Geodesic
Conway Groupoids, Regular Two-Graphs and Supersimple Designs
Séminaire lotharingien de combinatoire, Tome 79 (2018-2023)

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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. % 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. % The proof uses Hall's work on $3$-transposition groups of symplectic type, and Seidel's work on graphs that satisfy the triangle property. 

@article{SLC_2018-2023_79_a1,
     author = {Nick Gill and Neil I. Gillespie and Cheryl E. Praeger and Jason Semeraro},
     title = {Conway {Groupoids,} {Regular} {Two-Graphs} and {Supersimple} {Designs}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {79},
     year = {2018-2023},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018-2023_79_a1/}
}
TY  - JOUR
AU  - Nick Gill
AU  - Neil I. Gillespie
AU  - Cheryl E. Praeger
AU  - Jason Semeraro
TI  - Conway Groupoids, Regular Two-Graphs and Supersimple Designs
JO  - Séminaire lotharingien de combinatoire
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%0 Journal Article
%A Nick Gill
%A Neil I. Gillespie
%A Cheryl E. Praeger
%A Jason Semeraro
%T Conway Groupoids, Regular Two-Graphs and Supersimple Designs
%J Séminaire lotharingien de combinatoire
%D 2018-2023
%V 79
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2018-2023_79_a1/
%F SLC_2018-2023_79_a1
Nick Gill; Neil I. Gillespie; Cheryl E. Praeger; Jason Semeraro. Conway Groupoids, Regular Two-Graphs and Supersimple Designs. Séminaire lotharingien de combinatoire, Tome 79 (2018-2023). http://geodesic.mathdoc.fr/item/SLC_2018-2023_79_a1/