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ON TRANSITIVE PERMUTATION GROUPS WITH PRIMITIVE SUBCONSTITUENTS

Published online by Cambridge University Press:  01 May 1999

DMITRII V. PASECHNIK  and
CHERYL E. PRAEGER
DMITRII V. PASECHNIK
Affiliation:
SSOR, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
CHERYL E. PRAEGER
Affiliation:
Department of Mathematics, University of Western Australia, Perth, WA 6907, Australia
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Abstract

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω\{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 31 , Issue 3 , May 1999 , pp. 257 - 268
Copyright
© The London Mathematical Society 1999

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