{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:Z42UQF2YIQUPOAOQ56XOHLROW2","short_pith_number":"pith:Z42UQF2Y","schema_version":"1.0","canonical_sha256":"cf354817584428f701d0efaee3ae2eb692cff791484761feb665a222fc7787fc","source":{"kind":"arxiv","id":"1311.6148","version":2},"attestation_state":"computed","paper":{"title":"On finite groups in which coprime commutators are covered by few cyclic 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":"2013-11-24T17:49:27Z","abstract_excerpt":"The coprime commutators $\\gamma_j^*$ and $\\delta_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $\\gamma_1^*$-commutator and a $\\delta_0^*$-commutator. Now let $j\\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $\\gamma_{j-1}^*$-commutators. An element $g$ is a $\\gamma_j^*$-commutator if there exist $a\\in X$ and $b\\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\\geq 1$ let $Y$ "},"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":"1311.6148","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-11-24T17:49:27Z","cross_cats_sorted":[],"title_canon_sha256":"7ad8b437aa4d0abe661e99036e54caf85560b64591d6db56786b052ba7afa1ca","abstract_canon_sha256":"4180e30b515d7c7cad23e407e94df18b804e8e0dda4919e7aaedf2bdd69fb9f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:18.010510Z","signature_b64":"/pI1YoE9AnMg9zOa6wKCmtwfyfYGxKTqJvsM+g2EXVgS7/Nzxin4ShS20ZRj7Nm+jpfS0JH/Ewcu1MxYBEP3DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf354817584428f701d0efaee3ae2eb692cff791484761feb665a222fc7787fc","last_reissued_at":"2026-05-18T02:59:18.009740Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:18.009740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On finite groups in which coprime commutators are covered by few cyclic 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":"2013-11-24T17:49:27Z","abstract_excerpt":"The coprime commutators $\\gamma_j^*$ and $\\delta_j^*$ were recently introduced as a tool to study properties of finite groups that can be expressed in terms of commutators of elements of coprime orders. They are defined as follows. Let $G$ be a finite group. Every element of $G$ is both a $\\gamma_1^*$-commutator and a $\\delta_0^*$-commutator. Now let $j\\geq 2$ and let $X$ be the set of all elements of $G$ that are powers of $\\gamma_{j-1}^*$-commutators. An element $g$ is a $\\gamma_j^*$-commutator if there exist $a\\in X$ and $b\\in G$ such that $g=[a,b]$ and $(|a|,|b|)=1$. For $j\\geq 1$ let $Y$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6148","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":"1311.6148","created_at":"2026-05-18T02:59:18.009839+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6148v2","created_at":"2026-05-18T02:59:18.009839+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6148","created_at":"2026-05-18T02:59:18.009839+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z42UQF2YIQUP","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z42UQF2YIQUPOAOQ","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z42UQF2Y","created_at":"2026-05-18T12:28:09.283467+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/Z42UQF2YIQUPOAOQ56XOHLROW2","json":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2.json","graph_json":"https://pith.science/api/pith-number/Z42UQF2YIQUPOAOQ56XOHLROW2/graph.json","events_json":"https://pith.science/api/pith-number/Z42UQF2YIQUPOAOQ56XOHLROW2/events.json","paper":"https://pith.science/paper/Z42UQF2Y"},"agent_actions":{"view_html":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2","download_json":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2.json","view_paper":"https://pith.science/paper/Z42UQF2Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6148&json=true","fetch_graph":"https://pith.science/api/pith-number/Z42UQF2YIQUPOAOQ56XOHLROW2/graph.json","fetch_events":"https://pith.science/api/pith-number/Z42UQF2YIQUPOAOQ56XOHLROW2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2/action/storage_attestation","attest_author":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2/action/author_attestation","sign_citation":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2/action/citation_signature","submit_replication":"https://pith.science/pith/Z42UQF2YIQUPOAOQ56XOHLROW2/action/replication_record"}},"created_at":"2026-05-18T02:59:18.009839+00:00","updated_at":"2026-05-18T02:59:18.009839+00:00"}