Abstract
Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph \(\Gamma\) admitting a group of automorphisms \(G\) which is isomorphic to Aut\((Fi_{22})\) and has the following properties. First, the group \(G\) acts transitively on the set of vertices of \(\Gamma\), but intransitively on the set of \(3\)-arcs of \(\Gamma\). Second, the stabilizer in \(G\) of a vertex of \(\Gamma\) induces on the neighborhood of this vertex a group \(PSL_{3}(3)\) in its natural doubly transitive action. Third, the pointwise stabilizer in \(G\) of a ball of radius 2 in \(\Gamma\) is nontrivial. In this paper, we construct such a graph \(\Gamma\) with \(G=\mathrm{Aut}(\Gamma)\).
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Notes
In paper [1], there should be \(\mathbf{F}_{3}\) instead of \(\mathbf{F}_{2}\) in statement (b) of Section 5; in addition, there should be [20, I] instead of [20, II] on page 317, line 5 from the bottom.
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Funding
This work was performed as a part of the research conducted at the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. 075-02-2023-935).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 29, No. 4, pp. 274 - 278, 2023 https://doi.org/10.21538/0134-4889-2023-29-4-274-278.
Translated by E. Vasil’eva
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Trofimov, V.I. A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut(\(Fi_{22}\)) Which Has a Nontrivial Stabilizer of a Ball of Radius \(2\). Proc. Steklov Inst. Math. 323 (Suppl 1), S300–S304 (2023). https://doi.org/10.1134/S0081543823060238
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DOI: https://doi.org/10.1134/S0081543823060238