Rank type conditions on commutators in finite groups

For a subgroup S of a group G, let IG (S) denote the set of commutators g, s=g^-1gˢ, where g G and s S, so that G, S is the subgroup generated by IG (S). We prove that if G is a p-soluble finite group with a Sylow p-subgroup P such that any subgroup generated by a subset of IG (P) is r-generated, then G, P has r-bounded rank. We produce examples showing that such a result does not hold without the assumption of p-solubility. Instead, we prove that if a finite group G has a Sylow p-subgroup P such that (a) any subgroup generated by a subset of IG (P) is r-generated, and (b) for any x IG (P), any subgroup generated by a subset of IG (x) is r-generated, then G, P has r-bounded rank. We also prove that if G is a finite group such that for every prime p dividing |G| for any Sylow p-subgroup P, any subgroup generated by a subset of IG (P) can be generated by r elements, then the derived subgroup G' has r-bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group G admits a group of coprime automorphisms A such that any subgroup generated by a subset of IG (A) is r-generated, then the rank of G, A is r-bounded.