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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Volume 25, Number 4, Pages 88–98
DOI: https://doi.org/10.21538/0134-4889-2019-25-4-88-98
(Mi timm1673)
 

This article is cited in 1 scientific paper (total in 1 paper)

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

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Full-text PDF (231 kB) Citations (1)
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Abstract: 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: 19.09.2019
Revised: 18.11.2019
Accepted: 25.11.2019
Bibliographic databases:
Document Type: Article
UDC: 512.542
MSC: 20B15, 20D06
Language: Russian
Citation: 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 Inst. Mat. i Mekh. UrO RAN, 25, no. 4, 2019, 88–98
Citation in format AMSBIB
\Bibitem{Kon19}
\by A.~V.~Konygin
\paper 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 $E_8(q)$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 4
\pages 88--98
\mathnet{http://mi.mathnet.ru/timm1673}
\crossref{https://doi.org/10.21538/0134-4889-2019-25-4-88-98}
\elib{https://elibrary.ru/item.asp?id=41455524}
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  • https://www.mathnet.ru/eng/timm/v25/i4/p88
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Trudy Instituta Matematiki i Mekhaniki UrO RAN
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