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Cubic graphs admitting transitive non-abelian characteristically simple groups

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  19 January 2011

Xiao-Hui Hua  and
Yan-Quan Feng
Xiao-Hui Hua
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People's Republic of China ([email protected])
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People's Republic of China ([email protected])
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Abstract

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Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

Keywords

Cayley graphsymmetric graphquasiprimitive graph

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 113 - 123
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Baddeley, R. W. and Praeger, C. E., On primitive overgroups of quasiprimitive permutation groups, J. Alg. 263 (2003), 294–344.CrossRefGoogle Scholar
2.Baik, Y. G., Feng, Y.-Q., Sim, H. S. and Xu, M. Y., On the normality of Cayley graphs of abelian groups, Alg. Colloq. 5 (1998), 297–304.Google Scholar
3.Conder, M. D. E. and Dobcsányi, P., Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin. Comput. 40 (2002), 41–63.Google Scholar
4.Conway, H. J., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, 1985).Google Scholar
5.Dobson, E. and Witte, D., Transitive permutation groups of prime-squared degree, J. Algebraic Combin. 16 (2002), 43–69.CrossRefGoogle Scholar
6.Fang, X. G., Li, C. H., Wang, J. and Xu, M. Y., On cubic Cayley graphs of finite simple groups, Discr. Math. 244 (2002), 67–75.CrossRefGoogle Scholar
7.Fang, X. G., Li, C. H. and Xu, M. Y., On finite edge-transitive Cayley graphs of valency 4, Eur. J. Combin. 25 (2004), 1107–1116.CrossRefGoogle Scholar
8.Fang, X. G., Praeger, C. E. and Wang, J., On the automorphism groups of Cayley graphs of finite simple groups, J. Lond. Math. Soc. 66 (2002), 563–578.CrossRefGoogle Scholar
9.Feng, Y.-Q., Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets, J. Combin. Theory B 96 (2006), 67–72.CrossRefGoogle Scholar
10.Feng, Y.-Q., Lu, Z. P. and Xu, M. Y., Automorphism groups of Cayley digraphs, in Applications of group theory to combinatorics, pp. 13–26 (Taylor and Francis/CRC Press, Boca Raton, FL/London, 2008).Google Scholar
11.Feng, Y.-Q. and Xu, M. Y., Automorphism groups of tetravalent Cayley graphs on regular p-groups, Discr. Math. 305 (2005), 354–360.CrossRefGoogle Scholar
12.Godsil, C. D., On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256.CrossRefGoogle Scholar
13.Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982).CrossRefGoogle Scholar
14.Li, C. H., Isomorphisms of finite Cayley graphs, PhD thesis, University of Western Australia (1996).Google Scholar
15.Li, C. H., On isomorphisms of connected Cayley graphs, III, Bull. Austral. Math. Soc. 58 (1998), 137–145.CrossRefGoogle Scholar
16.Liebeck, M. W., Praeger, C. E. and Saxl, J., On the O'Nan-Scott Theorem for finite primitive permutation groups, J. Austral. Math. Soc. A 44 (1988), 389–396.CrossRefGoogle Scholar
17.Lorimer, P., Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory 8 (1984), 55–68.CrossRefGoogle Scholar
18.Praeger, C. E., Imprimitive symmetric graphs, Ars Combin. 19 (1985), 149–163.Google Scholar
19.Praeger, C. E., The inclusion problem for finite primitive permutation groups, Proc. Lond. Math. Soc. 60 (1990), 68–88.CrossRefGoogle Scholar
20.Praeger, C. E., An O'Nan–Scott Theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. Lond. Math. Soc. 47 (1993), 227–239.CrossRefGoogle Scholar
21.Praeger, C. E., Finite normal edge-transitive graphs, Bull. Austral. Math. Soc. 60 (1999), 207–220.CrossRefGoogle Scholar
22.Prager, C. E., Quotients and inclusions of finite quasiprimitive permutation groups, J. Alg. 269 (2003), 329–346.CrossRefGoogle Scholar
23.Robinson, D. J., A course in the theory of groups (Springer, 1982).CrossRefGoogle Scholar
24.Sabidussi, G. O., Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438.CrossRefGoogle Scholar
25.Tutte, W. T., On the symmetry of cubic graphs, Can. J. Math. 11 (1959), 621–624.CrossRefGoogle Scholar
26.Wielandt, H., Finite permutation groups (Academic Press, 1964).Google Scholar
27.Xu, M. Y., Automorphism groups and isomorphisms of Cayley digraphs, Discr. Math. 182 (1998), 309–319.CrossRefGoogle Scholar
28.Xu, S. J., Fang, X. G., Wang, J. and Xu, M. Y., On cubic s-arc transitive Cayley graphs of finite simple groups, Eur. J. Combin. 26 (2005), 133–143.CrossRefGoogle Scholar