University of IsfahanInternational Journal of Group Theory2251-76508220190601A classification of nilpotent $3$-BCI groups11242220210.22108/ijgt.2017.100795.1404ENHiroki KoikeNational Autonomous University of MexicoIstvan KovacsUniversity of PrimorskaJournal Article20161205Given 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.http://ijgt.ui.ac.ir/article_22202_277a0945cb23c54a90741a9b98909611.pdf