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Bounds on finite quasiprimitive permutation groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger  and
Aner Shalev
Cheryl E. Praeger
Affiliation:
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia e-mail: [email protected].
Aner Shalev
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel e-mail: [email protected]
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Abstract

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A permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

Keywords

primitive permutation groupquasiprimitive permutation group.

MSC classification

Secondary: 20B05: General theory for finite groups 20B15: Primitive groups 20B35: Subgroups of symmetric groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 2 , October 2001 , pp. 243 - 258
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Aschbacher, M. and Guralnick, R. M., ‘On abelian quotients of primitive groups’, Proc. Amer. Math. Soc. 107 (1989), 89–95.CrossRefGoogle Scholar
[2]Babai, L., ‘On the order of uniprimitive permutation groups’, Ann. of Math. (2) 113 (1981), 553–568.CrossRefGoogle Scholar
[3]Babai, L., ‘On the order of doubly transitive permutation groups’, Invent. Math. 65 (1982), 473–484.CrossRefGoogle Scholar
[4]Babai, L., Cameron, P. J. and Pálfy, P. P., ‘On the orders of primitive groups with restricted nonabelian composition factors’, J. Algebra 79 (1982), 161–168.CrossRefGoogle Scholar
[5]Babai, L., Kantor, W. M. and Luks, E. M., ‘Computational complexity and the classification of finite simple groups’, in: Proc. 24th IEEE FACS (1983) pp. 162–171.Google Scholar
[6]Bochert, A., ‘Uber die Zahl der verschiedenen Werthe, die eine Function gegebener Buchstaben durch Vertauschung derselben erlangen kann’, Math. Ann. 33 (1889), 584–590.CrossRefGoogle Scholar
[7]Cameron, P. J., ‘Finite permutation groups and finite simple groups’, Bull. London Math. Soc. 13 (1981), 1–22.CrossRefGoogle Scholar
[8]Cameron, P. J., ‘Some open problems on permutation groups’, in: Groups, combinatorics and geometry (eds. Liebeck, M. W. and Saxl, J.), London Math. Soc. Lecture Note Series 165 (Cambridge Univ. Press, Cambridge, 1992) pp. 340–350.CrossRefGoogle Scholar
[9]Cameron, P. J., ‘Permutation groups’, in: Handbook of combinatorics (eds. Graham, R., Grötschel, M. and Lovász, L.) (Elsevier Science B. V., 1995) chapter 12, pp. 611–645.Google Scholar
[10]Cameron, P. J. and Kantor, W. M., ‘Random permutations: some group-theoretic aspects’, Combinatorics, Probability and Computing 2 (1993), 257–262.CrossRefGoogle Scholar
[11]Cameron, P. J., Neumann, P. M. and Teague, D. N., ‘On the degrees of primitive permutation groups’, Math. Z. 180 (1982), 141–149.CrossRefGoogle Scholar
[12]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wison, R. A., An atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
[13]Dixon, J. D. and Mortimer, B., Permutation groups (Springer, New York, 1996).CrossRefGoogle Scholar
[14]Jordan, C., ‘Théorèmes sur les groupes primitifs’, J. Math. Pures Appl. 16 (1871), 383–408.Google Scholar
[15]Jordan, C., ‘Sur la limite de transitivité des groupes non alternés’, Bull. Soc. Math. France 1 (1873), 40–71.CrossRefGoogle Scholar
[16]Jordan, C., ‘Sur la limite du degré des groupes primitifs qui contiennent une substitution donnée’, J. Reine Angew. Math. 79 (1875), 248–258.Google Scholar
[17]Kantor, W. M., ‘Jordan groups’, J. Algebra 12 (1969), 471–493.CrossRefGoogle Scholar
[18]Kantor, W. M., ‘Homogeneous designs and geometric lattices’, J. Combin. Theory Ser. A 38 (1985), 66–74.CrossRefGoogle Scholar
[19]Knapp, W., Subkonstituenten und Struktur der Stabilisatoruntergruppe einer einfach transitiven Permutationsgruppe (Ph.D. Thesis, Eberhard-Karls-Universtät Tübingen, Tübingen, 1971).Google Scholar
[20]Knapp, W., ‘On the point stabilizer in a primitive permutation group’, Math. Z. 133 (1973), 137–168.CrossRefGoogle Scholar
[21]Liebeck, M. W., ‘On the minimal degrees and base sizes of primitive groups’, Arch. Math. 43 (1984), 11–15.CrossRefGoogle Scholar
[22]Liebeck, M. W. and Saxl, J., ‘The primitive permutation groups containing an element of large prime order’, J. London Math. Soc. (2) 31 (1985), 237–249.CrossRefGoogle Scholar
[23]Liebeck, M. W. and Shalev, A., ‘Simple groups, permutation groups, and probability’, J. Amer. Math. Soc. 12 (1999), 497–520.CrossRefGoogle Scholar
[24]Manning, W. A., ‘On the order of primitive groups, I–III’, Trans. Amer. Math. Soc. 10 (1909), 247–258: 16 (1915), 139–147: 19 (1918), 127–142.CrossRefGoogle Scholar
[25]Manning, W. A., ‘The degree and class of multiply transitive groups’, Trans. Amer. Math. Soc. 35 (1933), 585–599.Google Scholar
[26]Marggraf, B., Uber primitive Gruppen mit transitiven Untergruppen geringeren Grades (Dissertation, Giessen, 1892).Google Scholar
[27]Pálfy, P. P., ‘A polynomial bound for the orders of primitive soluble groups’, J. Algebra 77 (1982), 127–137.CrossRefGoogle Scholar
[28]Praeger, C. E., ‘On elements of prime order in primitive permutation groups’, J. Algebra 60 (1979), 126–157.CrossRefGoogle Scholar
[29]Praeger, C. E., ‘An O'Nan-Scott Theorem for finite quasiprimitive permutation groups, and an application to 2-arc transitive graphs’, J. London Math. Soc. (2) 47 (1993), 227–239.CrossRefGoogle Scholar
[30]Praeger, C. E. and Saxl, J., ‘On the orders of primitive permutation groups’, Bull. London Math. Soc. 12 (1980), 303–307.Google Scholar
[31]Praeger, C. E. and Shalev, A., ‘Indices of subgroups of finite simple groups and quasiprimitive permutation groups’, preprint, 2001.CrossRefGoogle Scholar
[32]Pyber, L., ‘Pálfy-Wolf type theorems for completely reducible subgroups of GL(n, pα)’, in preparation.Google Scholar
[33]Pyber, L., ‘Asymptotic results for permutation groups’, DIMACS Series in Discrete Math. and Computer Science 11 (1993), 197–219.CrossRefGoogle Scholar
[34]Pyber, L., ‘On the orders of doubly transitive permutation groups, elementary estimates’, J. Combin. Theory (A) 62 (1993), 361–366.CrossRefGoogle Scholar
[35]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar
[36]Wielandt, H., ‘Permutation groups through invariant relations and invariant functions’, Department of Mathematics, Ohio State University, Columbus, Ohio, 1969. (Reprinted in Mathematische Werke, Volume 1, de Gruyter, Berlin, 1994, pp. 237–296).Google Scholar
[37]Wolf, T. R., ‘Soluble and nilpotent subgroups of GL(n, qm)’, Canad. J. Math. 34 (1982), 1097–1111.CrossRefGoogle Scholar