Vertex stabilizers of locally \(s\)-arc transitive graphs of pushing up type (Q6604497)

This is another step in a continuing series of papers; see, for example, [\textit{A. Delgado} et al. [Groups and graphs: new results and methods. Basel-Boston-Stuttgart: Birkhäuser Verlag (1985; Zbl 0566.20013); \textit{J. van Bon} and \textit{B. Stellmacher}, J. Algebra 441, 243--293 (2015; Zbl 1328.05092)]. Let \(\Delta \) be a connected, undirected graph without loops or multiple edges and \(G\) be a subgroup of \(\Aut(G)\); the pair \((\Delta ,G)\) is called a \(G\)-graph. We denote the vertex and edge sets of \(\Delta \) by \(V\Delta \) and \(E\Delta \) and define an \(s\)-arc from \( x_{0}\in V\Delta \) to be a path \((x_{0},x_{1},\dots,x_{s})\) where \(x_{i-1}\neq x_{i+1}\) for each \(i\). For integers \(s\geq 1\), the present paper considers \(G\)-graphs with the following properties: (i) (thick) the valence of each vertex is \(\geq 3\); (ii) (locally finite) for each \(z\in V\Delta \) the stabilizer \(G_{z}\) of \(z\) is finite; and (iii) (locally \(s\)-arc transitive) \( G_{z}\) acts transitively on the \(s\)-arcs from \(z\). Let \(G_{z}^{[1]}\) be the kernel of the action of \(G_{z}\) on the set \(\Delta (z)\) of all vertices in \(\Delta \) adjacent to \(z\). Then the \(G\)-graph \(\Delta \) is said to be of pushing-up type with respect to a \(1\)-arc \((x,y)\) and a prime \(p\) if \(C_{G_{z}}(O_{p}(G_{z}^{[1]}))\leq \) \(O_{p}(G_{z}^{[1]})\) for all \(z\in V\Delta \) and \(O_{p}(G_{x}^{[1]})\leq O_{p}(G_{y}^{[1]})\). In this case, we write \(Q_{z}:=O_{p}(G_{z}^{[1]})\) and \(L_{z}:=\langle Q_{u}\mid u\in \Delta (z)\rangle Q_{z}\) for each \(z\in V\Delta \).\N\NThe main theorem of the paper is the following. Let \(s\geq 4\) and suppose that \((\Delta ,G)\) is a \(G\)-graph satisfying (i)-(iii) which is of the pushing-up type with respect to a \(1\)-arc \((x,y)\) and a prime \(p\). Then \(p\) is odd and the following hold: (a) \(G_{x}/G_{x}^{[1]}\cong X\) where \(\mathrm{PSL}_{2}(p^{a})\leq X\leq \mathrm{P}\Gamma\mathrm{L}_{2}(p^{a})\) for some integer \(a\) and \( \Delta (x)\ \)can be identified with the projective line of size \(p^{a}+1\); (b) \(L_{x}/Q_{x}\cong \) \(\mathrm{SL}_{2}(p^{a})\), \(O_{p}(L_{x})\cong \mathrm{ASL}_{2}(p^{a})^{\prime }\) and \(Q_{x}\) is an elementary abelian \(p\)-group.