A classification of nonabelian simple 3-BCI-groups

For a finite group GG and a subset S⊆GS⊆G (possibly, SS contains the identity of GG), the bi-Cayley graph BCay(G,S) of GG with respect to SS is the graph with vertex set G×{0,1}G×{0,1} and with edge set {(x,0),(sx,1)|x∈G,s∈S}{(x,0),(sx,1)|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=gSαT=gSα, for some g∈G,α∈Aut(G). A group GG is called an m-BCI-group  , if all bi-Cayley graphs of GG of valency at most mm are BCI-graphs. In this paper, we prove that a finite nonabelian simple group is a 3-BCI-group if and only if it is A5A5.