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Conjugacy of odd order Hall subgroups. (English) Zbl 0616.20007
If G is a finite simple group, the author defines an ordering \(>_ G\) on the set of odd prime divisors of \(| G|\) as follows. Let p, q be odd primes in \(\pi\) (G). If G is a Chevalley group over a field of characteristic p, then \(p>_ Gq\). If neither p nor q is the field characteristic, then \(p>_ Gq\) iff \(p>q\). If G is a sporadic group, then \(p>_ Gq\) if the order of an \(S_ p\)-subgroup of G is greater than that of an \(S_ q\)-subgroup of G. If G is an alternating group, choose \(>_ G\) arbitrarily.
The following result was proved previously by the author [Proc. Lond. Math. Soc., III. Ser. 52, 464-494 (1986; Zbl 0559.20014)] for all simple groups excepting \(E_ 6(q)\), \(E_ 7(q)\), \(E_ 8(q)\) and \({}^ 2E_ 6(q)\); here he completes the proof in the above remaining cases: Theorem. Let H be a Hall subgroup of odd order of the finite simple group G. Let \(\pi (H)=\{p_ 1,...,p_ n\}\), with \(p_ 1>_ Gp_ 2>_ G...>_ Gp_ n\). Then H has a Sylow tower with respect to this ordering of the \(p_ i's\). As a direct consequence of this theorem, it follows that if the finite group G has a Hall subgroup of odd order, then all Hall subgroups of G are conjugate (this last result was also announced, without details, by P. Cobb [in Abstracts Am. Math. Soc. 8, 36 (1987)].
Reviewer: M.Deaconescu

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D06 Simple groups: alternating groups and groups of Lie type
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