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\times\{۱,۲\} and edge set \{\{(x,۱),(sx,۲)\}\mid 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)‎, ‎whenever BCay(G,S)\cong BCay(G,T) we have T=gS^\alpha for some g\in G and \alpha\in 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.

Keywords:

Bi-Cayley graph‎ , ‎graph isomorphism‎ , ‎solvable group