AFFINE DISTANCE-TRANSITIVE GRAPHS AND EXCEPTIONAL CHEVALLEY GROUPS

Abstract

Suppose $r=p^b$, where $p$ is a prime. Let $V$ be an $n$-dimensional ${\rm GF}(r)$-space and $G$ a subgroup of ${\rm A\Gamma L}(V) \cong {\rm A\Gamma L}(n,r)$ containing all translations and acting primitively on the set of vectors in $V$. Denote by $G_0$ the stabilizer in $G$ of the zero vector, so that $G_0 \le {\rm \Gamma L}(V) \cong {\rm \Gamma L}(n,r)$ and $G$ is the semidirect product of $V$ and $G_0$. Suppose that the generalized Fitting subgroup $F^*(G_0)$ of $G_0$ is an exceptional (twisted or untwisted, quasisimple) Chevalley group and that $\Gamma$ is a graph structure on $V$ on which $G$ acts primitively and distance transitively. The content of this paper is that then $G$ and $\Gamma$ are known. This result solves an open case in the outstanding problem of classifying all finite primitive distance-transitive groups. 2000 Mathematics Subject Classification: primary 20B25; secondary 05C25, 20Gxx, 05E30.