Finite BCI-groups are solvable

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 G×{1,2} and edge set {{(x,1),(sx,2)}∣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)‎, ‎whenever BCay(G,S)≅BCay(G,T) we have T=gSα for some g∈G and α∈Aut(G)‎. ‎A group G is called a BCI-group if every bi-Cayley graph of GG is a BCI-graph‎. ‎In this paper‎, ‎we prove that every BCI-group is solvable‎.