Conway groupoids, regular two-graphs and supersimple designs

Nick Gill, Neil I. Gillespie, Cheryl E. Praeger, Jason Semeraro

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    Abstract

    A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid $M_{13}$. It turns out that several infinite families of groupoids arise in this way, some associated with 3-transposition groups, which have two additional properties. Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. In this paper, we show each of these properties corresponds to a group-theoretic property on the groupoid and we classify the Conway groupoids and the supersimple designs for which both of these two additional properties hold.
    Original languageEnglish
    Number of pages24
    JournalSéminaire Lotharingien de Combinatoire
    Issue number79
    Publication statusPublished - 2018

    Keywords

    • math.GR
    • 20B15, 20B25, 05B05

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