Generalized quaternion groups with the m-DCI property

TL;DR

This paper investigates the conditions under which generalized quaternion groups possess the m-DCI property, revealing that n must be odd and not divisible by p^2 for primes p ≤ m-1, with specific criteria for prime power cases.

Contribution

It establishes necessary and sufficient conditions for generalized quaternion groups to have the m-DCI property, extending understanding of Cayley digraph isomorphisms in these groups.

Findings

n is odd for the m-DCI property to hold

n is not divisible by p^2 for primes p ≤ m-1

Q_{4n} has the m-DCI property iff p is odd and n=p or 1≤m≤p when n is a prime power

Abstract

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 $1\leq m\leq p$.