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.
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