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Groups with the property of the title were considered by Chillag (1977); this paper completes his results by showing that, with known exceptions, they are triply transitive.

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.

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On the fixed points of Sylow subgroups of transitive permutation groups: corrigendum

On doubly transitive permutation groups of degree prime squared plus one

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

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143 results in 20Bxx

On the fixed points of Sylow subgroups of transitive permutation groups: corrigendum

On doubly transitive permutation groups of degree prime squared plus one

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