A classification of nilpotent $3$-BCI groups

Document Type : Research Paper

Authors

1 National Autonomous University of Mexico

2 University of Primorska

Abstract

‎‎Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $BCay(G,S)$ is the graph whose vertex‎ ‎set is $G \times \{0,1\}$ and edge set is‎ ‎$\{ \{(x,0),(s x,1)\}‎ : ‎x \in G‎, ‎s\in S \}$‎. ‎A bi-Cayley graph $BCay(G,S)$ is called a BCI-graph if for any bi-Cayley graph‎ ‎$BCay(G,T),$ $BCay(G,S) \cong BCay(G,T)$ implies that $T = g S^\alpha$ for some $g \in G$ and $\alpha \in aut(G)$‎. ‎A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs‎. ‎It was proved by Jin and Liu that‎, ‎if $G$ is a $3$-BCI-group‎, ‎then its Sylow $2$-subgroup is cyclic‎, ‎or elementary abelian‎, ‎or $Q_8$ [European J‎. ‎Combin‎. ‎31 (2010)‎ ‎1257--1264]‎, ‎and that a Sylow $p$-subgroup‎, ‎$p$ is an odd prime‎, ‎is homocyclic [Util‎. ‎Math‎. ‎86 (2011) 313--320]‎. ‎In this paper we show that the converse also holds in the‎ ‎case when $G$ is nilpotent‎, ‎and hence complete the classification of‎ ‎nilpotent $3$-BCI-groups‎.

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References

[1] M. Arezo omand and B. Taeri, Isomorphisms of nite semi-Cayley graphs, Acta Math. Sin. (Engl. Ser.) , 31 (2015)
715{730.
[2] M. Arezo omand and B. Taeri, Finite BCI-groups are solvable, Int. J. Group Theory , 5 no. 2 (2016) 1{6.
[3] L. Babai, Isomorphism problem for a class of p oint-symmetric structures, Acta Math. Acad. Sci. Hungar. , 29 (1977)
329{336.
[4] N. Biggs and M. Hoare, The sextet construction for cubic graphs, Combinatorica , 3 (1983) 153{165.
[5] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: The User Language, J. Symbolic Comput. ,
24 (1997) 235{265.
[6] M. D. E. Conder and P. Dob csanyi, Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin.
Comput. , 40 (2002) 41{63.
[7] M. D. E. Conder and R. Nedela, Symmetric cubic graphs of small girth, J. Combin. Theory Ser. B , 97 (2007)
757{768.
[8] J. D. Dixon and B. Mortimer, Permutation groups , Graduate Texts in Mathematics, 163 , Springer-Verlag, New
York, 1996.
[9] B. Hupp ert, End liche Gruppen I , Springer-Verlag, Berlin Heidelb erg New York, 1967.
[10] W. Jin and W. Liu, Two results on BCI-subset of nite groups, Ars Combin. , 93 (2009) 169{173.
[11] W. Jin and W. Liu, A classi cation of nonab elian simple 3-BCI-groups, European J. Combin. , 31 (2010) 1257{1264.
[12] W. Jin and W. Liu, On Sylow subgroups of BCI groups, Util. Math. , 86 (2011) 313{320.
[13] H. Koike and I. Kovacs, Isomorphic tetravalent cyclic Haar graphs, Ars Math. Contemp. , 7 (2014) 215{235.
[14] K. Kutnar and D. Marusic, A complete classi cation of cubic symmetric graphs of girth 6, J. Combin. Theory Ser.
B , 99 (2009) 162{184.
[15] C. H. Li, Isomorphism of nite Cayley digraphs of b ounded valency, I I, J. Combin. Theory. Ser. A , 87 (1999)
333{346.
[16] C. H. Li, The nite vertex-primitive and vertex-biprimitive s -transitive graphs for s  4, Trans. Amer. Math. Soc. ,
353 (2001) 3511{3529.
[17] C. H. Li, On isomorphisms of nite Cayley graphs-a survey, Discrete Math. , 256 (2002) 301{334.
[18] C. H. Li and C. E. Praeger, Finite groups in which any two elements of the same order are either fused or inverse
fused, Comm. Algebra , 25 (1997) 3081{3118.
[19] C. H. Li, C. E. Praeger and M. Y. Xu, Isomorphisms of nite Cayley digraphs of b ounded valency, J. Combin.
Theory. Ser. B , 73 (1998) 164{183.
[20] P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory , 8 (1984) 55{68.
[21] W. R. Scott, Group theory , Prentice-Hall, Inc., New Jersey, 1964.
[22] W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. , 43 (1947) 459{474.
[23] D. Wiedemann and M. E. Zieve, Equivalence of sparse circulants: the bipartite Adam problem, preprint
arXiv:0706.1567v1 [math. CO] (2007).
[24] S. J. Xu, W. Jin, Q. Shi and J. J. Li, The BCI-prop erty of the Bi-Cayley graphs, J. Guangxi Norm. Univ.: Nat.
Sci. Edition , 26 (2008) 33{36.

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History

  • Receive Date: 05 December 2016
  • Revise Date: 23 November 2017
  • Accept Date: 24 November 2017
  • Published Online: 01 June 2019

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