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=gs^α 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.