Observations about skew morphisms of cyclic groups

A permutation \(\varphi \) $\phi$ on a finite group G is a skew morphism of G if \(\varphi (1_G)=1_G\) $\phi (1_G)=1_G$ and there exists a function \(\pi \!: G \rightarrow \mathbb {Z}_{|\langle \varphi \rangle |}\) $\pi :G\to {\mathbb{Z}}_{|⟨\phi ⟩|}$ such that \(\varphi (ab) = \varphi (a)\varphi ^{\pi (a)}(b)\) $\phi (ab)=\phi (a){\phi }^{\pi (a)}(b)$ for all \(a,b\in G\) $a,b\in G$ . In this paper, we develop an efficient algorithm for generating skew morphisms of finite cyclic groups and use it to determine all skew morphisms for cyclic groups up to order 1175. We then use this census to prove a number of new theorems about skew morphisms of finite cyclic groups. For example, we prove that the set of all skew morphisms of \(\mathbb {Z}_n\) ${\mathbb{Z}}_n$ is itself a group if and only if all skew morphisms of \(\mathbb {Z}_n\) ${\mathbb{Z}}_n$ are automorphisms, and we also determine all n for which every non-trivial skew morphism of \(\mathbb {Z}_n\) ${\mathbb{Z}}_n$ gives rise to a regular Cayley map. In addition, we classify skew morphisms for all finite cyclic groups of orders \(p^2q\) $p^{2}q$ and \(p^3q\) $p^{3}q$ where p and q are primes satisfying \(p<q\) $p<q$ .