New paper accepted by “Journal of Algebra”

The paper “Local–global generation property of commutators in finite $\pi$ -soluble groups” by Cristina Acciarri, Robert M. Guralnick, Evgeny Khukhro, and Pavel Shumyatsky has been accepted for publication in “Journal of Algebra”. This research is a product of collaboration between group-theorists from Italy, USA, UK, and Brazil. The Lincoln algebraist Evgeny Khukhro was partially supported by the International Center for Mathematics at the Southern University for Science and Technology in Shenzhen, China, during his visit at the beginning of 2025.

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.