Generalized quaternion groups with the m-DCI property
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929249719418880 |
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| author | Xie, Jin-Hua Feng, Yan-Quan Xia, Binzhou |
| author_facet | Xie, Jin-Hua Feng, Yan-Quan Xia, Binzhou |
| contents | A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $σ$ of $G$ such that $S^σ=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if all $m$-valent Cayley digraphs of $G$ are CI-digraphs, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_16677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized quaternion groups with the m-DCI property Xie, Jin-Hua Feng, Yan-Quan Xia, Binzhou Combinatorics 05C25, 20B25 A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $σ$ of $G$ such that $S^σ=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if all $m$-valent Cayley digraphs of $G$ are CI-digraphs, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$. |
| title | Generalized quaternion groups with the m-DCI property |
| topic | Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2306.16677 |