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On the normal structure of a one-point stabilizer in a doubly transitive permutation group

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

M. D. Atkinson  and
Cheryl E. Praeger
M. D. Atkinson
Affiliation:
Department of Computing Mathematics University College Cardiff, Wales
Cheryl E. Praeger
Affiliation:
Department of Mathematics University of Western AustraliaNedlands, Australia6009
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Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

MSC classification

Secondary: 20B20: Multiply transitive finite groups 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 2 , March 1978 , pp. 145 - 166
Copyright
Copyright © Australian Mathematical Society 1978

References

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