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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1503.04527","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-16T05:48:09Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"dcee42d74a75a31129437f54178ff7d68d483245ab21238cdae49c96a1c17344","abstract_canon_sha256":"69c3de79d521f54a8d6151259c858421790e919c67c9ed08edeaddbe2ced97f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:23:24.729781Z","signature_b64":"Hh2etEZoGd7003vpL29VS2y85TNPemxH70f13dmi7/TsgOoMk2JT+8vF1GRWwucbxYOpezThOd5Oacp2p81RAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"436711307cc642a5ff8065acfb4571bf3950d4bc1453dd56c0b5cc72b5e47595","last_reissued_at":"2026-05-18T02:23:24.729165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:23:24.729165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1503.04527","created_at":"2026-05-18T02:23:24.729245+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.04527v1","created_at":"2026-05-18T02:23:24.729245+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.04527","created_at":"2026-05-18T02:23:24.729245+00:00"},{"alias_kind":"pith_short_12","alias_value":"INTRCMD4YZBK","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"INTRCMD4YZBKL74A","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"INTRCMD4","created_at":"2026-05-18T12:29:25.134429+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4","json":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4.json","graph_json":"https://pith.science/api/pith-number/INTRCMD4YZBKL74AMWWPWRLRX4/graph.json","events_json":"https://pith.science/api/pith-number/INTRCMD4YZBKL74AMWWPWRLRX4/events.json","paper":"https://pith.science/paper/INTRCMD4"},"agent_actions":{"view_html":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4","download_json":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4.json","view_paper":"https://pith.science/paper/INTRCMD4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.04527&json=true","fetch_graph":"https://pith.science/api/pith-number/INTRCMD4YZBKL74AMWWPWRLRX4/graph.json","fetch_events":"https://pith.science/api/pith-number/INTRCMD4YZBKL74AMWWPWRLRX4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4/action/storage_attestation","attest_author":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4/action/author_attestation","sign_citation":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4/action/citation_signature","submit_replication":"https://pith.science/pith/INTRCMD4YZBKL74AMWWPWRLRX4/action/replication_record"}},"created_at":"2026-05-18T02:23:24.729245+00:00","updated_at":"2026-05-18T02:23:24.729245+00:00"}