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

2025/05/05 by Cristina Acciarri, Robert M. Guralnick, Acciarri, Cristina +4
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2505.03017

openalex publication_date 2025/05/05 · openalex created_date 2025/10/16 · openalex updated_date 2026/08/01

Abstract

For a group A acting by automorphisms on a group G, let I_G(A) denote the set of commutators [g,a]=g^-1g^a, where g∈ G and a∈ 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.

Related

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• Coprime automorphisms of finite groups
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• A contribution to the theory of σ-properties of a finite group
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