Conway's groupoid and its relatives

Nick Gill, Neil I. Gillespie, Cheryl E. Praeger, Jason Semeraro

Research output: Chapter in Book/Conference paperConference paper

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.

LanguageEnglish
Title of host publicationFinite Simple Groups
Subtitle of host publicationThirty Years of the Atlas and Beyond
EditorsM Bhargava, R Guralnick, G Hiss, K Lux, PH Tiep
Place of PublicationUSA
PublisherAmerican Mathematical Society
Pages91-110
Number of pages20
ISBN (Electronic)9781470441685
ISBN (Print)9781470436780
DOIs
StatePublished - 2017
EventInternational Conference on Finite Simple Groups: Thirty Years of the Atlas and Beyond - Celebrating the Atlases and Honoring John Conway - Princeton
Duration: 2 Nov 20155 Nov 2015

Publication series

NameContemporary Mathematics
PublisherAMER MATHEMATICAL SOC
Volume694
ISSN (Print)0271-4132

Conference

ConferenceInternational Conference on Finite Simple Groups: Thirty Years of the Atlas and Beyond - Celebrating the Atlases and Honoring John Conway
CityPrinceton
Period2/11/155/11/15

Cite this

Gill, N., Gillespie, N. I., Praeger, C. E., & Semeraro, J. (2017). Conway's groupoid and its relatives. In M. Bhargava, R. Guralnick, G. Hiss, K. Lux, & PH. Tiep (Eds.), Finite Simple Groups: Thirty Years of the Atlas and Beyond (pp. 91-110). (Contemporary Mathematics; Vol. 694). USA: American Mathematical Society. DOI: 10.1090/conm/694/13962
Gill, Nick ; Gillespie, Neil I. ; Praeger, Cheryl E. ; Semeraro, Jason. / Conway's groupoid and its relatives. Finite Simple Groups: Thirty Years of the Atlas and Beyond. editor / M Bhargava ; R Guralnick ; G Hiss ; K Lux ; PH Tiep. USA : American Mathematical Society, 2017. pp. 91-110 (Contemporary Mathematics).
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Gill, N, Gillespie, NI, Praeger, CE & Semeraro, J 2017, Conway's groupoid and its relatives. in M Bhargava, R Guralnick, G Hiss, K Lux & PH Tiep (eds), 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. DOI: 10.1090/conm/694/13962

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

Finite Simple Groups: Thirty Years of the Atlas and Beyond. ed. / M Bhargava; R Guralnick; G Hiss; K Lux; PH Tiep. USA : American Mathematical Society, 2017. p. 91-110 (Contemporary Mathematics; Vol. 694).

Research output: Chapter in Book/Conference paperConference paper

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Gill N, Gillespie NI, Praeger CE, Semeraro J. Conway's groupoid and its relatives. In Bhargava M, Guralnick R, Hiss G, Lux K, Tiep PH, editors, Finite Simple Groups: Thirty Years of the Atlas and Beyond. USA: American Mathematical Society. 2017. p. 91-110. (Contemporary Mathematics). Available from, DOI: 10.1090/conm/694/13962