International Journal of Group Theory (Jun 2016)
Finite BCI-groups are solvable
Abstract
Let $S$ be a subset of a finite group $G$. The bi-Cayley graph ${rm BCay}(G,S)$ 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 ${rm BCay}(G,S)$ is called a BCI-graph if for any bi-Cayley graph ${rm BCay}(G,T)$, whenever ${rm BCay}(G,S)cong {rm BCay}(G,T)$ we have $T=gS^alpha$ for some $gin G$ and $alphain {rm Aut}(G)$. 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.
