Finite BCI-groups are solvable

Abstract

Let $S$ be a subset of a finite group $G$‎. ‎The bi-Cayley graph $BCay(G,S)$ of $G$ with respect to $S$ is an undirected graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin G‎, ‎ sin S}$‎. ‎A bi-Cayley graph $BCay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $BCay(G,T)$‎, ‎whenever $BCay(G,S)cong BCay(G,T)$ we have $T=gS^alpha$ for some $gin G$ and $alphain Aut(G)$‎. ‎A group $G$ is called a BCI-group if every bi-Cayley graph of $G$ is a BCI-graph‎. ‎In this paper‎, ‎we prove that every BCI-group is solvable‎.