New necessary conditions on the existence of Abelian difference sets
type: Item
instance of: scholarly article
title: New necessary conditions on the existence of Abelian difference sets
zbMATH Open document ID: 0765.05021
DOI: 10.1007/BF01191207
author: Alexander Pott
published in: Combinatorica
publication date: 1992-09-27T00:00:00Z
review text: It is still an unsolved conjecture that the order \(n\) of a cyclic (or abelian) finite projective plane (i.e. a plane with a cyclic or abelian automorphism group \(G\) acting sharply transitively on the points) has to be a prime power. \textit{H. A. Wilbrink} [J. Comb. Theory, Ser. A 38, 94-95 (1985; Zbl 0554.05012)] proved an interesting algebraic equation for the corresponding difference set \(D\) in \(\text{GF}(p)G\) provided that the order \(n\) is divisible by \(p\) but not by \(p^ 2\). One can use Wilbrink's equation to get evidence for the above mentioned prime power conjecture. This paper is an attempt to weaken the assumption on \(p\). We derive equations in \(\mathbb{Z}_{p^ i}G\) (using the group ring over the \(p\)-adic number field) if \(p^ i\) is the exact divisor of \(n\). Unfortunately, the equations are at present too complicated to have interesting applications.
reviewed by: Alexander Pott
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 64830
zbMATH Keywords: projective plane
zbMATH Keywords: difference set
zbMATH Keywords: Wilbrink's equation
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01191207
OpenAlex ID: W1975966127
description: scientific article; zbMATH DE number 64830