Smooth skew-morphisms of the dihedral groups

Abstract

A skew-morphism $\phi$ of a finite group $A$ is a permutation on $A$ such that $\phi (1)=1$ and $\phi (xy)=\phi (x){\phi }^{\pi (x)}(y)$ for all $x,y\in A$ where $\pi :A\to {\mathbb{Z}}_{|\phi |}$ is an integer function. A skew-morphism is smooth if $\pi (\phi (x))=\pi (x)$ for all $x\in A$. The concept of smooth skew-morphisms is a generalization of that of $t$-balanced skew-morphisms. The aim of the paper is to develop a general theory of smooth skew-morphisms. As an application we classify smooth skew-morphisms of the dihedral groups.

23pages