Camina $p$-groups that are generalized Frobenius complements

Abstract

Let $P$ be a Camina $p$-group that acts on a group $Q$ in such a way that $C_P (x) \subseteq P'$ for all nonidentity elements $x \in Q$. We show that $P$ must be isomorphic to the quaternion group $Q_8$. If $P$ has class $2$, this is a known result, and this paper corrects a previously published erroneous proof of the general case.