On the intersection of a class of maximal subgroups of a finite group. II (Q1078664)

In this paper the authors continue their study [Part I, to appear] of the influence of maximal subgroups of composite index by considering the generalized Frattini subgroup obtained by intersection. For example, if \(\pi\) is a set of primes (with complement \(\pi\) ') then \(S_{\pi}(G)\) is the subgroup of the finite group G obtained by intersecting all maximal M with [G:M] composite and \(\pi\)-free (or G if no such subgroups exist), and \(S_{\pi '}(G)\) is defined analogously. Theorem 5. If G is \(\pi\)- solvable then either \(S_{\pi}(G)\) or \(S_{\pi '}(G)\) is solvable. Other results of a similar nature are obtained.