On finite groups with bounded conjugacy classes of commutators

TL;DR

The paper addresses finiteness of the second derived subgroup $G''$ for finite groups $G$ in which every commutator of prime power order has a conjugacy class of size at most $m$, extending Neumann-type results. It introduces the set $X(G)$ of structured commutators and uses system normalizers to manage quotients, proving that $G''$ has $m$-bounded order. The argument proceeds by reducing to the stronger hypothesis $|x^G| m$ for all $x X(G)$, then treating soluble and semisimple components separately and bounding the semisimple part via the Fitting-related subgroup $F^*(G)$, ultimately obtaining a global $m$-bound on $G''$. The work clarifies the scope of finiteness phenomena for commutator-words and suggests directions for generalizing to other words, while noting limitations for extending Part (2) of the known results.

Abstract

In 1954 B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018 G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

Theorems & Definitions (9)

• proof : Proof of Propostition \ref{['easy']}
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• proof : Proof of Theorem \ref{['main']}