Conway groupoids, regular two-graphs and supersimple designs

TL;DR

This paper classifies Conway groupoids derived from supersimple 2-(n,4,λ) designs, linking their properties to group-theoretic features and exploring their connections with regular two-graphs and 3-transposition groups.

Contribution

It provides a classification of Conway groupoids and supersimple designs that exhibit specific combinatorial and algebraic properties, extending previous constructions.

Findings

Several infinite families of Conway groupoids are identified.

The properties of regular two-graphs and line differences are characterized group-theoretically.

Classification results for groupoids with both properties are established.

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.