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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

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