A classification of nonabelian simple 3-BCI-groups

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G ), the bi-Cayley graph BCay ( G , S ) of G with respect to S is the graph with vertex set G × { 0 , 1 } and with edge set { ( x , 0 ) , ( s x , 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 = g S α , for some g ∈ G , α ∈ 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 nonabelian simple group is a 3-BCI-group if and only if it is A 5 .