Classification of cyclic groups underlying only smooth skew morphisms

TL;DR

This paper characterizes when all skew morphisms of cyclic groups are smooth, showing it occurs precisely for groups of order with specific factorizations involving powers of 2 and odd square-free numbers.

Contribution

It provides a complete characterization of cyclic groups whose skew morphisms are all smooth, extending understanding of the structure of skew morphisms.

Findings

All skew morphisms of cyclic groups are smooth iff the order is 2^e n_1 with 0 ≤ e ≤ 4 and n_1 odd square-free.

Partial results are given for non-cyclic abelian groups.

The paper advances the classification of smooth skew morphisms in finite groups.

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.