Conway's groupoid and its relatives

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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In 1997, John Conway constructed a 6-fold transitive subset M13 of permutations on a set of size 13 for which the subset fixing any given point was isomorphic to the Mathieu group M12. The construction was via a "moving-counter puzzle" on the projective plane PG(2,3). We discuss consequences and generalisations of Conway's construction. In particular we explore how various designs and hypergraphs can be used instead of PG(2,3) to obtain interesting analogues of M13 we refer to these analogues as Conway groupoids. A number of open questions are presented.
Original languageEnglish
Title of host publicationFinite Simple Groups
Subtitle of host publicationThirty Years of the Atlas and Beyond
EditorsManjul Bhargava, Robert Guralnick, Gerard Hiss, Klaus Lux, Pham Huu Tiepp
PublisherAmerican Mathematical Society
Number of pages18
ISBN (Electronic)978-1-4704-4168-5
ISBN (Print)978-1-4704-3678-0
Publication statusPublished - 2017
EventFinite Simple Groups: Thirty Years of the Atlas and Beyond: Celebrating the Atlases and Honoring John Conway - Princeton University, Princeton, New Jersey, United States
Duration: 2 Nov 20155 Nov 2015

Publication series

NameContemporary Mathematics
PublisherAmerican Mathematical Society
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627


ConferenceFinite Simple Groups: Thirty Years of the Atlas and Beyond
Country/TerritoryUnited States
CityPrinceton, New Jersey


  • math.GR
  • 20B15 (Primary) 20B25, 05B05 (Secondary)
  • code
  • design
  • groupoid
  • hypergraph
  • M13
  • permutation group
  • projective plane
  • two-graph


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