Smooth skew morphisms of dicyclic groups

Abstract

A skew morphism of a finite group A is a permutation $\varphi$ on A fixing the identity element of A, and for which there exists an integer-valued function $\pi :A\rightarrow {\mathbb {Z}}_{|\varphi |}$ on A such that $\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)$ for all $a,b\in A$. Moreover, the period of $\varphi$ is the smallest positive integer d such that $\pi (\varphi ^d(a))\equiv \pi (a)\pmod {|\varphi |}$ for all $a\in A$. In the case where $d=1$, the skew morphism $\varphi$ is called smooth. It is well known that if $\varphi$ is a skew morphism of period d, then $\varphi ^d$ is a smooth skew morphism. Thus, every skew morphism of period d may be extracted as a dth root of a smooth skew morphism. In this paper, we introduce a new concept of average function to investigate skew morphisms and as an application we present a classification of smooth skew morphisms of the dicyclic groups.