A classification of nilpotent 3 -BCI groups

‎‎Given a finite group G and a subset S⊆G, the bi-Cayley graph \bcay(G,S) is the graph whose vertex‎ ‎set is G×{0,1} and edge set is‎ ‎{{(x,0),(sx,1)}‎:‎x∈G‎,‎s∈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)≅\bcay(G,T) implies that T=gSα for some g∈G and α∈\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.