Generalized quaternion groups with the m-DCI property

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 \(\sigma\) of \(G\) such that \(S^\sigma=T\). A finite group \(G\) is said to have the \(m\)-...

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Bibliographic Details
Published inarXiv.org
Main Authors Jin-Hua, Xie, Yan-Quan, Feng, Xia, Binzhou
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 21.02.2024
Subjects
Automorphisms
Graph theory
Group theory
Integers
Quaternions
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Summary: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 \(\sigma\) of \(G\) such that \(S^\sigma=T\). 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 \(1\leq m\leq p\).
ISSN:2331-8422