Recursive characterisation of skew morphisms of finite cyclic groups
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arXiv
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| Format: | Preprint |
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2025
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| author | Bachratý, Martin Hagara, Michal |
| author_facet | Bachratý, Martin Hagara, Michal |
| contents | A skew morphism of a finite group $G$ is an element $φ$ 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 $φ(ab)=φ(a)φ^{i_a}(b)$ for all $b\in G$. In this paper we show that if a skew morphism $φ$ of $\mathbb{Z}_n$ is not an automorphism of $\mathbb{Z}_n$, then it is uniquely determined by a triple $(h,α,β)$ where $h$ is an element of $\mathbb{Z}_n$, $α$ is a skew morphism of $\mathbb{Z}_a$ where $a<n$, and $β$ is a skew morphism of $\mathbb{Z}_b$ where either $b<n$, or $b=n$ and $|\langle β\rangle| <|\langle φ\rangle|$. Conversely, we also list necessary and sufficient conditions for a triple $(h,α,β)$ 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 $2000$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11626 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recursive characterisation of skew morphisms of finite cyclic groups Bachratý, Martin Hagara, Michal Group Theory Combinatorics 05E18 A skew morphism of a finite group $G$ is an element $φ$ 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 $φ(ab)=φ(a)φ^{i_a}(b)$ for all $b\in G$. In this paper we show that if a skew morphism $φ$ of $\mathbb{Z}_n$ is not an automorphism of $\mathbb{Z}_n$, then it is uniquely determined by a triple $(h,α,β)$ where $h$ is an element of $\mathbb{Z}_n$, $α$ is a skew morphism of $\mathbb{Z}_a$ where $a<n$, and $β$ is a skew morphism of $\mathbb{Z}_b$ where either $b<n$, or $b=n$ and $|\langle β\rangle| <|\langle φ\rangle|$. Conversely, we also list necessary and sufficient conditions for a triple $(h,α,β)$ 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 $2000$. |
| title | Recursive characterisation of skew morphisms of finite cyclic groups |
| topic | Group Theory Combinatorics 05E18 |
| url | https://arxiv.org/abs/2506.11626 |