Camina $p$-groups that are generalized Frobenius complements

TL;DR

This paper proves that certain Camina p-groups acting on groups must be isomorphic to the quaternion group Q_8, correcting a previous proof for the general case beyond class 2.

Contribution

It establishes that Camina p-groups with a specific action property are isomorphic to Q_8, extending known results and fixing an earlier proof error.

Findings

Camina p-groups under the given action are isomorphic to Q_8

The result holds for groups beyond class 2

Corrects a previous erroneous proof in the literature

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.