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 -1 g 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.