FINITE SYMMETRIC GRAPHS WITH 2-ARC TRANSITIVE QUOTIENTS

Let G be a finite group and Γ a G-Symmetric graph. Suppose that G is imprimitive on V (Γ) with B a block of imprimitivity and B := {B g ; g ∈ G} is a system of imprimitivity of G on V (Γ). Dfine ΓB to be the graph with vertex set B such that two blocks B, C ∈ B are adjacent if and only if there exists at least one edge of Γ joining a vertex in B and a vertex in C. Let U := ΓB(B) be the set of blocks of B adjacent to B in ΓB, v := |B| and k := |Γ(C) ∩ B| for C ∈ U. Assume that k = v − p ≥ 1 and ΓB is connected with valency b ≥ 2 where p is an odd prime. Suppose that ΓB is (G, 2)-arc-transitive, p = 2 n −1 is a Mersenne prime and v = 2 m p is a multiple of p, where n − 1 ≥ m ≥ 1 is an integer. Denote by G B the setwise stabiliser of B in G, and define H := G ΓB(B) B to be the quotient group of G B relative to the kernel of the induced action of G B on U. In this paper we show that if H is isomorphic to a 2-transitive subgroup of AGL(n, 2), then p = 3 = k and v = 6. In fact we show that there is a unique graph satisfying the conditions in the third row of table 2 in [2].