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 $$v=6$$ .