Finite groups in which every commutator has prime power order

Finite groups in which every element has prime power order (EPPO-groups) are nowadays fairly well understood. For instance, if G is a soluble EPPO-group, then the Fitting height of G is at most 3 and |pi(G)| <= 2 (Higman, 1957). Moreover, Suzuki showed that if G is insoluble, then the soluble radical of G is a 2-group and there are exactly eight nonabelian simple EPPO-groups. In the present work we concentrate on finite groups in which every commutator has prime power order (CPPO-groups). Roughly, we show that if G is a CPPO-group, then the structure of G' is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPOgroup is at most 3 and |pi(G')| <= 3. Moreover, if G is insoluble, then R(G') is a 2-group and G'/R(G') is isomorphic to a simple EPPO-group. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar