A classification of nilpotent 3-BCI groups

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Given a finite group G and a subset S G, the bi-Cayley graph (G,S) is the graph whose vertex set is G × {0,1} and edge set is {{(x,0),(s x,1)}: x ∈ G, s∈ S }. A bi-Cayley graph (G,S) is called a BCI-graph if for any bi-Cayley graph (G,T), (G,S) (G,T) implies that T = g Sα for some g ∈ G and α∈ (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.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form U × V, where U is a homocyclic group of odd order, and V is trivial or one of the groups ₂ᵣ, ₂ʳ and ₈.