{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:4XLGCETTLGMQYNWUOT3FLTQ37O","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d19059785ae8a17c46c6a650c487d9261977816e909f0a77ba5ea8f6f28800a0","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-08-03T10:25:18Z","title_canon_sha256":"fb53da8eb7c73523bd8b2defada444404311ece6fe69e64ed1b248ea5b5edcd3"},"schema_version":"1.0","source":{"id":"0808.0308","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.0308","created_at":"2026-05-18T04:30:59Z"},{"alias_kind":"arxiv_version","alias_value":"0808.0308v3","created_at":"2026-05-18T04:30:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.0308","created_at":"2026-05-18T04:30:59Z"},{"alias_kind":"pith_short_12","alias_value":"4XLGCETTLGMQ","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"4XLGCETTLGMQYNWU","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"4XLGCETT","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:b320ae9ffac73f179d993985ea40c14b3354e21433d4bd382c42f152a5c44b0d","target":"graph","created_at":"2026-05-18T04:30:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a Garside group with Garside element $\\Delta$. An element $g$ in $G$ is said to be \\emph{periodic} if some power of $g$ lies in the cyclic group generated by $\\Delta$. This paper shows the following. (i) 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. (ii) If $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$. (iii) Every finite subgroup of the quotient group $G/<\\Delta^m>$ is cyclic, where $\\Delta^m$ is the minimal positive central power of $\\Delta$.","authors_text":"Eon-Kyung Lee, Sang-Jin Lee","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-08-03T10:25:18Z","title":"Notes on periodic elements of Garside groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.0308","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a372a1dc52ba7ba26344c9ce86008cdae2f4ff4b42b1f1c23a232cc816257d30","target":"record","created_at":"2026-05-18T04:30:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d19059785ae8a17c46c6a650c487d9261977816e909f0a77ba5ea8f6f28800a0","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-08-03T10:25:18Z","title_canon_sha256":"fb53da8eb7c73523bd8b2defada444404311ece6fe69e64ed1b248ea5b5edcd3"},"schema_version":"1.0","source":{"id":"0808.0308","kind":"arxiv","version":3}},"canonical_sha256":"e5d661127359990c36d474f655ce1bfbab180845a1c036b96cc450183cc1b144","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5d661127359990c36d474f655ce1bfbab180845a1c036b96cc450183cc1b144","first_computed_at":"2026-05-18T04:30:59.965066Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:30:59.965066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Sj5GNOxzPneVbVa7EbG8lWR8CqWOLXG6z3ciRBbrEHx8dmU8pgG7DuHZYI+wlWL6pGKzV2n2a4bDLM/Cr+nTBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:30:59.965978Z","signed_message":"canonical_sha256_bytes"},"source_id":"0808.0308","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a372a1dc52ba7ba26344c9ce86008cdae2f4ff4b42b1f1c23a232cc816257d30","sha256:b320ae9ffac73f179d993985ea40c14b3354e21433d4bd382c42f152a5c44b0d"],"state_sha256":"00751cf8e04fc9ac9caf3aa098769aca6f94048cbd0dcf103c7d36a56ab13d9a"}