MAXIMAL-SUBGROUPS IN FINITE AND PROFINITE GROUPS

We prove that if a finitely generated profinite group G is not generat ed with positive probability by finitely many random elements, then ev ery finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups , as well as techniques from finite permutation groups and finite Chev alley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 2 03-220, we then prove that a finite group G has at most \G\(c) maximal soluble subgroups, and show that this result is rather useful in vari ous enumeration problems.