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On the fixed points of Sylow subgroups of transitive permutation groups

Published online by Cambridge University Press:  09 April 2009

Marcel Herzog  and
Cheryl E. Praeger
Marcel Herzog
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, Australia2600.
Cheryl E. Praeger
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, Australia2600.
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Abstract

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Let G be a transitive permutation group on a set Ω of n points, and let P be a Sylow p-subgroup of G for some prime p dividing ∣G∣. If P has t long orbits and f fixed points in Ω, then it is shown that f ≦ tp − ip(n), where ip(n) = p – rp(n), rp(n) denoting the residue of n modulo p. In addition, groups for which f attains the upper bound are classified.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 4 , June 1976 , pp. 428 - 437
Copyright
Copyright © Australian Mathematical Society 1976

References

Hall, M. Jr (1960), The Theory of Groups, (New York: Pergamon 1960).Google Scholar
Ito, N. (1960), ‘Über die Gruppen PSLn(q) die eine Untergruppe von Primzahlindex enthalten’, Acta Sci. Math. 21, 206–217.Google Scholar
Kantor, W. M. ‘Primitive groups having transitive subgroups of smaller, prime power degree’, (to appear).Google Scholar
Manning, W. A. (1909), ‘On the order of primitive groups’, Trans. Amer. Math. Soc. 10, 247–258.CrossRefGoogle Scholar
Manning, W. A. (1911), ‘On the limit of the degree of primitive groups’, Trans. Amer. Math. Soc. 12, 375–386.CrossRefGoogle Scholar
O'Nan, M. E., ‘Normal structure of the one-point stabiliser of a doubly transitive permutation group, II’, (to appear).Google Scholar
Praeger, C. E. (1973), ‘Sylow subgroups of transitive permutation groups’, Math. Z. 134, 179–180.CrossRefGoogle Scholar
Praeger, C. E. (1974), ‘On the Sylow subgroups of a doubly transitive permutation group’, Math. Z. 137, 155–171.CrossRefGoogle Scholar
Praeger, C. E., ‘Primitive permutation groups containing a p-element of small degree, p a prime’, (to appear), J. Algebra.Google Scholar
de Séguier, J. A. (1912), Théorie des Groupes finis, (Paris: Gauthier-Villars).Google Scholar
Weiss, M. J. (1928), ‘Primitive groups which contain substitutions of prime order p and of degree 6p or 7p’, Trans. Amer. Math. Soc. 30, 333–359.Google Scholar
Wielandt, H. (1964), Finite Permutation Groups (New Yorl-London: Academic Press 1964).Google Scholar