{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:5GQU5AGWJCEKXDJW5XHSOKRVHB","short_pith_number":"pith:5GQU5AGW","schema_version":"1.0","canonical_sha256":"e9a14e80d64888ab8d36edcf272a35386302ccc329265493f24349b9f0dc1718","source":{"kind":"arxiv","id":"1112.5879","version":1},"attestation_state":"computed","paper":{"title":"On profinite groups in which commutators are covered by finitely many subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cristina Acciarri, Pavel Shumyatsky","submitted_at":"2011-12-26T18:43:26Z","abstract_excerpt":"For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $\\gamma_k$-values in $G$, it is shown that $\\gamma_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose uni"},"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":"1112.5879","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-12-26T18:43:26Z","cross_cats_sorted":[],"title_canon_sha256":"b6891d2bad3894cf7d6384e308dc3bde5eaf63105eb03b8c2d4a249c4c46aff3","abstract_canon_sha256":"cb43bd7cab35007e51d86d8c3fe73a1209e5b9eea6f6595962f031dc8f109f52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:37.376248Z","signature_b64":"k/HP6VYwGssC5isyFdim213u8Ab+toUFeFz2r/89NjS+kAirX+gyYdWeyMsQ9IyAFpaWiV6sHcQO7Avjx/gPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9a14e80d64888ab8d36edcf272a35386302ccc329265493f24349b9f0dc1718","last_reissued_at":"2026-05-18T04:05:37.375458Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:37.375458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On profinite groups in which commutators are covered by finitely many subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cristina Acciarri, Pavel Shumyatsky","submitted_at":"2011-12-26T18:43:26Z","abstract_excerpt":"For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $\\gamma_k$-values in $G$, it is shown that $\\gamma_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5879","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":"1112.5879","created_at":"2026-05-18T04:05:37.375591+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.5879v1","created_at":"2026-05-18T04:05:37.375591+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5879","created_at":"2026-05-18T04:05:37.375591+00:00"},{"alias_kind":"pith_short_12","alias_value":"5GQU5AGWJCEK","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5GQU5AGWJCEKXDJW","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5GQU5AGW","created_at":"2026-05-18T12:26:20.644004+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/5GQU5AGWJCEKXDJW5XHSOKRVHB","json":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB.json","graph_json":"https://pith.science/api/pith-number/5GQU5AGWJCEKXDJW5XHSOKRVHB/graph.json","events_json":"https://pith.science/api/pith-number/5GQU5AGWJCEKXDJW5XHSOKRVHB/events.json","paper":"https://pith.science/paper/5GQU5AGW"},"agent_actions":{"view_html":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB","download_json":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB.json","view_paper":"https://pith.science/paper/5GQU5AGW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.5879&json=true","fetch_graph":"https://pith.science/api/pith-number/5GQU5AGWJCEKXDJW5XHSOKRVHB/graph.json","fetch_events":"https://pith.science/api/pith-number/5GQU5AGWJCEKXDJW5XHSOKRVHB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB/action/storage_attestation","attest_author":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB/action/author_attestation","sign_citation":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB/action/citation_signature","submit_replication":"https://pith.science/pith/5GQU5AGWJCEKXDJW5XHSOKRVHB/action/replication_record"}},"created_at":"2026-05-18T04:05:37.375591+00:00","updated_at":"2026-05-18T04:05:37.375591+00:00"}