Local--global generation property of commutators in finite $\pi$-soluble groups

TL;DR

This paper investigates the generation properties of commutators in finite groups under automorphisms, establishing bounds on their rank in the context of $ ext{pi}$-soluble groups.

Contribution

It proves a new bound on the rank of commutator subgroups in $ ext{pi}$-soluble groups acted upon by $ ext{pi}$-groups of automorphisms, extending previous results.

Findings

Bound on the rank of $[G,A]$ in $ ext{pi}$-soluble groups

Counterexamples without $ ext{pi}$-solubility assumption

Extension of earlier results for coprime automorphisms

Abstract

For a group $A$ acting by automorphisms on a group $G$, let $I_G(A)$ denote the set of commutators $[g,a]=g^{-1}g^a$, where $g\in G$ and $a\in A$, so that $[G,A]$ is the subgroup generated by $I_G(A)$. We prove that if $A$ is a $\pi$-group of automorphisms of a $\pi$-soluble finite group $G$ such that any subset of $I_G(A)$ 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 $\pi$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.