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Distance transitive graphs with symmetric or alternating automorphism group

Published online by Cambridge University Press:  17 April 2009

Martin W. Liebeck ,
Cheryl E. Praeger  and
Jan Saxl
Martin W. Liebeck
Affiliation:
Department of Mathematics, Imperial College, London SW7 2BZ, England
Cheryl E. Praeger
Affiliation:
Mathematics Department, University of Western Australia, Nedlands WA 6009, Western Australia
Jan Saxl
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge, England.
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Abstract

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The paper classifies all distance transitive graphs Γ such that An ≤ Aut Γ ≤ Aut An for some alternating group An, and Aut Γ acts primitively on the vertices of Γ. This result forms part of our programme for determining all finite primitive distance transitive graphs.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 35 , Issue 1 , February 1987 , pp. 1 - 25
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Biggs, N.L., Algebraic Graph Theory, Cambridge University Press, 1974.CrossRefGoogle Scholar
[2]Cameron, P.J., “Finite permutation groups and finite simple groups”, Bull. London Math. Soc. 13 (1981), 1–22.CrossRefGoogle Scholar
[3]Cameron, P.J., “There are finitely many distance transitive graphs of given valency greater than two”, Combinatorica 2 (1982), 9–13.CrossRefGoogle Scholar
[4]Conway, J.H., Curtis, R.T., Norton, S.P.. Parker, R.A. and Wilson, R.A., Atlas of finite Groups, Oxford University Press, 1985.Google Scholar
[5]Conway, J.H., “The permutations of S 12 modulo M 12”, unpublished.Google Scholar
[6]Halberstadt, E., “On certain maximal subgroups of symmetric or alternating groups”, Math, Z. 151 (1976), 117–125.CrossRefGoogle Scholar
[7]Inglis, N.F.J., Ph.D. Thesis, University of Cambridge, 1986.Google Scholar
[8]Ivanov, A.A., “Distance-transitive representations of the symmetric groups” (preprint).Google Scholar
[9]James, G.D. and Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley, 1981.Google Scholar
[10]Liebeck, M.W. and Saxl, J., “Some recent results on finite permutation groups”, in Proc. Rutgers Group Theory Year, 1983–4 (eds. Aschbacher, M., Gorenstein, D. et al. ), Cambridge Univ. Press, 1984, pp 53–62.Google Scholar
[11]Liebeck, M.W., Praeger, C.E. and Saxl, J., “A Classification of the maximal subgroups of the finite alternating and symmetric groups”, J. Algebra, (to appear).Google Scholar
[12]Liebeck, M.W., Praeger, C.E. and Saxl, J., “The affine distance transitive graphs of large degree”, in preparation.Google Scholar
[13]Littlewood, D.E., The Theory of Group Characters, Oxford University Press, 1940.Google Scholar
[14]Neumann, P.M., “Finite permutation groups, edge-coloured graphs and matrices”, in Topics in Group Theory and Computation (ed. Curran, M.P.J.) Academic Press, London, 1977, pp. 82–116.Google Scholar
[15]Praeger, C.E. and Saxl, J., “On the orders of primitive permutation groups”, Bull. London Math. Soc. 12 (1980), 303–307.CrossRefGoogle Scholar
[16]Praeger, C.E. and Saxl, J. and Yokoyama, K., “Distance transitive graphs and finite simple groups”, Proc. London Math. Soc., (to appear).Google Scholar
[17]Saxl, J., “On multiplicity-free permutation representations”, in Finite Geometries and Designs (eds Cameron, P.J., Hirschfeld, J.W.P. and Hughes, D.R.), London Math. Soc. Lecture Notes 49, Cambridge University Press, 1981, pp 337–353.CrossRefGoogle Scholar
[18]Sims, C.C., “Computational methods in the study of finite permutatin groups”, in Computational Problems in Abstract Algebra (ed. Leech, J.) Pergammon Press, 1970, pp 169–183.Google Scholar
[19]Smith, D.H., “Primitive and imprimitive graphs”, Quart. J. Math. (1971), pp. 551–557.CrossRefGoogle Scholar
[20]Tutte, W.T., Connectivity in Graphs, University of Toronto Press, 1966.CrossRefGoogle Scholar