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In this paper, we study periodic elements in Garside groups and their conjugacy classes.\n  We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$; every finite subgroup of the quotient group $G/<\\Delt"},"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":"1004.5308","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-04-29T14:41:17Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d6e3aac5fc23660a53a7f97841233d3c99ed452546dbb0788bf6589cd2f4179c","abstract_canon_sha256":"58ae3d07bc11dee2a4f4f7527758b0fe366c7aeeaf990d816059ec43b1fbd564"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:15.026613Z","signature_b64":"ETChH6royzEv/iMuK5Ml/AXvD2OYQAJ0CT2wY+KL1+wDFXhqUcMpMLCIVaZY5yuBYNoi31PDfocYy8qL4brXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8539f5a37bd0e439cdf9f94d33254ab8e3ec08e48216fc9b6ce5b2c9bb661f3","last_reissued_at":"2026-05-18T02:24:15.025951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:15.025951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Periodic elements in Garside groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Eon-Kyung Lee, Sang-Jin Lee","submitted_at":"2010-04-29T14:41:17Z","abstract_excerpt":"Let $G$ be a Garside group with Garside element $\\Delta$, and let $\\Delta^m$ be the minimal positive central power of $\\Delta$. 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In this paper, we study periodic elements in Garside groups and their conjugacy classes.\n  We show that the periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic; if $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$; every finite subgroup of the quotient group $G/<\\Delt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5308","kind":"arxiv","version":2},"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":"1004.5308","created_at":"2026-05-18T02:24:15.026045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1004.5308v2","created_at":"2026-05-18T02:24:15.026045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5308","created_at":"2026-05-18T02:24:15.026045+00:00"},{"alias_kind":"pith_short_12","alias_value":"XBJZ6WRXXUHE","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"XBJZ6WRXXUHEHHG7","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"XBJZ6WRX","created_at":"2026-05-18T12:26:17.028572+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/XBJZ6WRXXUHEHHG7T6KNGMSUVO","json":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO.json","graph_json":"https://pith.science/api/pith-number/XBJZ6WRXXUHEHHG7T6KNGMSUVO/graph.json","events_json":"https://pith.science/api/pith-number/XBJZ6WRXXUHEHHG7T6KNGMSUVO/events.json","paper":"https://pith.science/paper/XBJZ6WRX"},"agent_actions":{"view_html":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO","download_json":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO.json","view_paper":"https://pith.science/paper/XBJZ6WRX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1004.5308&json=true","fetch_graph":"https://pith.science/api/pith-number/XBJZ6WRXXUHEHHG7T6KNGMSUVO/graph.json","fetch_events":"https://pith.science/api/pith-number/XBJZ6WRXXUHEHHG7T6KNGMSUVO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/action/storage_attestation","attest_author":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/action/author_attestation","sign_citation":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/action/citation_signature","submit_replication":"https://pith.science/pith/XBJZ6WRXXUHEHHG7T6KNGMSUVO/action/replication_record"}},"created_at":"2026-05-18T02:24:15.026045+00:00","updated_at":"2026-05-18T02:24:15.026045+00:00"}