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Symmetric abelian group-invariant Steiner quadruple systems. (English) Zbl 1466.05014

Summary: Let \(K\) be an abelian group of order \(v\). A Steiner quadruple system of order \(v(\mathrm{SQS}(v))(K,\mathcal{B})\) is called symmetric \(K\)-invariant if for each \(B\in\mathcal{B}\), it holds that \(B+x\in\mathcal{B}\) for each \(x\in K\) and \(B=-B+y\) for some \(y\in K\). When the Sylow 2-subgroup of \(K\) is cyclic, a necessary and sufficient condition for the existence of a symmetric \(K\)-invariant \(\mathrm{SQS}(v)\) was given by A. Munemasa and M. Sawa [J. Stat. Theory Pract. 6, No. 1, 97–128 (2012; Zbl 1418.05028)], which is a generalization of a necessary and sufficient condition for the existence of a symmetric cyclic \(\mathrm{SQS}(v)\) shown in [W. Piotrowski. Untersuchungen über S-zyklische Quadrupelsysteme. Hamburg: Universität Hamburg (PhD Thesis) (1985). In this paper, we prove that a symmetric \(K\)-invariant \(\mathrm{SQS}(v)\) exists if and only if \(v\equiv 2,4\pmod 6\), the order of each element of \(K\) is not divisible by 8, and there exists a symmetric cyclic \(\mathrm{SQS}(2p)\) for any odd prime divisor \(p\) of \(v\).

MSC:

05B05 Combinatorial aspects of block designs
05B07 Triple systems

Citations:

Zbl 1418.05028
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References:

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