Skew-Morphisms of Elementary Abelian p-Groups

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

Decomposition of skew-morphisms of cyclic groups

A skew-morphism of a group ▫$H$▫ is a permutation ▫$sigma$▫ of its elements fixing the identity such that for every ▫$x, y in H$▫ there exists an integer ▫$k$▫ such that ▫$sigma (xy) = sigma (x)sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, varphi (n_{2})) = (varphi (n_{1}), n_{2}) = 1$▫ (▫$varphi$▫ denotes Euler\u27s function) then all skew-morphisms ▫$sigma$▫ of ▫$mathbb Z_n$▫ are obtained as ▫$sigma = sigma_1 times sigma_2$▫, where ▫$sigma_i$▫ are skew-morphisms of ▫$mathbb Z_{n_i},i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$mathbb Z_n$▫ are automorphisms of ▫$mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, varphi(n)) = 1$▫