A classification of nilpotent 3-bci groups

Abstract

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 ε G,s ε 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 ε G and α ε 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 Q8 [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.