{"paper":{"title":"Conway's groupoid and its relatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cheryl E. Praeger, Jason Semeraro, Neil I. Gillespie, Nick Gill","submitted_at":"2016-04-15T11:08:59Z","abstract_excerpt":"In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a \"moving-counter puzzle\" on the projective plane ${\\rm PG}(2,3)$. We discuss consequences and generalisations of Conway's construction. In particular we explore how various designs and hypergraphs can be used instead of ${\\rm PG}(2,3)$ to obtain interesting analogues of $M_{13}$; we refer to these analogues as Conway groupoids. A number of open questions are presented."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04429","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}