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.

Abstract

In 1987, John Horton Conway constructed a subset of permutations on a set of size for which
the subset fixing any given point is isomorphic to the Mathieu group . 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 .
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
to obtain interesting analogues of . In honour of John Conway, we refer to these analogues as . A number of
open questions are presented.

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