Twisted cyclic groups

Abstract

A finite group $G$ is said to be twisted cyclic if there exist $\phi \in \operatorname{Aut}(G)$ and $x \in G$ such that $G=\{(x^i)\phi^j\colon i,j \in \mathbb{Z}\}$. In this note, we classify all groups satisfying this property and determine that, if a finite group $G$ is twisted cyclic, then $G$ is isomorphic to $\mathbb{Z}_{p^n}$, $\mathbb{Z}_p\times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$, $Q_8$, $\mathbb{Z}_{p^n}\times \mathbb{Z}_{p^n}$ or direct products of these groups for some prime $p$ and some $n\in \mathbb{Z}^{+}$.