Séminaire Lotharingien de Combinatoire, B79b (2018), 24 pp.
Nick Gill, Neil I. Gillespie, Cheryl E. Praeger and Jason Semeraro
Conway Groupoids, Regular Two-Graphs and Supersimple Designs
Abstract.
A 2-(n,4,λ) design (Ω,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(Ω) called a "Conway
groupoid". The construction generalizes Conway's construction of the
groupoid M13.
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One would like to classify all of the Conway groupoids constructed
using supersimple designs. In this paper we classify a particular
subclass, consisting of those groupoids which satisfy 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.
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The proof uses Hall's work on $3$-transposition groups of symplectic
type, and Seidel's work on graphs that satisfy the triangle property.
Received: March 21, 2018.
Accepted: May 18, 2018.
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