPS AND ␣-CCP-GROUPS~~

~~In this section we give some basic properties of ␣-CCP-groups which we need later. But first we show the relation between the CCCP-groups and the ␣-CCP-groups. 372 ALEXANDER STEIN~~

LEMMA 2.1. Let G be a CCCP-group for g g G. Then G is an ␣-CCP- ␣ s group with i g , the inner automorphism induced by g. Let H be an ␣-CCP-group and let g g G with ␣Ž.x s xg for all x g G. Then G is a C 3P-group for g. s GGs s f Proof. By definition G CggGGŽ. ,soG gCŽ.Ž g.Ifx hg then y1 y1 x s hgf hy1 h s ghfh ..Ž So for x g G we can write xy1 . g s ghf for g g gy1 s y1 y1 f s y1 gy1 s some f G and h CgGŽ.. Then x hgŽ . and x gxg y1 y1 y1 f s y1wx y1 g ␣ Ž gh.ŽŽ g. g.Ž.hg f, g with Ž.hg CgGŽ.,soG is an - ␣ s g CCP-group for i g InnŽ.G . s ␣ wx␣ N g g On the other side let G CGŽ.Ä x, x G4. Then for y G we y1 g s wxwx␣ s s y1 f g ␣ s can write Ž y . hf, hf, g hgŽ . g with h CGŽ. y1 s y1 ffs y1 y1 s y1 fŽ gh.y1 CgGŽ.. Thus y ghŽ g . and y gh g Ž.gh g with y1 g Ž.gh CgG Ž.. The following proposition is the crucial tool for working with ␣-CCP- groups:

~~PROPOSITION 2.2. Let G be an ␣-CCP-group. Then the following hold:~~

wx␣ f g ␣ g wx␣ s Ž.i If g, CGŽ.for some f, g G, then g, 1. Ž.ii Let U F G with U␣ s U. ThenÄwx u, ␣ N u g U4Äs wxg, ␣ N g g G4 l U and U is an ␣-CCP-group. e ␣ s ␣ s ␣ r Ž.iii Let N G with N N. Then CG r NGŽ.C Ž.N N and GrNisan␣-CCP-group. s ␣ Proof. Let H CGŽ.. Ž.i Assume wxg, ␣ f g H.Then Hgfwx, ␣ s Hg wx, ␣ f wxf, ␣ s Hfwx, ␣ . Since Äw g, ␣ N g g G4 is a transversal, wxwxgf, ␣ s f, ␣ ; hence g g H. Ž.ii As each left coset of U l H is contained in exactly one left coset of H, it contains at most one G-commutator with ␣. But U contains as many U-commutators as U-cosets. Therefore U is an ␣-CCP-group and each G-commutator in U can be realized in U itself. Ž.iii Let fN g GrN. Then f s wxg, ␣ h for some g g G, h g H. s wx␣ g ␣ r ␣ Thus fN gN, hN and hN CG r N Ž..SoG N is an -CCP-group. g ␣ wx␣ g g Let eN CG r N Ž.,so e, N. ByŽ. ii there is an n N, such that wxwxe, ␣ s n, ␣ . Now wxwe, ␣ s eny1 n, ␣ xws eny1, ␣ xnwxn, ␣ ,so eny1 g H ␣ F r and therefore CG r N Ž. HN N. As the other inclusion is obvious, the lemma holds.

~~COROLLARY 2.3. Let ␣ g AutŽ.G , e g Inn Ž.G with ␣ / ␣ e, and w␣, ␣ e x s 1. Then G is not an ␣-CCP-group. CONJUGACY CLASS AS A TRANSVERSAL 373~~

/ ␣␣y1 e s wxwx␣ s ␣ g ␣ g Proof. Then 1 Ž . e, g, CGŽ.for some g G, so by PropositionŽ.Ž. 2.2 i , G is not an ␣-CCP-group. s = = иии = ␣ g LEMMA 2.4. Let G L12L Ln and let AutŽ.G with ␣ s ␣ s ␣ Ž.LiiLq1 and Ž.Ln L11. If G is an -CCP-group, then L is a ␤-CCP-group for ␤ s ␣ n. FurthermoreŽ< C Ž.␤ <, n. s 1. L1 g ␣ ␣ ny 1 Proof. For c12, c ,...,cn L112define Ž.c , c ,...,cn as cc12,...,cn . ␣ g s wx␣ Assume G is an -CCP-group, so for g L1 we have g x, h with x g G and h g C Ž.␣ . Let F s C Ž.␤ . GL1 s wx␣ s y1 ␤ y1 y1 If x Ž.a12, a ,...,an we have x, Žaa1 n , aa21,...,aanny1.. Furthermore h s Ž.b, b,...,b for some b g F. Thus comparing the unique y1 ␤ s y1 s - s iy1 factors of g we get a1 anib g and a q1 abii1 for i n.Soa ab1 s wx␤ n ␤ ¬ n and g a11, b . Thus L is a -CCP-group. But then the map p: y y is a surjection on F; thus Ž<

~~Type of G p~~

~~s A1Ž.2 3:2 2 2 s B2Ž.2 5:4 2 2 s 2 A28Ž.2 3:Q 2 s 2 D48Ž.2 3:D 2 s 2 A1Ž.3 2:3 3~~

␣ s ␣ / Proof. Assume acts nontrivially and let F FGŽ..IfCF Ž. 1, then wxF, ␣ s 1 byŽ.Ž. 2.3 i , as otherwise F contains a nontrivial commuta- ␣ ␣ tor with which is conjugate in G to some nontrivial element of CF Ž.. Now wxG, ␣, F s 1 by the three subgroup lemma; thus wxG, ␣ F F, but F does not contain nontrivial commutators with ␣ as seen, so wxG, ␣ s 1in this case. ␣ y ( s Therefore acts fixed point freely on the set F Ä41.If G Alt 4 2 2 : 3, thus ␣ has order 3 and the lemma holds. In the other cases G / OG2 Ž.and GrOG2 Ž.contains a unique involution in its center. By ␣ PropositionŽ.Ž. 2.2 iii CG Ž.contains an element covering this involution 374 ALEXANDER STEIN and the centralizer of this element is in all cases an ␣-invariant Sylow 2-subgroup. ⌺ n EXAMPLE 2.7. Let G be of type 4 or a dihedral group of order 2 , n ) 2. If G is ␣-CCP, then ␣ is trivial. Proof. If G is a dihedral group let T s ²:y be the cyclical normal ␣ r F ␣ subgroup. Now acts trivially on G T, so some ²:x CG Ž .covers this <

~~3. THE MINIMAL COUNTEREXAMPLE, PART I~~

~~For the rest of this article let M be a minimal counterexample to Theorem 1.1. In this section we show, using the classification of finite simple groups, that M is a simple group of Lie type.~~

LEMMA 3.1. FMŽ.s 1, M s SocŽ.M , and ␣ acts transiti¨ely on components. Proof. By PropositionŽ.Ž. 2.2 ii we have wxwM, ␣ s M, ␣, ␣ x. Therefore M s wxM, ␣ by minimality. Furthermore FMŽ.s 1 by minimality and s wxwx␣ s s ␣ PropositionŽ.Ž. 2.2 iii . Let C coreM Ž.H .As M, C, 1 C, , M we have wx␣, M, C s 1; thus C F ZMŽ.F FM Ž.s 1. Now 1 / w EMŽ., ␣ x s M, as otherwise w EMŽ., ␣ x would be a smaller counterexample. By minimality now ␣ acts transitively on the components.

LEMMA 3.2. M is simple. s = иии = ␣ Proof. By LemmaŽ. 2.4 M L1 Ln and acts transitively on the factors; by LemmaŽ. 2.4 either ␣ nns 1orn s 1. Assume ␣ s 1 and n ) 1. F = ␣ = иии = ␣ ny 1 Let E L1 nonsolvable. Then E E E is a smaller coun- s terexample; thus E L11and L is minimal simple. By the odd order wx paper of Feit and Thompson 9 , now L1 is of even order. By the work on wx N-groups of Thompson 21 , L1 contains a unique conjugacy class of involutions. CONJUGACY CLASS AS A TRANSVERSAL 375

~~By the work of Brauer and Suzuki,wx 3 and a classical theorem of~~

~~Burnside now a Sylow-2-subgroup of L1 is neither a quaternion nor a cyclic group.~~

Therefore L1 contains commuting involutions i, j such that i, j and ij are conjugate. ByŽ. 2.4 n is odd. In the notation ofŽ. 2.4 let x s Ž.i, j,...,i, j,1, so wxx, ␣ s Ž.i, ij,...,ij, ij, j which is conjugate to g ␣ Ž.Ž.i, i,...,i, i, i CM . But this is a contradiction to PropositionŽ. 2.2Ž i. , so n s 1. We now show that M is neither alternating nor sporadic. Our strategy is ␣ ␣ to show that CM Ž.contains a conjugate of , mostly a nontrivial power of ␣. We then use Corollary 1.3 to get a contradiction. G PROPOSITION 3.3. M is not Alt n for n 7. G s ⌺ / ␣ g ⌺ G Proof. As n 7 AutŽ. Alt nn. Let 1 n, n 5. We show by ␣ g induction over the number of orbits of that there is a g Altn with w␣, ␣ g x s 1 and ␣ / ␣ g. Then, by CorollaryŽ. 2.3 , the proposition holds. ␣ g ⌺ ␣ e s ␣y1 Assume acts transitively. Let e n with .If n is even, g ␣ g g ⌺ ␣ f s ␣ 2 e Altnn or e Alt . If n is odd let f nwith . Then one of the elements e, f, ef is in Altn . Assume now ␣ s ␤␥ with< MoveŽ.␤ <

~~␣ ng~~

Ž.1, 2, 3 5Ž.Ž. 1, 2 4, 5 Ž.Ž.1,2 3,4 5Ž. 1,2,3 Ž.Ž.1, 2 3, 4, 5 5Ž.Ž. 1, 2 4, 5 Ž.Ž.Ž.1, 2 3, 4 5, 6 6Ž. 1, 2, 3 Ž.Ž.1, 2, 3 4, 5, 6 6 Ž.Ž. 1, 2 4, 5 Ž.Ž.Ž.1, 2 3, 4 5, 6, 7 7Ž. 1, 2, 3

~~The remaining case ␣ s 1 is no counterexample as wxM, ␣ s 1 is solvable.~~

PROPOSITION 3.4. M is not a sporadic group. Proof. Assume otherwise. Let H [ ²Ž.:Inn M , ␣ . Now ␣ is not an ␣ ␤ / ␤ s ␣ h involution: Otherwise CH Ž.contains a a with for some g s ␫ g h H by Glauberman’s Z* theorem. In fact w.l.o.g. h g InnŽ.M as ␫ s ␣␫ g < < F ␣ h s ␣ g otherwise h g for some g M as OutŽ.M 2 and . Then / ␤␣ s ␣␣h s wx␫ ␣ g ␣ < < F / 1 g , CInnŽ M .Ž.as OutŽ.M 2. But then 1 wx␣ g ␣ g, CM Ž., a contradiction to PropositionŽ 2.2 .Ž. i . 376 ALEXANDER STEIN

Let n be the order of ␣ and let ␾ the Eulerian ␾-function. Then the following holds: Fact 1. All the generators of ²:␣ fall into ␾ Ž.n many conjugacy classes of H: Let h g H with ␣ i s Ž␣ j .h,0F i - j - n, Ž.i, n s 1 s Ž.j, n and ␣ jk s ␣ for some integer k. Then ␣ / ␣ iks ␣ h. As either h g InnŽ.M ␣ g g / ␣␣yik g ␣ or h InnŽ.M we may assume h Inn Ž.M . Then 1 CH Ž. ␣␣yik s ␣y h␣ s wx␣ g s ␫ g and Ž 1. h, InnŽ.M . Now h g for some g M / wx␣ g ␣ and 1 g, CM Ž.which contradicts Proposition Ž 2.2 .Ž. iii . We now look in the Atlaswx 7 for conjugacy classes of sporadic groups which satisfy the following conditions: There are at least ␾Ž.n many of them, all corresponding to elements of order n, their sizesŽ or the orders of a centralizer of one element. are the same, and all of them lie in the same InnŽ.M -coset. We give here the list of all these conjugacy classes and by Fact 1, ␣ is an element of one of these conjugacy classes.

~~࠻ M ²:␣ -classes Power map~~

~~1 M12 4A,4B A,A 2 M24 6A,6B AA,BB 3 Suz 6B,6C BA,BA 4 Suz 6H,6I CC,CD~~

~~5 Co2 4E,4F B,B 6 Fi22 6F,6G AC,BC 7 Fi22 6S,6T DD,DE 8 Co1 6C,6D BA,CA 9 Co1 12HIJK DC,EC,ED,FB 10 J 6B,6C AA,AB 4X 11 Fi24 6G,6H DA,DB~~

Note that in each case there are exactly ␾Ž.n many conjugacy classes given. First assume that ␣ is an element of one of the classes listed in cases 2, 4, 6, 7, 8, 9, 10 or 11. Then the generators of ²:␣ do not fall into ␾ Ž.n many different conjugacy classes of H, contradicting Fact 1: In each of these cases there are conjugacy classes x H / y H among the given ones, such that ²:x and ²:y contain nonconjugate subgroups which follows from the power map given in the Atlaswx 7 . So x cannot be conjugate to a power of y and vice versa, contradicting Fact 1 and the note above. So in the remaining cases M is of type M12 , Suz, or Co2 and ␣ F ␣ ²:CH Ž.contains an involution i of type 2A,2A resp. 2B. Then [ 1q41⌺ q6 1q64= ␣ C CiH Ž.is of type 2 3,2y U48Ž.2 resp. Ž 2 2. Alt . If has order 4, ␣ acts trivially on OCŽ.: Otherwise D [ C Ž.␣ is a proper 2 O 2 ŽC . CONJUGACY CLASS AS A TRANSVERSAL 377

␣ subgroup of OC2Ž.. But centralizes something in the nontrivial factor r group NDO ŽC .Ž.D. By PropositionŽ 2.2 .Ž. iii this is covered by something ␣ 2 ␣ in CC Ž., contradicting the definition of D. Now centralizes a Sylow-2- subgroup of C. ByŽ. 2.5 we have wxC, ␣ s 1. [ r In the case Suz now S C OC2Ž.is nonabelian simple and therefore ␣ r ␣ centralized by .AsOC2Ž.ZC Ž.is an absolute irreducible S-module, has to act trivially on OC2Ž.as it centralizes S. So also in this case wxC, ␣ s 1. But C contains an involution j / i conjugate to i by the Z* w ␣ x s theorem of Glauberman. Now CjM Ž., 1 by the same arguments as s for C, but M ²Ž.:C, CjM as C is a maximal subgroup of M by the Atlas wx7 ; therefore wM, ␣ xs 1, a contradiction to the order of ␣.

~~4. SEMISIMPLE ELEMENTS IN GROUPS OF LIE TYPE~~

This section consists mostly of technical lemmata used to prove Proposi- tionŽ. 4.1 , which will be needed in the last paragraph to get some informa- < ␣ < tion over the primes dividing CM Ž.in our minimal counterexample. However, we do not need our automorphism ␣ nor any facts about ␣-CCP-groups. So PropositionŽ. 4.1 can be seen as an independent result about simple groups of Lie type. Let LieŽ.p consist of all groups of Lie type in characteristic p as defined in the preface.

PROPOSITION 4.1. Let G be a finite simple group of Lie type in characteristic p and let x g G of order r, r an odd prime, r / p. Then one of the following cases holds: / Ž.i AutG Žx . 1. / p Ž.ii 1 OCxisin ŽG Ž..Lie Žp ..

Ž.iii NGG Ž C Ž.. x contains a characteristic subgroup of index 2. During this paragraph let Gˆ be a simply connected simple algebraic group of Lie type, defined over the algebraic closure of GFŽ.p , and let F be a Frobenius map of Gˆˆ, such that G F [ Ä x : x g Gˆ N FxŽ.s x4 is a central extension of G. Set G [ GˆF and identify G with GrZGŽ..

~~Remark. Let x be an element of the preimage of x. By a theorem of~~

Steinberg now CxGGˆˆŽ.is connected and either Cx Ž.is a maximal torus s F and x is regular or CxGGŽ.contains elements of order p, CxŽ.CxGˆ Ž., pЈ and OCxŽŽ..G is a central product of groups of Lie type in characteristic p.AsZGŽ.is coprime to p, caseŽ. ii of Ž 4.1 . holds in this case. 378 ALEXANDER STEIN

G 2 G G LEMMA 4.2. Let G be of type A1, Bn, n 2, B2 , Cn, n 3, D2 n, n 2, 222G D2 n, n 2, E78442, E , F , F , G , or G 2. Then case Ž.i holds. Proof. Let x g G be semisimple and let Tˆ be an F-stable maximal [ r torus of Gˆˆcontaining x. Define W NTGˆŽ.Tˆto be the Weyl group of Gˆ. Now W has a unique involution z in its center, which is therefore fixed by F. Now z acts on Y, the lattice generated by the co-roots, as y1. Byw 4, Proposition 3.2.2x TˆF is W-isomorphic to YrŽ.F y 1 Y. Thus each element z in the preimage of z inverts TˆˆF. Now the coset Tz is F-stable and by the ᎐ g Lang Steinberg theorem contains a fixed point z11. Thus z G inverts x / and AutGŽ.x 1. g [ LEMMA 4.3. Let x G be regular semisimple. Then Tˆ CGˆŽ. x is non- wxr degenerated in the sense of 4, Proposition 3.6.1 and NGGŽŽ..Ž. C x Cxis G wx isomorphic to CW , F Ž. w , as defined in 4, Sect. 3.3 . \ g F Proof. As Gˆˆis simply connected, CxGˆŽ. T is connected. Now x T and Tˆˆis the only maximal torus containing x. Thus T is the only maximal torus containing TˆF and therefore Tˆ is nondegenerated. s s s DEFINITION 4.4. Let r be a prime. Define drqŽ. 0if r 2orr p ) N i y and minÄi 0:r q 14 otherwise. Define lqŽ., r as 2 if drq Ž.is even and as drqŽ.otherwise. G 2 G DEFINITION 4.5. Let G be of type AqnnnŽ., n 1, Aq Ž., n 1, Dq Ž., ) 2 ) s ⍀qy⍀ n 1, or DqnnŽ., n 1. Let G˙ SL q1Ž.q ,SUnq12 Ž.q , n Ž.q resp. 2 n Ž.q such that G˙˙rZGŽ.( G. Let V be the natural G˙-module and let K [ GFŽq 2 .Žin the unitary case and K [ GF q.otherwise. Let r be a prime [ ) with d dr< K <Ž. 0. LEMMA 4.6. Assume Ž.4.5 . Let˙ x g G˙ be of order r. Then V admits a s H H иии H ¨ decomposition V U V1 Vk Ž for a tri ial bilinear form in the [ ) linear case.Ž where U CVi x. and the V , i 0 are nondegenerated x-in- ¨ ariant subspaces. The Vi are either of dimension d and x acts irreducibly on s [ Vii or V are of dimension 2 d and ViiiX Y with X ii, Yx-irreducible and totally singular. Proof. This is a basic result of representation theory.

LEMMA 4.7. Assume Ž.4.5 . Let x g G of order r, let d ) 1 in the linear and unitary case, and let d ) 2 in the orthogonal case. Then case Ž.i of Ž4.1 . holds. Proof. Let ˙x g G˙ be of order r in the preimage of x. We claim d < < < < divides AutG˙Ž.˙˙x in the linear and unitary case and lqŽ., r divides AutG˙Ž.x / in the orthogonal case. Therefore AutGŽ.x 1. Assume Ž.x, G is a counterexample and V is of minimal dimension. CONJUGACY CLASS AS A TRANSVERSAL 379

s Let U, Viii, X , Y as inŽ. 4.6 . As U CxV Ž.is nondegenerated we may s assume U 0 by minimality.Ž In the orthogonal case the Vi of dimension d are of minus type, and those of dimension 2 d are of plus type; thus U is of even dimension.. Further k s 1 by minimality. In the linear case ˙x acts nq1 y r y irreducibly on V, so by Schur’s lemma CxG˙Ž.˙ has order Žq 1.Žq 1 . which corresponds to a maximal torus T of type An in G. Now byŽ. 4.3 r q s q NCTGGŽŽ..Ž.CT G is cyclical of order n 1; therefore d n 1 divides AutG˙Ž.˙x . In the unitary case assume first that x acts irreducibly on V. Then s q ⌽ 2 << q d n 1 and as nq1Ž.q divides G˙˙now n 1 is odd. Now G contains a cyclical self-centralizing subgroup C˙ of order Ž.Ž.q nq1 q 1 r q q 1 , which contains a Sylow-r-subgroup by the group order formula. By Sylow’s g s theorem we may assume ˙˙x C˙˙and now C CxG˙Ž..AsC ˙corresponds to - < r < s q a maximal torus T G of type AnG, NTŽ.CT G Ž. n 1 byŽ. 4.3 . Assume now V s X [ Y with X, Yx-invariant and totally singular. Now [ 2 q r s StabŽ.X Y induces a GL Žnq1.r2 Žq .Žon X and Y and n 1.2 d < < divides AutG˙Ž.˙x as seen in the linear case. Now let G˙ be an orthogonal group. If x acts irreducibly on V, V is of ⌽ << s minus type as nŽ.q divides G˙ .If n is even we have lqŽ., r 2 and 2 < < ( divides AutG˙Ž.˙x byŽ. 4.2 . If n is odd, G˙˙contains a subgroup U GUn Ž.q which w.l.o.g. contains x by Sylow’s theorem. As seen in the unitary case s r now n d 2 divides AutG˙Ž.˙x ,asV restricted to U may be identified with the natural unitary module over GFŽq 2 .. If V s X [ Y with X, Y totally singular x-invariant, G˙ is of plus type as the Witt index has order n. Now the stabilizer of this decomposition s induces a GL nŽ.q on X and Y.Aslq Ž, r .divides d and n d divides AutG˙Ž.˙x by the linear case, the lemma holds.

LEMMA 4.8. Assume Ž.4.5 . Let x g G be of order r, d s 1, or d s 2 in the orthogonal case. Then Ž.4.1 holds. If r di¨ides< ZŽ. G˙ < in the linear or unitary case, then Ž.i or Žii .of Ž4.1 . holds.

Proof. Assume this is false and let ˙x g G˙ in the preimage of x. Assume first that ˙x is of order r. Let U, Viii, X , Y as inŽ. 4.6 . y In the unitary case let r divide q 1. Then the Vi are of dimension 2 and x N is inverted in SUŽ.V ( SL Ž.q . Thus x is inverted in G˙ and case ˙˙Vii 2 Ž.i of Ž 4.1 . holds. Now let r divide q y 1 in the linear case and let r divide q q 1 in the unitary case. So the Vi correspond to the nontrivial eigenvalues of ˙x and U is the eigenspace of the eigenvalue 1. If one of these eigenvalues has multiplicity greater than one, caseŽ. ii of Ž 4.1 . holds, as G˙ induces the full linear resp. unitary group on the corresponding eigenspace. Therefore all 380 ALEXANDER STEIN

y n eigenspaces are of dimension one, so CxG˙Ž.˙ is a group of order Žq 1 . resp. Ž.q q 1n corresponding to a maximally split torus in G. ByŽ. 4.3 now caseŽ. iii of Ž 4.1 . holds, if r does not divide < ZGŽ.˙ <.If r divides n q 1, then r s n q 1 as all eigenvalues have multiplicity one and so ˙˙x and xy1 have the same eigenvalues. Therefore caseŽ. i of Ž 4.1 . holds. Assume now V is orthogonal. Then U s 0 as otherwise U contains an anisotropic point P g \ and x StabŽ.P S˙˙. But S is isomorphic to Oq2 ny12Ž.resp. Sp ny2Ž.q , depending on q odd or even, so x is inverted in S˙ byŽ. 4.2 and case Ž. i of Ž.4.1 holds. N y N q Now the Vi are of dimension 2, which is clear if r q 1. If r q 1 and V are of type Oq , x N centralizes a SU Ž.q ( SL Ž.q and therefore i 4 ˙ Vi 22 caseŽ. ii of Ž 4.1 . holds. Now ˙x is contained in a subgroup of order Ž.q y 1 n or Ž.q q 1n , depending on r N q y 1orr N q q 1. Let x g G correspond to ˙x g G˙. Then either x is not regular and caseŽ. ii of Ž 4.1 . holds or x is regular and CxGŽ.is maximally split and soŽ.Ž.Ž. iii of 4.1 holds by 4.3 . Assume now that ˙˙x cannot be chosen of order r. Then x may be chosen of order r l / r, r divides n q 1, G˙ is linear or unitary, and ˙x r s ␭ IdŽ.V with ␭ g K. Assume first that r l N q y 1 resp. r l N q q 1 in the linear resp. unitary case. Then V admits a decomposition as an orthogonal sum of the eigenspaces as above and all the eigenspaces have multiplicity one as otherwiseŽ. ii of Ž 4.1 . would hold as above. Now r s n q 1, but det Ž.x s ␭ / 1 as the eigenvalues are exactly the solutions of the equation y r s ␭ as ˙x r g ZGŽ.˙ . Assume now that r l does not divide q y 1 resp. q q 1, but r ly1 divides Ž.q y 1 resp. Ž.q q 1as˙x r s ␭ Id Ž.Ž.V g ZG˙ . Then the polynomial pyŽ. [ y r y ␭ is irreducible over K, as it has no solutions in K and fully splits [ <

inii N Cases ⌽6 ( / 1 115AqŽ.Aq Ž. r 3, 5 Ž.q y 1 6 1 ⌽633 WEŽ. rs 3 163 ⌽6 ( s 1 1155 AqŽ.Aq Ž. r 5 ⌽4 / 2 24FqŽ. r 3 ⌽ 42 s 2 243 FqŽ. r 3 ⌽3 ( ( 3 3Aq 222Ž.Aq Ž.Aq Ž. ⌽2 4 44FqŽ. ⌽ ( y 5 55DqŽ.Žq 1 . ⌽2 6 64FqŽ. ⌽ 8 84FqŽ. ⌽ 3 9 92AqŽ. ⌽ 12 12 Fq4Ž.

2 LEMMA 4.9. Assume G is of type E66Ž. q or E Ž. q . Then Ž4.1 .holds. << s Proof. The r-part of G divides niqif drŽ. i, where niis defined. A Sylow-r-subgroup of G is contained in a subgroup Mi of isomorphism type g Niiand w.l.o.g. x M . 2 In case Eq6Ž.these are the nii’s and M ’s: The ni can be calculated from the group order. The existence of the wx ( ⑀ subgroups of type Ni follows from 16 for types F41, AqŽ. Aq 5Ž., ⑀ ( y ⑀ wx Dq5 Ž.Žq .and from 6 for the other types. If Mi is of type Fq4Ž.the statement holds byŽ. 4.2 . If Mi is of type ( y 2 ( q 3 2 3 ( ( Dq5Ž.Žq 1, . Dq5Ž.Žq 1, .Aq2 Ž ., Aq 2Ž .Ž.Ž.Ž., Aq 222Aq Aq,or 2 (2 (2 N y Aq222Ž. AqŽ. AqŽ., case Ž.Ž. i of 4.1 holds by Ž. 4.5 . Now r q 1 in case N q 2 / Eq66Ž.and r q 1 in case EqŽ..If r 3, 5 the r-part of G is the r-part

inii M Cases ⌽4 / 1 14FqŽ. r 3 ⌽42 s 1 143 FqŽ. r 3 ⌽6 (2 / 2 215AqŽ.Aq Ž. r 3, 5 Ž.q q 1 6 2 ⌽633 WEŽ. rs 3 263 ⌽6 (2 s 2 215 AqŽ.Aq5 Ž. r 5 ⌽ 2 3 34FqŽ. ⌽2 4 44FqŽ. ⌽ 2 ( q 10 10 Dq5Ž.Žq 1 . ⌽ 3 222( ( 6 6Aq 222Ž.Aq Ž.Aq Ž. ⌽ 8 84FqŽ. ⌽ 2 3 18 18 Aq2Ž. ⌽ 12 12 Fq4Ž. 382 ALEXANDER STEIN of a maximal split torus. Then either x is regular and caseŽ. iii of Ž 4.1 . holds byŽ. 4.3 or x is not regular and so case Ž. ii of G holds. If r s 5 and r N q y 1 resp. q q 1, a Sylow-5-subgroup is contained in a ( (2 subgroup M of type Aq15Ž.Aq Ž.resp. Aq 15 Ž. AqŽ.. Now x cannot be s regular: Let EMŽ. CC12be the product of its components and C1be of s g type Aq11Ž.. Then x xx2with xiiC . Now x22cannot be regular in C as x2 can have only five different eigenvalues but the dimension of the corresponding vector space is 6. Therefore caseŽ. ii of Ž 4.1 . holds. So now r s 3 divides q y 1 resp. q q 1 and ZGŽ./ 1. We claim that x cannot be regular, so caseŽ. ii of Ž 4.1 . holds: Otherwise x is contained in some T [ TˆF for some F-stable maximal torus Tˆ. Bywx 4 the G-classes of F-stable maximal tori correspond to the F-conjugacy classes of WGŽ.and the orders of TˆF are given inwx 5 for G 2 of type Eq66Ž.. The orders for G of type EqŽ.are obtained from those for y wx Eq6Ž.by replacing q with q by 19 . If T is contained in a subgroup ( ( s U Aq15Ž.Aq Ž., U does not contain regular elements x with oxŽ. 3 byŽ. 4.8 . The remaining tori are of type D441551, DaŽ., D , Da Ž.,3A 26, E , Ea61Ž.,orEa 62 Ž.. In the first four cases T can be found in a subgroup of ( y 2 ( q s type Dq55Ž.Žq 1 . resp. Dq Ž.Žq 1 . . Now an element x with oxŽ. 3 lying in these tori cannot be regular. In the last three cases 9 does not << << 2 q q 3 divide T , so finally T is of type 3 A2 and T is Žq q 1. resp. 2 y q 3 [ r 1q2 Žq q 1.If. x is regular, N NTGŽ.T is of type 3 SL2Ž. 3 by 3 Ž.4.3 . Now OT3 Ž.is elementary abelian of order 3 and contains ZGŽ.. N cannot act faithfully on OT3Ž.,as N cannot be embedded into a point 2 ⌺ - stabilizer of GL34Ž. 3 of type 3 2 ; therefore T CxGŽ., contradicting regularity of x and simple connectedness of G.

~~5. THE MINIMAL COUNTEREXAMPLE, PART II~~

In this section we prove Theorem C, continuing at the end of Section 3. By the classification of finite simple groups now M is a group of Lie type. s ␣ ␲ Let p be the characteristic of M, H CMHŽ.and let be the set of all primes dividing <direct product of groups of Lie type in characteristic p, generated by its elements of order p. CONJUGACY CLASS AS A TRANSVERSAL 383

␣ ␣ If CDŽ.contains elements of order p then so does CA r B Ž.by PropositionŽ.Ž. 2.2 iii . If D contains an ␣-invariant Sylow-p-subgroup P then so does ArB as the preimage of P in ArB contains a unique Sylow-p-subgroup. s s s Let E1 EDŽ., F CED Ž1 ., E CFD Ž.. F is the direct product of all solvable groups of Lie type and ZFŽ.s 1 by definition of D. E is the s product of all nonsolvable factors and E1 F*Ž.E .As M is a minimal ␣ s counterexample is trivial on E1 and on E.Soif F 1 the lemma holds. / s s s s иии Assume F 1; thus p 2orp 3. First let p 2, so F L1 Lk [ s [2 s with Li of one of the following types: T11A Ž.2 3:2, T 2B 2Ž.2 [2 s 2 [ s 2 ␣ 5:4, T32A Ž.2 3:Q 842, T D Ž.2 3:D 8. Now acts on these factors by permutation: g Let x Ljjbe an involution if L is of type T1 and an element of s F ␣ s иии order 4 otherwise. Then Li ²ON2 ŽF Ž² x :.. :. Let x y1 yk with < << ␣ < k < < y g L .As F : CxŽ.s F : CxŽ . s Ł s L : CyŽ.all but one y are ii F F i1 iLii i ␣ s ␣ F 1. Thus Li ²ŽŽ²ON2 F x:.. : is a direct factor. ␣ Now let Fijbe the product of all factors L of type Ti. As shown s permutes the factors of FiiŽ1, 2, 3, 4 . . By Ž. 2.4 and Ž. 2.6 now each of the ␣ Fi’s contains an -invariant Sylow-2-subgroup and centralizes involutions. ␣ ␣ So F and D contain an -invariant Sylow-2-subgroup and CDŽ.contains involutions. s s 2 Finally let p 3soF is a direct product of groups of type Alt 4 2 :3. A similar argument as in the case p s 2 shows that ␣ permutes the direct factors. Now F and therefore D contain an ␣-invariant Sylow-3-subgroup ␣ and CF Ž.contains elements of order 3 byŽ. 2.4 and Ž. 2.6 . g ␲ g ␲ LEMMA 5.2. If 2 HH, then p or M is of type A1.

Proof. We may assume p / 2. Let i g H be an involution. The iso- wx morphism type of CiM Ž.is listed in 12 for all groups of Lie type with p pЈ g odd. It turns out that OCiŽŽ..Ž.M Lie p if M is not of type A1. ByŽ. 5.1 now the lemma holds. g ␲ g s LEMMA 5.3. Assume p Hp. Then there is a P Syl Ž.M , B NPM Ž. such that B ␣ s B. g s s s Proof. Let x H, oxŽ.p. Set N0 CxMi Ž.and N q1 NONMpi Ž Ž ... [ s As M is finite the chain Nikends in a stationary subgroup P N Nkq1. By a theorem of Borel and Titswx 2 the group P is a parabolic subgroup of ␣ s f M. By construction P P, so assume OPppŽ.Syl ŽM .. [ pЈ r g Let L OPŽŽ..OPp ,so L Lie Ž.Ž.p . By 5.1 L contains an invariant Sylow-p-subgroup and so does M. 384 ALEXANDER STEIN

g ␲ ␣ LEMMA 5.4. Assume p H . Let B be the -stable Borel subgroup from ␣ ¨ Ž.5.3 . Then either fixes all o ergroups of B or M is one of A22Ž. q , Bq Ž., ␣ G2Ž. q and interchanges the maximal parabolics containing B. Proof. Assume otherwise. Then ␣ induces a graph automorphism on ⌬ ) the Dynkin diagram corresponding to M.So M is one of AqnŽ., n 1, ) s f ) Bq2Ž., q even, DqnŽ., n 3, Eq62 Ž.,orGq Ž., q 3 . Let P B be a parabolic of M corresponding to all but the ending nodes of ⌬ if M is of ) type An, n 3orE634, to all ending nodes if M is of type A or D , and to ) a subdiagram of type D4 if M is of type Dm, m 4. s p, pЈ r s wx␣ s Let L OPŽŽ.OPp .If L EL Ž., L, 1 by minimality of M, contradicting the action of ␣ on ⌬.

So L contains solvable factors and M is of type A334Ž.2, A Ž.3,D Ž.2,or ( D43Ž.3 . Case A Ž.2 contradicts Ž 2.4 . and Ž 2.6 . . Assume M A3Ž.3 . Then P 4 = ␣ has structure 3 : 2.Ž. Alt 44Alt .2 and normalizes a section S of type = ␣ r 2 g ␣ ŽŽ.Ž..PSL22 3 PSL 3 .2. As is trivial on S OSŽ.there is an x CSŽ. by PropositionŽ.Ž. 2.2 iii which covers this factor. Elements in S y OS2 Ž. [ s induce a diagonal automorphism on S; thus C CxO ŽS.Ž.CxCxNN Ž. Ž., Ј 212 where N12, N are the factors of S of type Alt4 which are interchanged by ␣. Thus ␣ induces an automorphism of order 2 on C, contradicting ␣ PropositionŽ.Ž. 2.2 i as C is an -CCP-group. So let M be of type Dq4 Ž., q s 2, 3. If ␥ has order 2 let P ) B be the maximal parabolic corresponding to all nodes except the ending node fixed by ␣. As above we get a contradiction. So ␣ induces a symmetry of order 3 on ⌬. Let P corre- s 3Ј r s r spond to all ending nodes. Let L OPŽŽ..OPp and let L L ZLŽ.. ␣ Then acts on a direct product of three groups of type Alt4 by permuting the factors transitively. This contradictsŽ. 2.4 and Ž. 2.6 . s 1q8 ⌺ = ⌺ = ⌺ s r If q 2, P has structure 2 .Ž.333. Set P P OP2 Ž.and ␣ ⌺ ␣ assume permutes the 3’s transitively. ByŽ. 2.4 and Ž. 2.6 then CP Ž.is not divisible by 3 but contains involutions and an ␣-invariant Sylow-2-sub- s r s r group S. Let S be the full preimage of S. Set Z ZSŽ., Z2 Z ZS ŽZ ., s F ␣ <

~~Proof. For each Lie group M let P12and P be the maximal parabolics containing B of type given below:~~

Type of M Type of P12Type of P ) AqnnŽ., n 1 Aqy1Ž. Aqny1Ž. 22) 2 Aq2 nŽ., n 1 Aq2 ny2Ž. Aqny1Ž. 22) 2 Aq2 nq12Ž., n 0 Aqny1Ž. Aqn Ž . ) BqnnŽ., n 1 Bqy1Ž. Aqny1Ž. ) CqnnŽ., n 2 Cqy1Ž. Aqny1Ž. ) DqnnŽ., n 3 Dqy1Ž. Aqny1Ž. 22) DqnnŽ., n 3 Dqy1Ž. Any2Ž. q 3 3 Dq411Ž. AqŽ. AqŽ. Eq655Ž. DqŽ. AqŽ. 222 Eq645Ž. DqŽ. AqŽ. Eq766Ž. EqŽ. AqŽ. Eq877Ž. EqŽ. AqŽ. Fq433Ž. BqŽ. CqŽ. 22 Fq421Ž. BqŽ. AqŽ. Gq211Ž. AqŽ. AqŽ.

We claim that C Ž.␣ acts transitively on flags containing E . This Pii s ␣ [ pЈ r / follows if PiPBC Ž.. Set L iOPŽŽ..Ž. ipiiOP.If EL 1 we have in s i w ␣ x s fact F*Ž.LiiEL Ž.and by minimality of M therefore F*Ž.Li, 1; wx␣ s ␣ thus LiP, 1. By PropositionŽ.Ž. 2.2 iii now C Ž.covers this factor s ␣ s i group, so PiPBC Ž.. So let EP Ž.i1 which means that Piis one of i 2 2 the following groups: A11Ž.2, A Ž.3, A 2Ž.2,or B 2Ž.2 . Then M is either an exception of the lemma or M is one of the following groups: A22Ž.2, A Ž.3, 2 3 s3 wx A34Ž.3, D Ž.2,orG 2 Ž.3 . First let M D4Ž.2 . By 7 the involved parabolic 2 w 9 x has structure 2 . 2 : A11Ž.2 in Atlas notation. Note that the factor A Ž.2 acts faithfully on the normal subgroup N of order 4. By assumption we have P ␣ s P . So let x g C Ž.␣ be an element which covers the factor 11 Pi r 2 < < s ␣ P11OPŽ.. Now CxNNŽ. 2 and both CxŽ.and N are -invariant; thus wx␣ s w␣ xs N, 1. Now by the three subgroup lemma we get P1, , N 1 which means wxP , ␣ F OPŽ.F B; thus P s BC Ž.␣ in this case too. 1211Pi In the other cases we see bywx 7 that all the involved parabolics contain a r ( ⌺ - characteristic subgroup U with Pi U 4 and U B. Now byŽ. 2.7 wxP rU, ␣ s 1 and therefore P s BC Ž.␣ . iiPi Now flag transitivity follows from the existence of a Ä4i, j -path between any two flags. g ␲ g ␲ LEMMA 5.6. If 2 HHor p then one of the following holds: Ž.i The Lie rank of M is 1. s ␣ Ž.ii M Aq22 Ž., Bq Ž., or G 2 Ž. q and interchanges two maximal parabolics and fixes their common Borel subgroup. 386 ALEXANDER STEIN

s2 2 Ј 3 2 Ј Ј Ž.iii M A342244Ž.2, A Ž.2,B Ž.2 , B Ž.3, D Ž.3, F Ž.2 , or G 2 Ž.2 . g ␲ Proof. Assume that the Lie rank of M is greater than one and 2 H g ␲ g ␲ or p HH. ByŽ. 5.2 we may assume that p . ByŽ. 5.3 we get a Borel subgroup B with B ␣ s B. ByŽ. 5.4 M is either one of the exceptionsŽ. ii or ␣ fixes all overgroups of B. So we can applyŽ. 5.5 and M is either one of the exceptionsŽ. iii or H acts flag transitively on M. Now the main theorem ofwx 18 gives all flag transitive subgroups of the finite simple groups of Lie type. Thus either H s M, contradicting wxM, ␣ s M,orM possesses a proper subgroup F acting flag transitively on the building and Ž.M, F s 3 4 ␣ ŽŽ.ŽAq20, Aq..Ž.3 , B 2 Ž.3,2 :A 1Ž..4,or ŽA 3 Ž.2 , Alt 7 . . But CM Ž .is not contained in such a subgroup in the first case. The second case is excluded byŽ. iii and the third case contradictsŽ. 3.3 . g ␲ g ␲ s2 Ј LEMMA 5.7. 2 HHor p or M F4Ž.2 . wx␲ / л Proof. By 17 H as finite simple groups do not possess fixed point free automorphisms. Let x g H be of prime order r and assume r / 2, p.So x is semisimple of odd order and byŽ. 4.1 one of the cases Ž. i , pЈ Ž.ii , or Ž iii . holds. If caseŽ. ii holds, OCxŽŽ..M is a central product of ␣ g ␲ groups of Lie type, which is -invariant. Now byŽ. 5.1 p H . IfŽ. iii holds, ␣ acts trivially on a factor group of order 2; therefore g ␲ ␣ 2 HM. If finally caseŽ i . holds, acts trivially on Aut Ž²x :.as Aut Ž²x :. < < g ␲ is abelian. If s divides Aut MHŽ²x :. then s byŽ. 2.5 and we can g ␣ - repeat this process with an element y CM Ž.of order s.As s r this process terminates and the lemma is proved. ) LEMMA 5.8. M is none of the following groups: Aq12Ž., q 3, Aq Ž., 2 ) ) 2 ) ) 2 ) Aq22Ž., q 2, Bq Ž., q 2, Bq 2Ž., q 2, Gq 2 Ž., q 3, Gq 2Ž., q 3. Proof. Let x g H be of prime order r. ByŽ. 5.7 we may assume r s 2 s g ␲ g ␣ or r p.If p Hplet P Syl Ž.M be -invariant as given in Ž. 5.3 . g ␲ We show now 2 H Ž.or the lemma holds : Otherwise p is odd. If s N y 2 2 / M Aq12Ž.,4 q 1, AqŽ., Gq2Ž., Aq2 Ž., Bq22 Ž.,orGq Ž., p 3, then r ␣ NPMMŽ.CZP ŽŽ..is cyclic of even order; therefore acts trivially on a g ␲ subgroup of index 2 and 2 H by PropositionŽ.Ž. 2.2 iii . s N q If M Aq1Ž.,4 q 1, NPM Ž.does not contain involutions, so no element of order p is conjugate to its inverse and all subgroups of order p are conjugate in M. Thus wxP, ␣ s 1 by PropositionŽ.Ž. 2.2 i and now F NPM Ž.H by Ž 2.5 . . As H contains now semisimple elements, which are g ␲ s inverted in M,2 H Ž.and therefore H M . s s So M Gq2Ž.and p 3. If H contains semisimple elements, then g ␲ ␣ 2 H as all semisimple elements are inverted byŽ. 4.2 . If does not ␣ interchange the maximal parabolics P12, P containing NPM Ž., then acts r g ␲ trivially on Pi OP3Ž.iHby minimality of M. So in this case 2 by CONJUGACY CLASS AS A TRANSVERSAL 387

PropositionŽ.Ž. 2.2 iii . So assume otherwise. Now there are exactly three 6 subgroups of order q .2 containing P, one normal in P1 and another one normal in P2 , and both Pi contain exactly one such normal subgroup. Thus ␣ g ␲ has to interchange these two and fix the third. Now 2 H by PropositionŽ.Ž. 2.2 iii . If now p is odd, we show H s M, a contradiction: Let i g H be an s 2 2 involution and C CiM Ž.. In the cases Aq2222Ž., GqŽ., Aq Ž., Bq Ž.,or w ␣ x s < 2Ј < s Gq2Ž.we have ECŽ., 1 by minimality of M and OCŽ.: EC Ž. 2, so ␣ fixes each involution of C. As the commuting graph of involutions is connected we have H s M, a contradiction. s s s f If M Aq11Ž.we show that H Aq Ž0 .with q q0for some integer f: g s Let i H be an involution and C CiM Ž.. Assume first that 8 does not divide <

In both cases now H contains a V4. ByŽ. 2.5 now H contains an element x of order 3 and an involution j inverting x. By Dixon now ²:i, x, j is either of type Aq10Ž.or of type Alt5 and H contains elements of order 5. If all elements of order 2, 3, and 5 are semisimple, M contains elements of order 6, 10, or 15, as there are only two classes of maximal tori. Now H contains also such elements by PropositionŽ.Ž. 2.2 i and H is of type s s Aq10Ž..If p 5, subgroups of type Alt5 are of type Aq10 Ž..If p 3 and 8 does not divide <

~~␣ the permutation module of Alt5 which is absolutely irreducible. As now wx␣ s s wx␣ centralizes Alt52 it centralizes OPŽ.1; thus P1, 1 S, .As~~

6. THE MAIN THEOREMS Proof of Theorem C. Assume M is a minimal counterexample. ByŽ. 3.2 M is simple. By the classification of finite simple groups G is either alternating, sporadic, or of Lie type. ByŽ. 3.3 and Ž. 3.4 M is of Lie type. But byŽ.Ž. 5.6 , 5.8 , and Ž. 5.9 this cannot happen. Therefore we get a contradiction and no counterexample exists. Proof of Theorem A. Let G be a counterexample. By the first remark in the introduction now G is a CCCP-group for some g g G with G s ² g G :Ž.. By 2.1 this nonsolvable group G corresponds to a nonsolvable ␣-CCP-group H with wxH, ␣ s H. But this contradicts Theorem C, so G does not exist. 390 ALEXANDER STEIN

Proof of Ž.1.4 . Let Q be a counterexample. By Ž. 1.5 now GQr Ž.is a nonsolvable CCCP-group for some g g G with G s ² g G :. By Theorem A such a group does not exist and we get a contradiction. y1 GGl s Proof of Theorem B. The condition Ž gg. CgGŽ. Ä41 is exactly G the condition that g is a transversal to CgGŽ.. Therefore Theorem B is equivalent to Theorem A.

~~REFERENCES~~

1. M. Aschbacher, ‘‘Finite Group Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1986. 2. A. Borel and J. Tits, Elements´ ´ unipotents et sous-groupes paraboliques de groupes reductifs.´ I, In¨ent. Math. 12 Ž.1971 , 95᎐104. 3. R. Brauer and M. Suzuki, On finite groups of even order whose 2-Sylowsubgroup is a quaternion group, Proc. Nat. Acad. Sci. 45 Ž.1959 , 175᎐179. 4. R. W. Carter, ‘‘Finite Groups of Lie Type, Conjugacy Classes, and Complex Characters,’’ Wiley᎐Interscience, New York, 1985. 5. R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 Ž.1972 , 1᎐59. 6. A. M. Cohen, M. W. Liebeck, J. Saxl, and G. M. Seitz, ‘‘The Local Maximal Subgroups of the Exceptional Groups of Lie Type,’’ preprint. 7. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘An ATLAS of Finite Groups,’’ Oxford Univ. Press, London. n ᎐ 8. B. N. Cooperstein, Maximal subgroups of G2Ž2,.Ž.J. Algebra 70 1981 , 23 36. 9. W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 Ž.1963 , 775᎐1029. 10. B. Fischer, Distributive Quasigruppen endlicher Ordnung, Math. Z. 83 Ž.1964 , 267᎐303. 11. G. Glauberman, Central elements in core free groups, J. Algebra 4 Ž.1966 , 403᎐420. 12. D. Gorenstein, R. Lyons, and R. Solomon, ‘‘The Classification of Finite Simple Groups III,’’ Math. Surveys and Monographs, Vol. 40, No. 3, Am. Math. Soc., Providence. 13. P. Kleidman and M. Liebeck, ‘‘The Subgroup Structure of the Finite Classical Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1990. 3 14. P. Kleidman, The maximal subgroups of the Steinberg triality groups Dq4Ž.and of their automorphism groups, J. Algebra 115 Ž.1988 , 182᎐199.

15. P. Kleidman, The maximal subgroups of the Chevalley groups Gq2Ž.with q odd, of the 2 ᎐ Ree groups Gq2Ž., and of their automorphism groups, J. Algebra 117 Ž.1988 , 30 71. 16. M. Liebeck and J. Saxl, On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proc. London Math. SocŽ. 3 , 55 Ž1987 . , 299᎐330. 17. P. Rowley, Finite groups admitting a fixed-point-free automorphism group, J. Algebra 174 Ž.1995 , 724᎐727. 18. G. M. Seitz, Flag-transitive subgroups of Chevalley groups, Ann. of Math. Ž.2 Ser. 97 Ž.1973 , 27᎐56. 19. T. A. Springer and R. Steinberg, Conjugacy classes, in ‘‘Seminar on Algebraic Groups and Related Finite Groups’’Ž. A. Borel et al., eds. , Lecture Notes in Mathematics, Vol. 131, pp. 168᎐266, Springer-Verlag, Berlin, 1970. 20. J. G. Thompson, Finite groups with fixed-point-free automorphism of prime order, Proc. Nat. Acad. Sci. U.S. A. 45 Ž.1959 , 578᎐581. 21. J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable I-VI, Bull. Amer. Math. Soc. 74 Ž.1968 , 383᎐437; Pacific J. Math. 33 Ž.1970 , 451᎐536; 39 Ž.1971 , 483᎐534; 48 Ž.1973 , 511᎐592; 50 Ž.1974 , 215᎐297; 51 Ž.1974 , 573᎐630.