Criteria for solubility and nilpotency of finite groups with automorphisms

TL;DR

This paper establishes criteria linking the solubility and nilpotency of finite groups with coprime automorphisms to properties of subgroups generated by specific commutators, providing new characterizations of these group properties.

Contribution

It introduces new criteria for solubility and nilpotency of finite groups with coprime automorphisms based on subgroup generation by commutators.

Findings

$[G, \alpha]$ is soluble or nilpotent if and only if certain subgroups are soluble or nilpotent

Characterizes solubility and nilpotency via commutator-generated subgroups

Provides necessary and sufficient conditions for group properties based on automorphism commutators

Abstract

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 $G$. We show that $[G,\alpha]$ is soluble or nilpotent if and only if any subgroup generated by a pair of elements of coprime orders from the set $J_G(\alpha)$ is soluble or nilpotent, respectively.