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On the Sylow subgroups of a doubly transitive permutation group III

Published online by Cambridge University Press:  17 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Let G be a 2-transitive permutation group of a set Ω of n points and let P be a Sylow p-subgroup of G where p is a prime dividing |G|. If we restrict the lengths of the orbits of P, can we correspondingly restrict the order of P? In the previous two papers of this series we were concerned with the case in which all P–orbits have length at most p; in the second paper we looked at Sylow p–subgroups of a two point stabiliser. We showed that either P had order p, or G ≥ An, G = PSL(2, 5) with p = 2, or G = M11 of degree 12 with p = 3. In this paper we assume that P has a subgroup Q of index p and all orbits of Q have length at most p. We conclude that either P has order at most p2, or the groups are known; namely PSL(3, p) ≤ G ≤ PGL(3, p), ASL(2, p) ≤ G ≤ AGL(2, p), G = PΓL,(2, 8) with p = 3, G = M12 with p = 3, G = PGL(2, 5) with p = 2, or G ≥ An with 3p ≤ n < 2p2; all in their natural representations.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 2 , October 1975 , pp. 211 - 240
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Bochert, Alfred, “Ueber die Classe der transitiven Substitutionengruppen”, Math. Ann. 40 (1892), 176–193.CrossRefGoogle Scholar
[2]Burnside, W., Theory of groups of finite order, 2nd ed. (Cambridge University Press, Cambridge, 1911; reprinted Dover, New York, 1955).Google Scholar
[3]Gorenstein, Daniel, Finite groups (Harper and Row, New York, Evanston, London, 1968).Google Scholar
[4]Greenberg, Philip J., Mathieu Groups (Lecture Notes, Courant Institute of Mathematical Sciences, New York University, New York, 1973).Google Scholar
[5]Itô, Noboru, “Über die Gruppen PSLn (q) die eine Untergruppe von Primzahlindex enthalten”, Acta Soc. Math. 21 (1960), 206–217.Google Scholar
[6]Kantor, William M., “Jordan groups”, J. Algebra 12 (1969), 471–493.CrossRefGoogle Scholar
[7]Kantor, William M., “Primitive groups having transitive subgroups of smaller, prime power degree”, Israel J. Math, (to appear).Google Scholar
[8]Praeger, Cheryl E., “Sylow subgroups of transitive permutation groups”, Math. Z. 134 (1973), 179–180.CrossRefGoogle Scholar
[9]Praeger, Cheryl E., “On the Sylow subgroups of a doubly transitive permutation group”, Math. Z. 137 (1974), 155–171.CrossRefGoogle Scholar
[10]Praeger, Cheryl E., “On the Sylow subgroups of a doubly transitive permutation group II”, Math. Z. (to appear).Google Scholar
[11]Praeger, Cheryl E., “Primitive permutation groups containing a p–element of small degree, p a prime”, J. Algebra 34 (1975), 540–546.CrossRefGoogle Scholar
[12]de Séguier, J.-A., Théorie des groupes finis (Gauthier-Villars, Paris, 1912).Google Scholar
[13]Sims, Charles C., “Computational methods in the study of permutation groups”, Computational problems in abstract algebra 169–183 (Proc. Conf. Oxford, 1967. Pergamon, Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1970).Google Scholar
[14]Wielandt, Helmut, Finite permutation groups (translated by Bercov, R.. Academic Press, New York, London, 1964).Google Scholar
[15]Witt, Ernst, “Die 5-fach transitiven Gruppen von Mathieu”, Abh. Math. Sem. Univ. Hamburg 12 (1938), 256–264.CrossRefGoogle Scholar