具有半径为 $$2$$ 的球的非小稳定子的局部投影顶点-传递自整定群 Aut( $$Fi_{22}$$ ) 的图形

Pub Date : 2024-02-12 DOI:10.1134/s0081543823060238

V. I. Trofimov

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引用次数: 0

摘要

早些时候，为了证实具有投影子边的图的顶点稳定器结构的可能性之一是可实现的，我们宣布存在一个连通图 $\Gamma$ ，它容许一个自变量群 $G$，该群与 Aut$(Fi_{22})$ 同构，并具有以下性质。首先，群 $G$ 在 $\Gamma$ 的顶点集合上起传递作用，但是在 $\Gamma$ 的 $3$-arcs 集合上起非传递作用。第二， $\Gamma$ 的一个顶点在 $G$ 中的稳定器在这个顶点的邻域上引起了一个群 $PSL_{3}(3)$ 的自然双传递作用。第三，在 $\Gamma$ 中半径为 2 的球在$G$ 中的点稳定器是非微观的。在本文中，我们构建了这样一个图（G=\mathrm{Aut}(\Gamma)\）。

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A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut( $$Fi_{22}$$ ) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

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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