Vertex stabilizers of locally s-arc transitive graphs of pushing up type

期刊论文
出版模式：

开放获取
发表日期：

2024年5月8日

查看全文

John van Bon,
Chris Parker

摘要

Suppose that $\Delta $ $Δ$ is a thick, locally finite and locally s-arc transitive G-graph with $s \ge 4$ $s \geq 4$ . For a vertex z in $\Delta $ $Δ$ , let $G_z$ $G_{z}$ be the stabilizer of z and $G_z^{[1]}$ $G_{z}^{[1]}$ the kernel of the action of $G_z$ $G_{z}$ on the neighbours of z. We say $\Delta $ $Δ$ is of pushing up type provided there exist a prime p and a 1-arc (x, y) such that $C_{G_z}(O_p(G_z^{[1]})) \le O_p(G_z^{[1]})$ $C_{G_{z}} (O_{p} (G_{z}^{[1]})) \leq O_{p} (G_{z}^{[1]})$ for $z \in \{x,y\}$ $z \in {x, y}$ and $O_p(G_x^{[1]}) \le O_p(G_y^{[1]})$ $O_{p} (G_{x}^{[1]}) \leq O_{p} (G_{y}^{[1]})$ . We show that if $\Delta $ $Δ$ is of pushing up type, then $O_p(G_x^{[1]})$ $O_{p} (G_{x}^{[1]})$ is elementary abelian and $G_x/G_x^{[1]}\cong X$ $G_{x} / G_{x}^{[1]} ≅ X$ with $ \textrm{PSL}_2(p^a)\le X \le \mathrm{P\Gamma L}_2(p^a)$ ${PSL}_{2} (p^{a}) \leq X \leq {P Γ L}_{2} (p^{a})$ .