{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:VKHRRIGMUEEQQY2ZWNQXESWRT5","short_pith_number":"pith:VKHRRIGM","schema_version":"1.0","canonical_sha256":"aa8f18a0cca109086359b361724ad19f5acf8119f23f5f6127e60b42a80a4d9f","source":{"kind":"arxiv","id":"1512.06097","version":2},"attestation_state":"computed","paper":{"title":"Almost Engel finite and profinite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. I. Khukhro, P. Shumyatsky","submitted_at":"2015-12-18T20:21:14Z","abstract_excerpt":"Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\\dots ,g]$ over $x\\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\\leq m$ for all $g\\in G$,"},"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":"1512.06097","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-12-18T20:21:14Z","cross_cats_sorted":[],"title_canon_sha256":"2d3255c7ab91c87dd3002e1826567be761af9a6909529b9f3887f9652064d88a","abstract_canon_sha256":"eb80d36e372ba6dec10638510c05f3538062b4bf45f434abd681949df4792ffe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:09.514226Z","signature_b64":"oFd7s/fUQdRmO6Rj0D9ePMkwE3gptiuIYxScI5shCtKC4zv/dLsg8PSwACXXi3flYboUF/28SLDFd+tTBJBkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa8f18a0cca109086359b361724ad19f5acf8119f23f5f6127e60b42a80a4d9f","last_reissued_at":"2026-05-18T01:13:09.513889Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:09.513889Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost Engel finite and profinite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"E. I. Khukhro, P. Shumyatsky","submitted_at":"2015-12-18T20:21:14Z","abstract_excerpt":"Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\\dots ,g]$ over $x\\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\\leq m$ for all $g\\in G$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06097","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":"1512.06097","created_at":"2026-05-18T01:13:09.513944+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.06097v2","created_at":"2026-05-18T01:13:09.513944+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06097","created_at":"2026-05-18T01:13:09.513944+00:00"},{"alias_kind":"pith_short_12","alias_value":"VKHRRIGMUEEQ","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VKHRRIGMUEEQQY2Z","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VKHRRIGM","created_at":"2026-05-18T12:29:44.643036+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/VKHRRIGMUEEQQY2ZWNQXESWRT5","json":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5.json","graph_json":"https://pith.science/api/pith-number/VKHRRIGMUEEQQY2ZWNQXESWRT5/graph.json","events_json":"https://pith.science/api/pith-number/VKHRRIGMUEEQQY2ZWNQXESWRT5/events.json","paper":"https://pith.science/paper/VKHRRIGM"},"agent_actions":{"view_html":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5","download_json":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5.json","view_paper":"https://pith.science/paper/VKHRRIGM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.06097&json=true","fetch_graph":"https://pith.science/api/pith-number/VKHRRIGMUEEQQY2ZWNQXESWRT5/graph.json","fetch_events":"https://pith.science/api/pith-number/VKHRRIGMUEEQQY2ZWNQXESWRT5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5/action/storage_attestation","attest_author":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5/action/author_attestation","sign_citation":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5/action/citation_signature","submit_replication":"https://pith.science/pith/VKHRRIGMUEEQQY2ZWNQXESWRT5/action/replication_record"}},"created_at":"2026-05-18T01:13:09.513944+00:00","updated_at":"2026-05-18T01:13:09.513944+00:00"}