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

TL;DR

The paper proves a local-to-global rank generation property for commutators $[G,A]$ where $A$ is a $\pi$-group of automorphisms of a finite $\pi$-soluble group $G$. Under the hypothesis that every subset of the commutator set $I_G(A)$ generates an $r$-generator subgroup, the authors show that $[G,A]$ has rank bounded in terms of $r$ when $G$ is $\pi$-soluble and $A$ is a $\pi$-group; this extends prior results for coprime automorphisms and Sylow subgroups. The approach combines nilpotent-case analysis (via Thompson's critical subgroup and powerful $p$-groups), coprime-action techniques (Burnside basis, generation through sections), a $p$-group case for $p$-soluble groups (Hall–Higman bounds and induction on $p$-length), and a culmination for general $\pi$-groups using a big-prime/small-prime decomposition. The results rely on the generalized Fitting theory and the classification of finite simple groups, yielding a robust framework for rank bounds on commutator subgroups arising from automorphism actions in soluble and semisimple contexts.

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 $π$-group of automorphisms of a $π$-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 $π$-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow $p$-subgroups of $p$-soluble groups.

Theorems & Definitions (87)

• Proposition 1
• proof
• Example 2
• Lemma 1.1
• proof
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Lemma 1.6