Local–global generation property of commutators in finite π-soluble groups

For a group A acting by automorphisms on a group G, let IG(A) denote the set of commutators [g,a]=g−1ga, where g∈G and a∈A, so that [G,A] is the subgroup generated by IG(A). We prove that if A is a π-group of automorphisms of a π-soluble finite group G such that any subset of IG(A) generates a subgroup that can be generated by r elements, then the rank of [G,A] is bounded in terms of r. Examples show that such a result does not hold without the assumption of π-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow p-subgroups of p-soluble groups.