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

Published online by Cambridge University Press:  24 August 2009

V. I. TROFIMOV  and
R. M. WEISS
V. I. TROFIMOV
Affiliation:
Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ural Branch, 620219 Ekaterinburg, Russia. e-mail: trofimov@imm.uran.ru
R. M. WEISS
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, U.S.A. e-mail: richard.weiss@tufts.edu
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Abstract

Let 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 1 - 32
Copyright
Copyright © Cambridge Philosophical Society 2009

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