Affine distance-transitive groups of dimension one
\(V\) is an \(n\)-dimensional vector space over \(\text{F}_q\) and \(G\) is a subgroup in \(\text{A}\Gamma\text{L}(v)\cong \text{A}\Gamma\text{L}(n, q)\) containing all translations with \(G_0\subseteq G\) the stabilizer of the zero vector. Hypothesis DTG says that \(\Gamma\) is a graph structure on \(V\) of diameter \(d\) such that \(G\) acts primitively and distance-transitively on \(\Gamma\). By \textit{J. van Bon} [Affine distance-transitive groups, Proc. Lond. Math. Soc., III. Ser. 67, No. 1, 1-52 (1993; Zbl 0795.05074)] this hypothesis implies at least one of three cases. The authors describe the case \(n=1\) for \(d\geq 3\). (The cases \(d= 1, 2\) are in the literature.) The conclusion reads: \(q= 64\), \(G_0\cong Z_9: Z_3\) or \(Z_9: Z_6\), and \(\Gamma\) is the Hamming graph \(H(4,3)\) of diameter \(3\).
scientific article; zbMATH DE number 1423287