Affine Distance‐Transitive Graphs and Exceptional Chevalley Groups

Suppose r = p b , where p is a prime. Let V be an n ‐dimensional GF( r )‐space and G a subgroup of AΓL( V ) ≅ AΓL( 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 ⩽ Γ L( V ) ≅ Γ 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 Γ is a graph structure on V on which G acts primitively and distance transitively. The content of this paper is that then G and Γ 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.