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

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 B: = {B-g : g is an element of G} is a system of imprimitivity of G on v(Gamma). Define Gamma(B) to be the graph with vertex set B, such that two blocks B, C is an element of B are adjacent if and only if there exists at least one edge of joining a vertex in B and a vertex in C. Set nu = vertical bar B vertical bar and k: = vertical bar Gamma(C) boolean AND B vertical bar where C is adjacent to B in Gamma(B) and Gamma(c) denotes the set of vertices of Gamma adjacent to at least one vertex in C. Assume that k = v - p >= 1, where p is an odd prime, and Gamma(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.