A translation plane of order \(7^ 2\) with a very small translation complement
type: Item
instance of: scholarly article
title: A translation plane of order \(7^ 2\) with a very small translation complement
zbMATH Open document ID: 0786.51003
DOI: 10.1007/BF02988314
author: M. L. Narayana Rao
author: K. Kuppu Swamy Rao
author: Vinod Joshi
published in: Graphs and Combinatorics
publication date: 1993-10-28T00:00:00Z
review text: The authors construct a translation plane of order 49\(=7^ 2\) whose translation complement \({\mathcal C}\), the group of collineations fixing the origin (0,0), has order 72 and its factor group modulo the kernel \(K\) is the dihedral group of order 12. The construction is accomplished by exhibiting a set \(S\) of forty-nine 2 by 2 matrices over \(GF(7)\) forming a matrix spread. All calculations are performed on the set \(S\). The authors claim that the plane obtained has the smallest translation complement among all known planes of order 49. No proof is given of this statement, and it may well have been true in 1986 when the article was originally submitted. Since then however several additional translation planes of order 49 have been found.
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 436562
zbMATH Keywords: translation plane of order 49
zbMATH Keywords: smallest translation complement
reviewed by: Michael J. Kallaher
MaRDI profile type: Publication
cites work: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe
cites work: Q5545191
cites work: A new class of square order planes
cites work: Q3227201
full work available at URL: https://doi.org/10.1007/bf02988314
OpenAlex ID: W1978396951
description: scientific article; zbMATH DE number 436562