Classification of cyclic groups underlying only smooth skew morphisms

Abstract

A skew morphism of a finite group A is a permutation \(\varphi \) of A fixing the identity element and for which there is an integer-valued function \(\pi \) on A such that \(\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)\) for all \(a, b \in A\) . A skew morphism \(\varphi \) of A is smooth if the associated power function \(\pi \) is constant on the orbits of \(\varphi \) , that is, \(\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}\) for all \(a\in A\) . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if \(n=2^en_1\) , where \(0 \le e \le 4\) and \(n_1\) is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.