COPRIME AUTOMORPHISMS OF FINITE GROUPS

Let G be a finite group admitting a coprime automorphism α of order e. Denote by IG(α) the set of commutators g−1gα, where g ∈ G, and by [G, α] the subgroup generated by IG(α). We study the impact of IG(α) on the structure of [G, α]. Suppose that each subgroup generated by a subset of IG(α) can be generated by at most r elements. We show that the rank of [G, α] is (e, r)-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of IG(α) has odd order, then [G, α] has odd order too. Further, if every pair of elements from IG(α) generates a soluble, or nilpotent, subgroup, then [G, α] is soluble, or respectively nilpotent.