### 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 Conway groupoids. A number of open questions are presented.

Language | English |
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Title of host publication | Finite Simple Groups |

Subtitle of host publication | Thirty Years of the Atlas and Beyond |

Editors | M Bhargava, R Guralnick, G Hiss, K Lux, PH Tiep |

Place of Publication | USA |

Publisher | American Mathematical Society |

Pages | 91-110 |

Number of pages | 20 |

ISBN (Electronic) | 9781470441685 |

ISBN (Print) | 9781470436780 |

DOIs | |

Publication status | Published - 2017 |

Event | International Conference on Finite Simple Groups: Thirty Years of the Atlas and Beyond - Celebrating the Atlases and Honoring John Conway - Princeton Duration: 2 Nov 2015 → 5 Nov 2015 |

### Publication series

Name | Contemporary Mathematics |
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Publisher | AMER MATHEMATICAL SOC |

Volume | 694 |

ISSN (Print) | 0271-4132 |

### Conference

Conference | International Conference on Finite Simple Groups: Thirty Years of the Atlas and Beyond - Celebrating the Atlases and Honoring John Conway |
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City | Princeton |

Period | 2/11/15 → 5/11/15 |

### Cite this

*Finite Simple Groups: Thirty Years of the Atlas and Beyond*(pp. 91-110). (Contemporary Mathematics; Vol. 694). USA: American Mathematical Society. https://doi.org/10.1090/conm/694/13962

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*Finite Simple Groups: Thirty Years of the Atlas and Beyond.*Contemporary Mathematics, vol. 694, American Mathematical Society, USA, pp. 91-110, International Conference on Finite Simple Groups: Thirty Years of the Atlas and Beyond - Celebrating the Atlases and Honoring John Conway, Princeton, 2/11/15. https://doi.org/10.1090/conm/694/13962

**Conway's groupoid and its relatives.** / Gill, Nick; Gillespie, Neil I.; Praeger, Cheryl E.; Semeraro, Jason.

Research output: Chapter in Book/Conference paper › Conference paper

TY - GEN

T1 - Conway's groupoid and its relatives

AU - Gill, Nick

AU - Gillespie, Neil I.

AU - Praeger, Cheryl E.

AU - Semeraro, Jason

PY - 2017

Y1 - 2017

N2 - 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 Conway groupoids. A number of open questions are presented.

AB - 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 Conway groupoids. A number of open questions are presented.

KW - M-13

KW - projective plane

KW - design

KW - permutation group

KW - groupoid

KW - code

KW - hypergraph

KW - two-graph

KW - REGULAR CODES

KW - TRANSITIVE CODES

KW - PERMUTATION-GROUPS

KW - PRIMITIVE GROUPS

KW - GRAPHS

KW - FAMILIES

KW - ORDER

U2 - 10.1090/conm/694/13962

DO - 10.1090/conm/694/13962

M3 - Conference paper

SN - 9781470436780

T3 - Contemporary Mathematics

SP - 91

EP - 110

BT - Finite Simple Groups

A2 - Bhargava, M

A2 - Guralnick, R

A2 - Hiss, G

A2 - Lux, K

A2 - Tiep, PH

PB - American Mathematical Society

CY - USA

ER -