Abstract:A binary operation ``$\cdot$" which satisfies the identities $x\cdot e = x$, $x \cdot x = e$, $(x \cdot y) \cdot x = y$ and $x \cdot y = y \cdot x$ is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order~$n$ with centre of order~$m$ and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be {\it maxi-Pasch}. We show that loop factorization preserves the maxi-Pasch property and find that the Steiner loops of all currently known maxi-Pasch Steiner triple systems have centre of maximum possible order.
Keywords: Steiner loop; centre; nucleus; Steiner triple system; Pasch configuration; quadrilateral
DOI: DOI 10.14712/1213-7243.2020.035
AMS Subject Classification: 05B07 20N05