This presentation is based on joint work with P. Shumyatsky. We study the class of all finite groups in which every commutator has prime power order. A group in this class is called a CPPO-group. Our interest in this class arose from the observation that it includes, as a subclass, all finite groups in which every element has prime power order - commonly known as EPPO-groups. These were the subject of foundational work by G. Higman and M. Suzuki: Higman described the soluble EPPO-groups, and subsequently, Suzuki classified the (nonabelian) simple EPPO-groups, proving that only eight isomorphism classes of such groups exist.

Although the class of CPPO-groups is significantly broader than that of EPPO-groups, we demonstrate that the structure of the derived subgroup of a CPPO-group exhibits notable similarities to that of an EPPO-group. Specifically, we prove that the Fitting height of a soluble EPPO-group $G$ is at most 3, and that the order of $G'$ is divisible by no more than two distinct primes. Furthermore, if $G$ is a nonsoluble CPPO-group, then $G'$ is perfect, its soluble radical $R(G')$ is a 2-group, and the quotient group $G'/R(G')$ is isomorphic to a (nonabelian) simple EPPO-group.

A key component of our approach involved techniques developed by A. Turull, particularly the use of Turull towers, which played a crucial role in establishing the upper bound on the Fitting height.