A.V. Konygin. On primitive permutation groups with the stabilizer of two points normal in the stabilizer of one of them: The case when the socle is a power of a group $E_8(q)$ ... P. 88-98

Assume that $G$ is a primitive permutation group on a finite set $X$, $x\in X\setminus\{x\}$, and $G_{x, y}\trianglelefteq G_x$. P. Cameron raised the question about the validity of the equality $G_{x, y} = 1$ in this case. The author proved earlier that, if the socle of $G$ is not a power of a group isomorphic to $E_8(q)$ for a prime power $q$, then $G_{x, y}=1$. In the present paper, we consider the case where the socle of $G$ is a power of a group isomorphic to $E_8(q)$. Together with the previous result, we establish the following two statements. 1. Let $G$ be an almost simple primitive permutation group on a finite set $X$. Assume that, if the socle of $G$ is isomorphic to $E_8(q)$, then $G_x$ for $x \in X$ is not the Borovik subgroup of $G$. Then the answer to Cameron's question for such primitive permutation groups is affirmative. 2. Let $G$ be a primitive permutation group on a finite set $X$ with the property $G\leq H\mathrm{ wr } S_m$. Assume that, if the socle of $H$ is isomorphic to $E_8(q)$, then the stabilizer of a point in the group $H$ is not the Borovik subgroup of $H$. Then the answer to Cameron's question for such primitive permutation groups is also affirmative.

Keywords: primitive permutation group, regular suborbit

Received September 19, 2019

Revised November 18, 2019

Accepted November 25, 2019

Anton Vladimirovich Konygin, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia e-mail: konygin@imm.uran.ru

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Cite this article as: A.V.Konygin. On primitive permutation groups with the stabilizer of two points normal in the stabilizer of one of them: The case when the socle is a power of a group $E_8(q)$, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 88–98.