Conway groupoids, regular two-graphs and supersimple designs

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

Research output: Contribution to journalArticle

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 language English 24 Séminaire Lotharingien de Combinatoire 79 Published - 2018

Keywords

• math.GR
• 20B15, 20B25, 05B05

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