Quotients of skew morphisms of cyclic groups

TL;DR

This paper advances the understanding of skew morphisms of cyclic groups by classifying those of certain orders, developing an algorithm for their enumeration, and fixing previous theoretical flaws in related literature.

Contribution

It provides a classification of skew morphisms for cyclic groups of specific orders, introduces an algorithm for their enumeration, and corrects earlier theoretical errors.

Findings

Classified skew morphisms for cyclic groups of order 2^em with e in {0,1,2,3,4} and m odd square-free.

Developed and implemented an algorithm to find all skew morphisms of cyclic groups up to order 161.

Identified and corrected flaws in previous theoretical results related to skew morphisms.

Abstract

A skew morphism of a finite group $B$ is a permutation $\varphi$ of $B$ that preserves the identity element of $B$ and has the property that for every $a\in B$ there exists a positive integer $i_a$ such that $\varphi(ab) = \varphi(a)\varphi^{i_a}(b)$ for all $b\in B$. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism $\varphi$ of $\mathbb{Z}_n$ is closely related to a specific skew morphism of $\mathbb{Z}_{|\!\langle \varphi \rangle\!|}$, called the quotient of $\varphi$. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order $2^em$ with $e\in \{0,1,2,3,4\}$ and $m$ odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to $161$. During the preparation of this paper we noticed a few flaws in Section~5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68-100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.