On finite groups with bounded conjugacy classes of commutators

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 |xG| ≤ m for any commutator x∈ G , then the second derived group $G''$ is finite and has m -bounded order. This paper deals with finite groups in which |xG|≤ m whenever x∈ G is a commutator of prime power order. The main result is that $G''$ has m -bounded order.