Recursive characterisation of skew morphisms of finite cyclic groups

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 $2000$.