Finite BCI-groups are solvable

Document Type : Research Paper


Isfahan University of Technology


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 $G\times\{1,2\}$ and edge set $\{\{(x,1),(sx,2)\}\mid x\in G‎, ‎\ s\in S\}$‎. ‎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.


Main Subjects

M‎. ‎Arezoomand and B‎. ‎Taeri (2015). ‎Isomorphisms of finite semi-Cayley graphs. Acta Math‎. ‎Sin‎. ‎(Engl‎. ‎Ser.). 31, 715-730 M‎. ‎Conder and C‎. ‎H‎. ‎Li (1998). ‎On isomorphisms of finite Cayley graphs. European J‎. ‎Combin.. 19, 911-919 J‎. ‎H‎. ‎Conway‎, ‎R‎. ‎T‎. ‎Curtis‎, ‎S‎. ‎P‎. ‎Norton‎, ‎R‎. ‎A‎. ‎Parker and R‎. ‎A‎. ‎Wilson (1985). ATLAS of Finite Groups. ‎Oxford Univ‎. ‎Press‎, ‎New York. J‎. ‎D‎. ‎Dixon and B‎. ‎Mortimer (1996). Permutation groups. ‎New York‎, ‎Springer-Verlag. E‎. ‎Dobson and J‎. ‎Morris (2015). ‎Quotients of CI-groups are CI-groups. Graphs Combin.. 31, 547-550 F‎. ‎Harary (1969). Graph theory. ‎Addison-Welsey Publishing Company‎, ‎Reading‎, ‎Mass.-Menlo Park‎, ‎Calif.-London. W‎. ‎Jin and W‎. ‎Liu (2010). ‎A classification of nonabelian simple 3-BCI-groups. European J‎. ‎Combin.. 31, 1257-1264 W‎. ‎Jin and W‎. ‎Liu (2011). ‎On Sylow subgroups of BCI groups. Util‎. ‎Math.. 86, 313-320 H‎. ‎Koike and I‎. ‎Kov'acs (2014). ‎Isomorphic tetravalent cyclic Haar graphs. Ars Math‎. ‎Contemp.. 7, 215-235 H‎. ‎Koike and I‎. ‎Kov'acs (2013). ‎A classification of nilpotent 3-BCI-groups. ‎preprint arXiv‎: ‎1308.6812v1 [math.GR]. C‎. ‎H‎. ‎Li and C‎. ‎E‎. ‎Praeger (1997). ‎Finite groups in which any two elements of the same order are either fused or inverse-fused.. Comm‎. ‎Algebra. 25, 3081-3118 C‎. ‎H‎. ‎Li (1999). ‎Finite CI-groups are soluble. Bull‎. ‎London Math‎. ‎Soc.. 31, 419-423 C‎. ‎H‎. ‎Li (2002). ‎On isomorphisms of finite Cayley graphs-a survay. Discrete Math.. 256, 301-334 S‎. ‎J‎. ‎Xu‎, ‎W‎. ‎Jin‎, ‎Q‎. ‎Shi‎, ‎Y‎. ‎Zhu and J‎. ‎J‎. ‎Li (2008). ‎The BCI-property of the Bi-Cayley graphs. J‎. ‎Guangxi Norm‎. ‎Univ.‎: ‎Nat‎. ‎Sci‎. ‎Edition. 26, 33-36