A classification of nilpotent 3-BCI groups

Abstract

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex set is $G\times \{0,1\}$ and edge set is $\{\{(x,0),(sx,1)\}:x\in G,s\in S\}$. A bi-Cayley graph $\bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $\bcay(G,T),$ $\bcay(G,S) \cong \bcay(G,T)$ implies that $T=g{S}^{\alpha }$ for some $g\in G$ and $α\in \aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U\times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups ${\mathbb{Z}}_{2^r},$ ${\mathbb{Z}}_2^r$ and $\Q_8$.

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• Equivalence of sparse circulants: the bipartite Ádám problem