Classification of cyclic groups underlying only smooth skew morphisms

Abstract

A skew morphism of a finite group $A$ is a permutation $\phi$ of $A$ fixing the identity element and for which there is an integer-valued function $\pi$ on $A$ such that $\phi (ab)=\phi (a){\phi }^{\pi (a)}(b)$ for all $a,b\in A$. A skew morphism $\phi$ of $A$ is smooth if the associated power function $\pi$ is constant on the orbits of $\phi$, that is, $\pi (\phi (a))\equiv \pi (a)(\mathrm{mod}|\phi |)$ 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 $n=2^e{n}_1$, 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.