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 $E_8(q)$

A. V. Konygin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 88-98

Résumé

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.

Détail

Mots-clés : primitive permutation group
Keywords: regular suborbit.