Symmetric Cayley graphs on non-abelian simple groups of valency 7
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| Format: | Preprint |
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2024
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| _version_ | 1866910636572672000 |
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| author | Zhang, Xing Feng, Yan-Quan Yin, Fu-Gang Wang, Hong |
| author_facet | Zhang, Xing Feng, Yan-Quan Yin, Fu-Gang Wang, Hong |
| contents | Let $Γ$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $Γ$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022) 1097-1118] characterised the group pairs $(\mathrm{soc}(\mathrm{Aut}(Γ)/K),GK/K)$, where $K$ is a maximal intransitive normal subgroup of $\mathrm{Aut}(Γ)$. In this paper, we improve this result by proving that if $Γ$ is not normal, then $\mathrm{Aut}(Γ)$ contains an arc-transitive non-abelian simple normal subgroup $T$ such that $G<T$ and $(T,G)=(\mathrm{A}_{n},\mathrm{A}_{n-1})$ with $n=7$, $3\cdot 7$, $3^2\cdot 7$, $2^2\cdot 3\cdot 7$, $2^3\cdot3\cdot7$, $2^3\cdot3^2\cdot5\cdot7$, $2^4\cdot3^2\cdot5\cdot7$, $2^6\cdot3\cdot7$, $2^7\cdot3\cdot7$, $2^6\cdot3^2\cdot7$, $2^6\cdot3^4\cdot5^2\cdot7$, $2^8\cdot3^4\cdot5^2\cdot7$, $2^7\cdot3^4\cdot5^2\cdot7$, $2^{10}\cdot3^2\cdot7$, $2^{24}\cdot3^2\cdot7$. Furthermore, $\mathrm{soc}(\mathrm{Aut}(Γ)/R)=(T\times R)/R$, where $R$ is the largest solvable normal subgroup of $\mathrm{Aut}(Γ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_19225 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric Cayley graphs on non-abelian simple groups of valency 7 Zhang, Xing Feng, Yan-Quan Yin, Fu-Gang Wang, Hong Group Theory Combinatorics 05C25, 20B25 Let $Γ$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $Γ$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022) 1097-1118] characterised the group pairs $(\mathrm{soc}(\mathrm{Aut}(Γ)/K),GK/K)$, where $K$ is a maximal intransitive normal subgroup of $\mathrm{Aut}(Γ)$. In this paper, we improve this result by proving that if $Γ$ is not normal, then $\mathrm{Aut}(Γ)$ contains an arc-transitive non-abelian simple normal subgroup $T$ such that $G<T$ and $(T,G)=(\mathrm{A}_{n},\mathrm{A}_{n-1})$ with $n=7$, $3\cdot 7$, $3^2\cdot 7$, $2^2\cdot 3\cdot 7$, $2^3\cdot3\cdot7$, $2^3\cdot3^2\cdot5\cdot7$, $2^4\cdot3^2\cdot5\cdot7$, $2^6\cdot3\cdot7$, $2^7\cdot3\cdot7$, $2^6\cdot3^2\cdot7$, $2^6\cdot3^4\cdot5^2\cdot7$, $2^8\cdot3^4\cdot5^2\cdot7$, $2^7\cdot3^4\cdot5^2\cdot7$, $2^{10}\cdot3^2\cdot7$, $2^{24}\cdot3^2\cdot7$. Furthermore, $\mathrm{soc}(\mathrm{Aut}(Γ)/R)=(T\times R)/R$, where $R$ is the largest solvable normal subgroup of $\mathrm{Aut}(Γ)$. |
| title | Symmetric Cayley graphs on non-abelian simple groups of valency 7 |
| topic | Group Theory Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2409.19225 |