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On primitive permutation groups with a stabilizer of two points that is normal in the stabilizer of one of them: Case when the socle is a power of a sporadic simple group

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Abstract

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G x,y \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } \) G x . P. Cameron raised the question about the validity of the equality G x,y = 1 in this case. If the group G is of type I, type III(a), or type III(c) (according to the O’Nan-Scott classification) or G is of type II and soc(G) is not an exceptional group of Lie type, then it is proved that G x,y = 1. In addition, if the group G is of type III(b) and soc(G) is not a direct product of exceptional groups of Lie type, then it is proved that G x,y = 1.

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  1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    A. V. Konygin

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  1. A. V. Konygin
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Correspondence to A. V. Konygin.

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Original Russian Text © A.V. Konygin, 2010, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2010, Vol. 16, No. 3.

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Konygin, A.V. On primitive permutation groups with a stabilizer of two points that is normal in the stabilizer of one of them: Case when the socle is a power of a sporadic simple group. Proc. Steklov Inst. Math. 272 (Suppl 1), 65–73 (2011). https://doi.org/10.1134/S0081543811020064

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