Hall \(\pi\)-subgroups which are direct products of non-Abelian simple groups (Q1115966)

The main result of this paper is Corollary B: If H is characteristically simple, G is simple, \(H<G\), and H is a Hall \(\pi\)-subgroup of G, then the inclusion is one of the following: (i) \(A_ 5<L_ 2(q)\), (ii) \(A_{p- 1}<A_ p\) for p prime, \(p\geq 7\), (iii) \(M_{22}<M_{23}\), (iv) \(PSp_ 6(2)<P\Omega_ 7(q)\), or (v) \(P\Omega^+_ 8(2)<P\Omega^+_ 8(q)\). In cases (i), (iv) and (v) there are of course restrictions on the possible values of q. Theorem A is a straightforward generalization to arbitrary finite groups G, but is too complicated to state here. Since every finite simple group has even order, it follows that \(| G:H|\) is odd. Now \textit{M. Liebeck} and \textit{J. Saxl} [J. Lond. Math. Soc., II. Ser. 31, 250-264 (1985; Zbl 0573.20004)] have completely classified the primitive permutation groups of odd degree. The proof of Corollary B now goes by considering the various cases in turn. In particular, it depends heavily on the classification of finite simple groups.