Let $\Gamma$ be a connected finite graph and $G$ a vertex-transitive group of automorphisms of $\Gamma$ such that the stabilizer $G_x$ in $G$ of a vertex $x$ of $\Gamma$ induces on the neighborhood $\Gamma(x)$ of $x$ a primitive permutation group $G_x^{\Gamma(x)}$. The Weiss conjecture says that, under this assumption, the order of $G_x$ is bounded from above by a number depending only on the degree $|\Gamma(x)|$ of $\Gamma$. In the work whose first part is the present paper we show that some results of the theory of finite groups can be used to provide unified considerations of a number of cases of the Weiss conjecture (including a number of cases not considered before). Although this first part is introductory, it makes possible to use certain previous results to confirm the Weiss conjecture for all primitive groups $G_x^{\Gamma(x)}$ different from groups of AS type and from groups of PA type (constructed on the basis of groups of AS type).
Keywords: graph, group of automorphisms, Weiss conjecture
Received October 29, 2021
Revised November 19, 2021
Accepted December 13, 2021
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. № 20-01-00456).
Trofimov Vladimir Ivanovich, Dr. Phys.-Math. Sci., Leading researcher, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Prof., Ural Federal University, Yekaterinburg, 620000 Russia,
e-mail: trofimov@imm.uran.ru
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Cite this article as: V.I. Trofimov. On the Weiss Conjecture. I, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 247–256; Proc. Steklov Inst. Math. (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S281–S290.
