Isomorphisms of bi-Cayley graphs on generalized quaternion groups

Abstract

Let $m$ be a positive integer. A group $G$ is said to be an $m$-BCI-group if $G$ has the $k$-BCI property for all positive integers $k$ at most $m$. Let $G$ be a generalized quaternion group of order $4n$ with $n\geq 2$. It is shown that $G$ is a 3-BCI-group if and only if $G$ is a $2$-BCI-group if and only if $n=2$ or $n$ is odd.