Conway’s groupoid and its relatives

Gill, Nick; Gillespie, Neil; Praeger, Cheryl and Semeraro, Jason (2017). Conway’s groupoid and its relatives. In: Bhargava, Manjul; Guralnick, Robert; Hiss, Gerhard; Lux, Klaus and Tiep, Pham Huu eds. Finite Simple Groups: Thirty Years of the Atlas and Beyond. Contemporary Mathematics, 694. American Mathematical Society, pp. 91–110.

DOI: https://doi.org/10.1090/conm/694/13962

URL: http://doi.org/10.1090/conm/694/13962

Abstract

In 1987, John Horton Conway constructed a subset $M_{13}$ of permutations on a set of size $13$ for which
the subset fixing any given point is isomorphic to the Mathieu group $M_{12}$ . The construction has fascinated mathematicians
for the past thirty years, and remains remarkable in its mathematical isolation. It is based on a
``moving-counter puzzle'' on the projective plane $\PG(2,3)$ .
This survey, a homage to John Conway and his mathematics,
discusses consequences and generalisations
of Conway's construction. In particular it explores how various designs and hypergraphs can be used instead of
$\PG(2,3)$ to obtain interesting analogues of $M_{13}$ . In honour of John Conway, we refer to these analogues as ${\it Conway groupoids}$ . A number of
open questions are presented.