A classification of nilpotent $3$-BCI groups

Abstract

‎‎Given a finite group $G$ and a subset $Ssubseteq G,$ the bi-Cayley graph $bcay(G,S)$ is the graph whose vertex‎ ‎set is $G times {0,1}$ and edge set is‎ ‎${ {(x,0),(s x,1)}‎ : ‎x in G‎, ‎sin S }$‎. ‎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 $3$-BCI-group‎, ‎then its Sylow $2$-subgroup is cyclic‎, ‎or elementary abelian‎, ‎or $Q$ [European J‎. ‎Combin‎. ‎31 (2010)‎ ‎1257--1264]‎, ‎and that a Sylow $p$-subgroup‎, ‎$p$ is an odd prime‎, ‎is homocyclic [Util‎. ‎Math‎. ‎86 (2011) 313--320]‎. ‎In this paper we show that the converse also holds in the‎ ‎case when $G$ is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent $3$-BCI-groups‎.