A classification of nilpotent $۳$-BCI groups

‎‎Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex‎ ‎set is $G \times \{۰,۱\}$ and edge set is‎ ‎$\{ \{(x,۰),(s x,۱)\}‎ : ‎x \in G‎, ‎s\in S \}$‎. ‎A bi-Cayley graph $\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph‎ ‎$\bcay(G,T),$ $\bcay(G,S) \cong \bcay(G,T)$ implies that $T = g S^\alpha$ for some $g \in G$ and $\alpha \in \aut(G)$‎. ‎A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs‎. ‎It was proved by Jin and Liu that‎, ‎if $G$ is a $۳$-BCI-group‎, ‎then its Sylow $۲$-subgroup is cyclic‎, ‎or elementary abelian‎, ‎or $\Q$ [European J‎. ‎Combin‎. ‎۳۱ (۲۰۱۰)‎ ‎۱۲۵۷--۱۲۶۴]‎, ‎and that a Sylow $p$-subgroup‎, ‎$p$ is an odd prime‎, ‎is homocyclic [Util‎. ‎Math‎. ‎۸۶ (۲۰۱۱) ۳۱۳--۳۲۰]‎. ‎In this paper we show that the converse also holds in the‎ ‎case when $G$ is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent $۳$-BCI-groups‎.

Keywords:

‎bi-Cayley graph‎ , ‎BCI-group‎ , ‎graph isomorphism