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References (5)
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I. Isaacs (1990)
Subgroups close to all of their conjugatesArchiv der Mathematik, 55
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I. Isaacs, C. Praeger (1993)
Permutation Group Subdegrees and the Common Divisor GraphJournal of Algebra, 159
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G. Bergman, H. Lenstra (1989)
Subgroups close to normal subgroupsJournal of Algebra, 127
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G. Schlichting (1980)
Operationen mit periodischen StabilisatorenArchiv der Mathematik, 34
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V. Trofimov (1985)
GRAPHS WITH POLYNOMIAL GROWTHMathematics of The Ussr-sbornik, 51
- Publisher
- Wiley
- Copyright
- © London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/S0024609398005669
- Publisher site
- See Article on Publisher Site
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. 1991 Mathematics Subject Classification 20B07, 20B05.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: May 1, 1999