Some remarks on the construction of cyclic Steiner quadruple systems
type: Item
instance of: scholarly article
title: Some remarks on the construction of cyclic Steiner quadruple systems
zbMATH Open document ID: 0601.05006
DOI: 10.1007/BF01200480
author: Helmut Siemon
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Steiner Quadruple Systems SQS(v) with parameter v are considered where the automorphism group contains a cyclic subgroup \(C_ v\) of order v so that the orbits of \(C_ v\) are of equal length v (strictly cyclic SQS(v)'s). In a natural way we can define a graph GS(v) the vertices of which being certain difference triples and two difference triples form an edge if they are contained in a certain difference quadruple. If this graph has a 1-factor then a strictly cyclic SQS(v) exists. Now GS(v) splits essentially up into two components \(GS_ 1(v)\), \(GS_ 2(v)\). We show that \(GS_ 1(v)\) has always as 1-factor. In order to determine a 1- factor of \(GS_ 2(v)\) we make a detailed analysis of how the unit group \(E(2p^{\alpha}) mod 2p^{\alpha},\) p prime number \(\equiv 1\) or 5 (12), operates as an automorphism group on the graph \(GS_ 2(2p^{\alpha})\) and consider an embedding process to generate an ascending series of subgraphs of \(GS_ 2(2p^{\alpha})\) which reaches \(GS_ 2(2p^{\alpha})\). By means of this devise we obtain a Lemma on the number of length of the orbits of a certain subgroup of \(E(2p^{\alpha})\). This allows us to determine a 1-factor of a certain subgraph ḠS\({}_ 2(2\cdot 5^{\alpha})\) which occurs in the decomposition of \(GS_ 2(2\cdot 5^{\alpha})\). Then we use induction to establish that \(GS_ 2(2\cdot 5^{\alpha})\) has a 1-factor, so constructing an infinite series of strictly \(SQS(2\cdot 5^{\alpha})\). Further SQS(v) are explicitly listed.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 3968595
zbMATH Keywords: graph
zbMATH Keywords: Steiner quadruple systems
zbMATH Keywords: SQS
zbMATH Keywords: automorphism group
zbMATH Keywords: difference triples
zbMATH Keywords: difference quadruple
zbMATH Keywords: 1-factor
zbMATH Keywords: cyclic SQS
MaRDI profile type: Publication
cites work: Q3897062
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description: scientific article; zbMATH DE number 3968595