On finite groups with bounded conjugacy classes of commutators

TL;DR

This paper extends classical results on finite groups with bounded conjugacy classes of commutators, showing that the second derived group has bounded order under prime power conditions.

Contribution

It proves that for finite groups where commutators of prime power order have bounded conjugacy classes, the second derived group is finite with bounded order.

Findings

The second derived group $G''$ has $m$-bounded order under the given conditions.

Generalizes previous results by considering commutators of prime power order.

Provides new bounds for the structure of finite groups based on conjugacy class sizes.

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.