Existence results for translation nets. II
type: Item
instance of: scholarly article
title: Existence results for translation nets. II
zbMATH Open document ID: 0671.05016
DOI: 10.1016/0021-8693(89)90217-2
author: Dieter Jungnickel
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let D be a translation (Bruck) net of degree s and order r (i.e., D admits a point regular group respecting parallel classes). If \(s=q_ 1...q_ n\) is the prime power factorization of s, then \(r\leq \min \{q_ i+1:\) \(i=1,...,n\}\); moreover, equality can be realized (e.g. for direct products of elementary abelian groups). Similar results are obtained for translation (s,r,\(\mu)\)-nets and, more generally, for groups with large systems of large subgroups. As a corollary, we considerably strengthen a result on symmetric translation nets by showing that every (s,r,\(\mu)\)- translation net with \(r\geq s\mu\) has parameters of the form \(s=p^ i\) and \(\mu =p^ j\) for some prime p and also necessarily has an elementary abelian translation group. These results settle some problems and conjectures from part I of the paper [Finite geometries and designs, Proc. 2nd Isle of Thorns Conf. 1980, Lond. Math. Soc., Lect. Note Ser. 49, 172-196 (1981; Zbl 0457.05011)].
reviewed by: Dieter Jungnickel
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4097393
zbMATH Keywords: disjoint subgroups
zbMATH Keywords: translation nets
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90217-2
OpenAlex ID: W2056097979
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description: scientific article; zbMATH DE number 4097393