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On transitive permutation groups with a subgroup satisfying a certain conjugacy condition

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of Mathematics University of Western AustraliaNedlands, W. A. 6009, Australia
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Abstract

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Let G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

MSC classification

Secondary: 20B05: General theory for finite groups 20B10: Characterization theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 69 - 86
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Aschbacher, M., ‘The nonexistance of rank three permutation groups of degree 3250 and subdegree 57’, J. Algebra 19 (1971), 538–540.CrossRefGoogle Scholar
[2]Bannai, E., ‘Doubly transitive permutation representations of the finite projective special linear groups PSL (n, q)’, Osako J. Math. 8 (1971), 437–445.Google Scholar
[3]Burnside, W., Theory of groups of finite order (Cambridge University Press, London, 1911, reprinted Dover, New York, 1955).Google Scholar
[4]Cameron, P. J., ‘Permutation groups with multiply transitive suborbits’, Proc. London Math. Soc. (3) 25 (1972), 427–440.CrossRefGoogle Scholar
[5]Cameron, P. J., ‘Primitive groups with most suborbits doubly transitive’, Geometricae Dedicata 1 (1973), 434–446.Google Scholar
[6]Dembowski, P., Finite geometries (Ergebnisse der Mathematik 44, Springer-Verlag, Berlin-Heidelberg-New York, 1968).CrossRefGoogle Scholar
[7]Higman, D. G., ‘Finite permutation groups of rank 3’, Math. Z. 86 (1964), 145–156.CrossRefGoogle Scholar
[8]Ito, N., ‘Über die Gruppenn PSLn(q), die eine Untergruppe von Primzahlindex enthalthen’, Acta Sci. Math. (Szeged) 21 (1960), 206–217.Google Scholar
[9]Kantor, W. M., ‘2-Transitive designs’, Combinatorics, Part 3, Combinatortial group theory, ed. by Hall, M. Jr, and. H. van Lint (Math. Centre Tracts 57, Amsterdam, 1974, 44–97).Google Scholar
[10]Maillet, E., ‘Sur les isomorphes holoédriques et transitifs des groups symétriques ou alternés’, J. Math. Pures Appl. Ser. (5) 1 (1895), 5–34.Google Scholar
[11]O'Nan, M., “A characterisation of Ln(q) as a permutation group”, Math. Z. 127 (1972), 301–314.CrossRefGoogle Scholar
[12]O'Nan, M., “Normal structure of the one-point stabilizer of a doubly transitive permutation group II’, Trans. Amer. Math. Soc. 214 (1975), 43–74.CrossRefGoogle Scholar
[13]O'Nan, M., ‘Estimation of Sylow subgroups in primitive permutation groups’, Math. Z. 147 (1976), 101–111.CrossRefGoogle Scholar
[14]Praeger, C. E., ‘Doubly transitive permutation group which are not doubly primitive’, J. Algebra 44 (1977), 389–395.CrossRefGoogle Scholar
[15]Prarger, C. E., ‘Doubly transitive automorphism groups of block designs’, J. Combinatorial Theory Ser. A 25 (1978), 258–266.CrossRefGoogle Scholar
[16]Praeger, C. E., ‘Sylow subgroups of transitive permutation groups’, Math. Z. 134 (1973), 179–180.CrossRefGoogle Scholar
[17]Praeger, C. E., ‘On primitive permutation groups with a doubly transitive suborbit’, J. London Math. Soc. (2) 17 (1978), 67–73.CrossRefGoogle Scholar
[18]Sims, C. C., ‘Computational methods on the study of permutation groups’, Computational problems in abstract algebra (Oxford, 1967), ed., by Leech, J. (Pergamon Press, Oxford-London-Edinburgh, 1970, 164–184).Google Scholar
[19]Weiss, M. J., ‘On simply transitive groups’, Bull. Amer. Math. Soc. 30 (1928), 333–359.Google Scholar
[20]Wielandt, H., Finite permutation groups (Academic Press, New York-London, 1964).Google Scholar
[21]Wong, W. J., ‘Determination of a class of primitive permutation groups’, Math. Z. 99 (1967), 235–246.CrossRefGoogle Scholar