Difference sets in abelian groups of \(p\)-rank two
type: Item
instance of: scholarly article
title: Difference sets in abelian groups of \(p\)-rank two
zbMATH Open document ID: 0822.05005
DOI: 10.1007/BF01388500
author: K. T. Arasu
author: Surinder K. Sehgal
published in: Designs, Codes and Cryptography
publication date: 1995-10-05T00:00:00Z
review text: A \((v, k, \lambda)\)-difference set \(D\) is a \(k\)-subset of a group \(G\) such that the list of differences covers every non-identity group element exactly \(\lambda\) times. One of the open problems in finite geometry is to decide for which parameter triples difference sets can exist. In the case of abelian groups, a powerful approach to prove non-existence results is the use of algebraic number theory. Using this method, one can develop rather strong results if a Sylow \(p\)-subgroup of \(G\) is cyclic. In this paper, the authors study the case of Sylow \(p\)-subgroups of rank 2. In particular, they prove the non-existence of (96, 20, 4)-difference sets in \(\mathbb{Z}_ 4\times \mathbb{Z}_ 8\times \mathbb{Z}_ 3\), (640, 72, 8)- difference sets in \(\mathbb{Z}_ 8\times \mathbb{Z}_{16}\times \mathbb{Z}_ 5\), and (320, 88, 24)-difference sets in \(\mathbb{Z}_ 8\times \mathbb{Z}_ 8\times \mathbb{Z}_ 5\). Recently, a new difference set with parameters (96, 20, 4) in \(\mathbb{Z}_ 4\times \mathbb{Z}_ 4\times \mathbb{Z}_ 2\times \mathbb{Z}_ 3\) has been constructed by the same authors [J. Comb. Theory, Ser. 69, No. 1, 170-172 (1995; see the immediately preceding item Zbl 0822.05004)].
reviewed by: Alexander Pott
Mathematics Subject Classification ID: 05B10
zbMATH DE Number: 722114
zbMATH Keywords: difference sets
zbMATH Keywords: abelian groups
zbMATH Keywords: Sylow \(p\)-subgroup
MaRDI profile type: Publication
cites work: Q3972075
cites work: Q5285556
cites work: Q3686719
cites work: Q4026149
cites work: Difference sets in noncyclic Abelian groups
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cites work: A family of difference sets in non-cyclic groups
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full work available at URL: https://doi.org/10.1007/bf01388500
OpenAlex ID: W2026090923
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description: scientific article; zbMATH DE number 722114