Criteria for solubility and nilpotency of finite groups with automorphisms

Cristina Acciarri, Robert M. Guralnick, Pavel Shumyatsky

Let $G$ be a finite group admitting a coprime automorphism $\alpha$. Let $J_G(\alpha )$ denote the set of all commutators $[x,\alpha ]$, where $x$ belongs to an $\alpha$-invariant Sylow subgroup of $G$.…