New paper accepted by a high-ranking journal

The paper “Rank type conditions on commutators in finite groups” by Cristina Acciarri, Robert M. Guralnick, Evgeny Khukhro, and Pavel Shumyatsky has been accepted for publication in a high-ranking (Q1) journal “Annali della Scuola Normale Superiore di Pisa – Classe di Scienze“. 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 in 2026.

Abstract: For a subset $S$ of a group $G$ , let $I_G(S)$ denote the set of commutators $[g,s]=g^{-1}g^s$ , where $g\in G$ and $s\in S$ , so that $[G,S]$ is the subgroup generated by $I_G(S)$ . We prove that if $G$ is a $p$ -soluble finite group with a Sylow $p$ -subgroup $P$ such that any subgroup generated by a subset of $I_G(P)$ is $r$ -generated, then $[G,P]$ has $r$ -bounded rank. We produce examples showing that such a result does not hold without the assumption of $p$ -solubility. Instead, we prove that if a finite group $G$ has a Sylow $p$ -subgroup $P$ such that (a) any subgroup generated by a subset of $I_G(P)$ is $r$ -generated, and (b) for any $x\in I_G(P)$ , any subgroup generated by a subset of $I_G(x)$ is $r$ -generated, then $[G,P]$ has $r$ -bounded rank. We also prove that if $G$ is a finite group such that for every prime $p$ dividing $|G|$ , for any Sylow $p$ -subgroup $P$ , any subgroup generated by a subset of $I_G(P)$ can be generated by $r$ elements, then the derived subgroup $G'$ has $r$ -bounded rank. As an important tool in the proofs, we prove the following result, which is also of independent interest: if a finite group $G$ admits a group of coprime automorphisms $A$ such that any subgroup generated by a subset of $I_G(A)$ is $r$ -generated, then the rank of $[G,A]$ is $r$ -bounded.