Affine distance-transitive groups of dimension one
type: Item
instance of: scholarly article
title: Affine distance-transitive groups of dimension one
zbMATH Open document ID: 0951.05050
author: Arjeh M. Cohen
author: Alexander A. Ivanov
published in: European Journal of Combinatorics
publication date: 2000-12-13T00:00:00Z
review text: \(V\) 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 \(3\).
reviewed by: Steven C. Althoen
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1423287
zbMATH Keywords: vector space
zbMATH Keywords: stabilizer
zbMATH Keywords: Hamming graph
MaRDI profile type: Publication
full work available at URL: eujc.1999.0277
OpenAlex ID: W2047525956
cites work: How many P-polynomial structures can an association scheme have?
cites work: Affine Distance-Transitive Groups
cites work: Q4505247
cites work: Q3992965
cites work: Affine distance-transitive graphs: The cross characteristic case
cites work: Solvable, flag-transitive, rank 3 collineation groups
cites work: The Affine Permutation Groups of Rank Three
cites work: Affine distance-transitive groups with alternating or symmetric point stabiliser
cites work: Equations over finite fields. An elementary approach
DOI: 10.1006/EUJC.1999.0277
description: scientific article; zbMATH DE number 1423287