On boundedly generated subgroups of profinite groups

Abstract

In this paper we investigate the following general problem. Let G be a group and let i(G) be a property of G. Is there an integer d such that G contains a d-generated subgroup H with i(H) = i(G)? Here we consider the case where G is a profinite group and H is a closed subgroup, extending earlier work of Lucchini and others on finite groups. For example, we prove that d = 3 if i(G) is the prime graph of G, which is best possible, and we show that d = 2 if i(G) is the exponent of a finitely generated prosupersolvable group G.