Some remarks on the construction of cyclic Steiner quadruple systems
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.
scientific article; zbMATH DE number 3968595