Combinatorial geometries representable over GF(3) and GF(q). II: Dowling geometries
type: Item
instance of: scholarly article
title: Combinatorial geometries representable over GF(3) and GF(q). II: Dowling geometries
zbMATH Open document ID: 0702.51004
DOI: 10.1007/BF01864171
author: Joseph P. S. Kung
author: James Oxley
published in: Graphs and Combinatorics
publication date: 1988-01-01T00:00:00Z
review text: [For part I see the first author, Discrete Comput. Geom. 5, No.1, 83-95 (1990; Zbl 0697.51007).] Let \(\{v_ 1,v_ 2,...,v_ n\}\) be a basis of an n-dimensional vector space of GF(3). The ternary Dowling geometry \(Q_ n(GF(3)^*)\) \((Q_ n\) for short) is the ternary geometry of rank n consisting of the points \(v_ 1,v_ 2,...,v_ n\) and the points \(v_ i-v_ j\) and \(v_ i+v_ j\), where \(i<j\). The Dowling geometry \(Q_ n\) contains \(n^ 2\) points and is representable over any field whose multiplicative group contains the cyclic group of order 2 as a subgroup. In the present paper the authors show that, with the exception of \(n=3\), a rank n geometry that is representable over GF(3) and GF(q) and contains exactly \(n^ 2\) points is isomorphic to the rank n Dowling geometry based on the multiplicative group of GF(3).
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 4151236
zbMATH Keywords: combinatorial geometries
zbMATH Keywords: ternary geometry
zbMATH Keywords: rank n geometry
zbMATH Keywords: rank n Dowling geometry
reviewed by: Manfred Stern
MaRDI profile type: Publication
cites work: Q5675735
cites work: A class of geometric lattices based on finite groups
cites work: On linear systems with integral valued solutions
cites work: Dimension and embedding theorems for geometric lattices
cites work: Combinatorial geometries representable over GF(3) and GF(q). I: The number of points
cites work: Matroid representation over GF(3)
cites work: Decomposition of regular matroids
cites work: Q3249424
cites work: Q4111952
cites work: Theory of Matroids
cites work: Signed graphs
full work available at URL: https://doi.org/10.1007/bf01864171
OpenAlex ID: W2064482790
description: scientific article; zbMATH DE number 4151236