Finite groups in which every commutator has prime power order

Abstract

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 $G^{\prime }$ is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPO-group is at most 3 and $|\pi (G^{\prime })|⩽3$. Moreover, if $G$ is insoluble, then $R(G^{\prime })$ is a 2-group and $G^{\prime }/R(G^{\prime })$ is isomorphic to a simple EPPO-group.