Recursive characterisation of skew morphisms of finite cyclic groups

TL;DR

This paper provides a recursive characterization of skew morphisms of finite cyclic groups, enabling the enumeration and analysis of these structures up to order 2000, and introduces new theoretical insights.

Contribution

It introduces a recursive method to uniquely determine skew morphisms of cyclic groups using triples, advancing understanding and classification of these morphisms.

Findings

Recursive characterization of skew morphisms for all finite cyclic groups.

Complete census of skew morphisms for cyclic groups up to order 2000.

New theorems about the structure of skew morphisms.

Abstract

A skew morphism of a finite group $G$ is an element $\varphi$ of $\mathrm{Sym}(G)$ preserving the identity element of $G$ and having the property that for each $a\in G$ there exists a non-negative integer $i_a$ such that $\varphi(ab)=\varphi(a)\varphi^{i_a}(b)$ for all $b\in G$. In this paper we show that if a skew morphism $\varphi$ of $\mathbb{Z}_n$ is not an automorphism of $\mathbb{Z}_n$, then it is uniquely determined by a triple $(h,\alpha,\beta)$ where $h$ is an element of $\mathbb{Z}_n$, $\alpha$ is a skew morphism of $\mathbb{Z}_a$ where $a<n$, and $\beta$ is a skew morphism of $\mathbb{Z}_b$ where either $b<n$, or $b=n$ and $|\langle \beta\rangle| <|\langle \varphi\rangle|$. Conversely, we also list necessary and sufficient conditions for a triple $(h,\alpha,\beta)$ to define a skew morphism of a given cyclic group. In particular, this gives a recursive characterisation of skew morphisms for all finite cyclic groups. We use this characterisation to prove new theorems about skew morphisms of cyclic groups and to generate a census of all skew morphisms for cyclic groups of order up to $2000$.