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The centre of a Steiner loop and the maxi-Pasch problem. (English) Zbl 07332727

Associated with every Steiner triple system (STS) of order \(n-1\) is a Steiner loop of order \(n\). This paper explores the necessary and sufficient conditions on \(n\) and \(m\) [D. Donovan and A. Rahilly, Southeast Asian Bull. Math. 16, No. 2, 115–121 (1992; Zbl 0789.20078)] for the existence of a Steiner loop of order \(n\) with centre of order \(m\). Connections to the determination of the maximum number of Pasch configurations in a Steiner triple system (STS) of a given order are established. An STS whose number of Pasch configurations is maximum for its order is called maxi-Pasch. In the paper, it is shown that loop factorization preserves the maxi-Pasch property. Perhaps surprisingly, the Steiner loops of all currently known maxi-Pasch Steiner triple systems have centre of maximum possible order.

MSC:

05B07 Triple systems
20N05 Loops, quasigroups

Citations:

Zbl 0789.20078

Software:

Mace4
PDFBibTeX XMLCite
Full Text: DOI

References:

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