2 research outputs found
Finitely generated groups with polynomial index growth
We prove that a finitely generated soluble residually finite group has
polynomial index growth if and only if it is a minimax group. We also show that
if a finitely generated group with PIG is residually finite-soluble then it is
a linear group.
These results apply in particular to boundedly generated groups; they imply
that every infinite BG residually finite group has an infinite linear quotient.Comment: To appear in Crelle's Journa
Maximal subgroups in finite and profinite groups
Abstract. We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every 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 Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203–220, 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 various enumeration problems. 1