Skew-Morphisms of Elementary Abelian p-Groups

Abstract

A skew-morphism of a finite group $G$ is a permutation $\ddot{I}ƒ$ on $G$ fixing the identity element, and for which there exists an integer function $\ddot{I}€$ on $G$ such that $\ddot{I}ƒ(xy)=\ddot{I}ƒ(x)\ddot{I}{ƒ}^{\ddot{I}€(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $\ddot{I}ƒ$ of $G$, the product of $⟨\ddot{I}ƒ⟩$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $\ddot{I}ƒ$. In this paper, the skew-product groups of skew-morphisms of finite elementary abelian $p$-groups are investigated. Some properties, characterizations and constructions about that are obtained.