A CLASSIFICATION OF NILPOTENT 3-BCI GROUPS

Abstract. Given a nite 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 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 classication of nilpotent 3-BCI-groups.