Skew-product groups of finite abelian $p$-groups

Abstract

A skew-morphism of a finite group $G$ is a permutation $\sigma$ of $G$ fixing the identity element such that there exists a function $\pi\colon G\to\mathbb{Z}$ satisfying $\sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y)$ for all $x,y\in G$. For a given skew-morphism $\sigma$ of $G$, the product of the left regular representation of $G$ and the cyclic group $\langle\sigma\rangle$ forms a permutation group on $G$, called a skew-product group of $G$. Skew-product groups of finite elementary abelian $p$-groups were investigated in \cite{DLYZ}. In the present paper, we extend this study to all finite abelian $p$-groups.