Skew-Morphisms of Elementary Abelian p-Groups

TL;DR

This paper studies skew-morphisms of elementary abelian p-groups, exploring their properties, characterizations, and constructions, and how they form permutation groups called skew-product groups.

Contribution

It provides new insights into the structure, properties, and construction methods of skew-morphisms and their associated skew-product groups for elementary abelian p-groups.

Findings

Characterized skew-morphisms of elementary abelian p-groups.

Established properties of skew-product groups.

Developed methods to construct skew-morphisms.

Abstract

A skew-morphism of a finite group $G$ is a permutation $\sigma$ on $G$ fixing the identity element, and for which there exists an integer function $\pi$ on $G$ such that $\sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $\sigma $ of $G$, the product of $\langle \sigma \rangle$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $\sigma$. 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.