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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.20590 |
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| _version_ | 1866914498964619264 |
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| author | Bachratý, Martin |
| author_facet | Bachratý, Martin |
| contents | A skew morphism of a finite group $B$ is a permutation of $B$ fixing the identity and satisfying $φ(xy) = φ(x)φ^{i_x}(y)$ for some integers $i_x$ indexed by $x \in B$. The enumeration of skew morphisms of finite cyclic groups remains an open problem. The most substantial progress to date concerns cyclic $p$-groups with $p$ odd, for which a full classification and enumeration was obtained by Kovács and Nedela. In this paper we treat the remaining case $p = 2$, giving a complete classification and enumeration of skew morphisms of finite cyclic $2$-groups. Writing $\mathrm{Skew}(n)$ for the number of skew morphisms of $\mathbb{Z}_n$, we prove that $\mathrm{Skew}(2^e) = 4\,\mathrm{Skew}(2^{e-1}) - 4$ for each $e \geq 4$, and that $\mathrm{Skew}(2^e) = (7 \cdot 4^{e-2} + 8)/6$ for each $e \geq 3$. This completes the enumeration of skew morphisms for all cyclic $p$-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20590 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Enumeration of skew morphisms of cyclic $2$-groups Bachratý, Martin Group Theory 20B25 A skew morphism of a finite group $B$ is a permutation of $B$ fixing the identity and satisfying $φ(xy) = φ(x)φ^{i_x}(y)$ for some integers $i_x$ indexed by $x \in B$. The enumeration of skew morphisms of finite cyclic groups remains an open problem. The most substantial progress to date concerns cyclic $p$-groups with $p$ odd, for which a full classification and enumeration was obtained by Kovács and Nedela. In this paper we treat the remaining case $p = 2$, giving a complete classification and enumeration of skew morphisms of finite cyclic $2$-groups. Writing $\mathrm{Skew}(n)$ for the number of skew morphisms of $\mathbb{Z}_n$, we prove that $\mathrm{Skew}(2^e) = 4\,\mathrm{Skew}(2^{e-1}) - 4$ for each $e \geq 4$, and that $\mathrm{Skew}(2^e) = (7 \cdot 4^{e-2} + 8)/6$ for each $e \geq 3$. This completes the enumeration of skew morphisms for all cyclic $p$-groups. |
| title | Enumeration of skew morphisms of cyclic $2$-groups |
| topic | Group Theory 20B25 |
| url | https://arxiv.org/abs/2604.20590 |