{"paper":{"title":"Quotients of the Artin braid groups and crystallographic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Daciberg Lima Gon\\c{c}alves, John Guaschi (LMNO, Oscar Ocampo (UFBA), UNICAEN)","submitted_at":"2015-03-16T05:48:09Z","abstract_excerpt":"Let n be greater than or equal to 3. We study the quotient group B\\_n/[P n,P\\_n] of the Artin braid group B\\_n by the commutator subgroup of its pure Artin braid group P\\_n. We show that B\\_n/[P n,P\\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\\_n/[P n,P\\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\\_n/[P n,P\\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\\_n, and that the isomorphism class "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04527","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"}