Criteria for solubility and nilpotency of finite groups with automorphisms

Abstract

Let 𝐺 be a finite group admitting a coprime automorphism 𝛼. Let 𝐽 𝐺 (𝛼) denote the set of all commutators [𝑥, 𝛼], where 𝑥 belongs to an 𝛼-invariant Sylow subgroup of 𝐺. We show that [𝐺, 𝛼] is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set 𝐽 𝐺 (𝛼) is soluble or nilpotent, respectively.