Rank type conditions on commutators in finite groups

For a subset S of a group G, let I_G(S) denote the set of commutators [g,s] = g⁻¹g^s, where g ∈ G and s ∈ S, so that [G,S] is the subgroup generated by I_G(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 I_G(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 I_G(P) is r-generated, and (b) for any x ∈ I_G(P), any subgroup generated by a subset of I_G(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 I_G(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 I_G(A) is r-generated, then the rank of [G,A] is r-bounded.