On boundedly generated subgroups of profinite groups

In this paper we investigate the following general problem. Let G   be a group and let i(G)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)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=3d=3 if i(G)i(G) is the prime graph of G  , which is best possible, and we show that d=2d=2 if i(G)i(G) is the exponent of a finitely generated prosupersolvable group G.