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Extensions of a theorem of Jordan on primitive permutation groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Martin W. Liebeck
Martin W. Liebeck
Affiliation:
Department of Pure Mathematics University CollegeCardiff CF1 1XL, Wales, U. K.
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Abstract

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Let G be a primitive permutation group of finite degree n containing a subgroup H which fixes k points and has r orbits on Δ, the set of points it moves. An old and important theorem of Jordan says that if r = 1 and k ≥ 1 then G is 2-transitive; moreover if H acts primitively on Δ then G is (k + 1)-transitive. Three extensions of this result are proved here: (i) if r = 2 and k ≥ 2 then G is 2-transitive, (ii) if r = 2, n > 9 and H acts primitively on both of its two nontrivial orbits then G is k-primitive, (iii) if r = 3, n > 13 and H acts primitively on each of its three nontrivial orbits, all of which have size at least 3, then G is (k − 1)-primitive.

MSC classification

Secondary: 20B20: Multiply transitive finite groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 34 , Issue 2 , April 1983 , pp. 155 - 171
Copyright
Copyright © Australian Mathematical Society 1983

References

Antopolski, V. D. (1971), ‘A certain condition for double transitivity of a primitive permutation group’, Math. USSR-Sb. 14, 582–586.CrossRefGoogle Scholar
Atkinson, M. D. (1973), ‘Two theorems on doubly transitive permutation groups’, J. London Math. Soc. 6, 269–274.CrossRefGoogle Scholar
Atkinson, M. D. (1977), ‘On rank 3 groups having λ = 0’, Canad. J. Math. 29, 845–847.CrossRefGoogle Scholar
Higman, D. G. and Sims, C. C. (1968), ‘A simple group of order 44,352,000’, Math. Z. 105, 110–13.CrossRefGoogle Scholar
Jordan, C. (1971), ‘Théorèmes sur les groupes primitifs’, J. Math. Pures Appl. (Liouville) 16, 383–408.Google Scholar
Liebeck, M. W. (1977), Primitive permutation groups: a problem of Wielandt (M. Sc. Dissertation. Oxford).Google Scholar
Marggraff, B. (1892), Über primitive Gruppen mit transitiven Untergruppen geringeren Grades (Dissertation, Giessen).Google Scholar
Neumann, P. M. (1977), ‘Finite permutation groups, edge-coloured graphs and matrices’, Topics in group theory and computation, Proc. Summer School (Galway 1973), pp. 82–118, (London, Academic Press).Google Scholar
O'Nan, M. E. (1975), ‘Normal structure of the one-point stabiliser of a doubly transitive permutation group. II’, Trans. Amer. Math. Soc. 214, 43–74.CrossRefGoogle Scholar
Sims, C. C. (1970), ‘Computational methods in the study of permutation groups’, Computational problems in abstract algebra, Proc. Conf. (Oxford 1967), ed. Leech, J., pp. 169–183. (Pergamon Press, Oxford-New York).Google Scholar
Wielandt, H. (1971), Subnormal subgroups and permutation groups (Lecture notes, Ohio State University).Google Scholar
Wielandt, H. (1964), Finite permutation groups (Academic Press, London-New York).Google Scholar