5 research outputs found
Classification of nilpotent 3-BCI-groups
‎‎Given a finite group and a subset the bi-Cayley graph is the graph whose vertex‎ ‎set is and edge set is‎ ‎‎. ‎A bi-Cayley graph is called a BCI-graph if for any bi-Cayley graph‎ ‎ implies that for some and ‎. ‎A group is called an -BCI-group if all bi-Cayley graphs of of valency at most are BCI-graphs‎. ‎It was proved by Jin and Liu that‎, ‎if is a -BCI-group‎, ‎then its Sylow -subgroup is cyclic‎, ‎or elementary abelian‎, ‎or [European J‎. ‎Combin‎. ‎31 (2010)‎ ‎1257--1264]‎, ‎and that a Sylow -subgroup‎, ‎ 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 is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent -BCI-groups‎