Finite Symmetric Graphs with 2-Arc-Transitive Quotients: General Affine Case

Abstract

Let G be a finite group and $\Gamma$ a G-symmetric graph. Suppose that G is imprimitive on $V(\Gamma )$ with B a block of imprimitivity and $\mathcal {B} := \{B^g: g\in G\}$ is a system of imprimitivity of G on $V(\Gamma )$. Define $\Gamma _{\mathcal {B}}$ to be the graph with vertex set $\mathcal {B}$, such that two blocks $B, C \in \mathcal {B}$ are adjacent if and only if there exists at least one edge of $\Gamma$ joining a vertex in B and a vertex in C. Set $v=|B|$ and $k := |\Gamma (C)\cap B|$ where C is adjacent to B in $\Gamma _{\mathcal {B}}$ and $\Gamma (C)$ denotes the set of vertices of $\Gamma$ adjacent to at least one vertex in C. Assume that $k=v-p\ge 1$, where p is an odd prime, and $\Gamma _{\mathcal {B}}$ is (G, 2)-arc-transitive. In this paper , we show that if the group induced on each block is an affine group then $v=6$.