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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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