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Conjugacy Class As a Transversal in a Finite Group

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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 <<<s G < s g g . Therefore H CgGŽ .. On the other hand G : H g < a < s a aGg G : CgGGŽ . ; therefore H CgŽ . for some g g . Ž.2Ifg G is a transversal, then it is both a left and a right transversal as g eh s hg eh if g, e g G, h g H and therefore G s g GH s Hg G. As an abbreviation we define: 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 solv- able. 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 central- izer product. The relation between ␣-CCP-groups and CCCP-groups is very simple: ␫ ␫ Every CCCP-group is an gg-CCP-group, where is the inner automor- phism 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 quasi- groups. 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 counterex- ample 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-sub- group for any prime r dividing <<G . An analogue statement for ␣-CCP- groups does not hold. However, using a famous theorem of Borel and Tits wx ␣ ␣ 2 , one can show that fixes a Borel 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.
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