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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
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

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