Normal and non-normal Cayley digraphs on cyclic and dihedral groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909537154367488 |
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| author | Yang, Jun-Feng Feng, Yan-Quan Yin, Fu-Gang Zhou, Jin-Xin |
| author_facet | Yang, Jun-Feng Feng, Yan-Quan Yin, Fu-Gang Zhou, Jin-Xin |
| contents | A Cayley digraph on a group $G$ is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to $G$. A group is called NNND-group or NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order $2n$ is an NNND-group or an NNN-group if and only if $n\ge 6$ is even and $n\not=8$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal and non-normal Cayley digraphs on cyclic and dihedral groups Yang, Jun-Feng Feng, Yan-Quan Yin, Fu-Gang Zhou, Jin-Xin Group Theory Combinatorics 05C25, 20B25 A Cayley digraph on a group $G$ is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to $G$. A group is called NNND-group or NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order $2n$ is an NNND-group or an NNN-group if and only if $n\ge 6$ is even and $n\not=8$. |
| title | Normal and non-normal Cayley digraphs on cyclic and dihedral groups |
| topic | Group Theory Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2503.10994 |