The group E6(q) and graphs with a locally linear group of automorphisms

Abstract

AbstractLet q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. It is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. (These special situations are eliminated in a companion paper by the first author.) An example coming from the exceptional group E6(q) shows that the kernel of the action of Gx on the ball of radius 2 around x can be non-trivial.