On finite groups with bounded conjugacy classes of commutators

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.