Twisted cyclic groups

Abstract

A finite group $G$ is said to be twisted cyclic if there exist $\varphi \in \mathrm{Aut}(G)$ and $x\in G$ such that $G=\{(x^i){\varphi }^j:i,j\in \mathbb{Z}\}$. 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 $n\in {\mathbb{Z}}^{+}$.