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The following results are proved.\n  If $\\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\\in A^{#}$, then $\\gamma_{k-2}(G)$ is nilpotent and has $\\{c,k,p\\}$-bounded nilpotency class.\n  If, for some integer $d$ such that $2^{d}+2\\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\\{c,k,p\\}$-bounded nilpotency class.\n  Earlier this was known only in the case wher"},"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":"1112.5880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-12-26T18:52:08Z","cross_cats_sorted":[],"title_canon_sha256":"2fd55c6d055a60b989e00a7b0b824ed94c318b979d82e01700ac46503bb5cc58","abstract_canon_sha256":"27bbbe50cc8b9f5687c5888ee3e974f9f335dac125bec3b69b682fbb74db4d65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:37.365600Z","signature_b64":"n96YvOMOvLPE6zew1UwY+5hmZlXeNczK7t/fuWJyGoNFGR0g3oe1fImkCpUyv5vDCAhFPP1ahoL0RfJiuajhAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e054a9b157ca5415aa6df9e8df53b83dd3a8fbfbc3dae21be3ce312a9bd4bc8e","last_reissued_at":"2026-05-18T04:05:37.364968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:37.364968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Centralizers of coprime automorphisms of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Cristina Acciarri, Pavel Shumyatsky","submitted_at":"2011-12-26T18:52:08Z","abstract_excerpt":"Let $A$ be an elementary abelian group of order $p^{k}$ with $k\\geq 3$ acting on a finite $p'$-group $G$. The following results are proved.\n  If $\\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\\in A^{#}$, then $\\gamma_{k-2}(G)$ is nilpotent and has $\\{c,k,p\\}$-bounded nilpotency class.\n  If, for some integer $d$ such that $2^{d}+2\\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\\{c,k,p\\}$-bounded nilpotency class.\n  Earlier this was known only in the case wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5880","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":"1112.5880","created_at":"2026-05-18T04:05:37.365087+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.5880v1","created_at":"2026-05-18T04:05:37.365087+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5880","created_at":"2026-05-18T04:05:37.365087+00:00"},{"alias_kind":"pith_short_12","alias_value":"4BKKTMKXZJKB","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"4BKKTMKXZJKBLKTN","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"4BKKTMKX","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/4BKKTMKXZJKBLKTN7HUN6U5YHX","json":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX.json","graph_json":"https://pith.science/api/pith-number/4BKKTMKXZJKBLKTN7HUN6U5YHX/graph.json","events_json":"https://pith.science/api/pith-number/4BKKTMKXZJKBLKTN7HUN6U5YHX/events.json","paper":"https://pith.science/paper/4BKKTMKX"},"agent_actions":{"view_html":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX","download_json":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX.json","view_paper":"https://pith.science/paper/4BKKTMKX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.5880&json=true","fetch_graph":"https://pith.science/api/pith-number/4BKKTMKXZJKBLKTN7HUN6U5YHX/graph.json","fetch_events":"https://pith.science/api/pith-number/4BKKTMKXZJKBLKTN7HUN6U5YHX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX/action/storage_attestation","attest_author":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX/action/author_attestation","sign_citation":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX/action/citation_signature","submit_replication":"https://pith.science/pith/4BKKTMKXZJKBLKTN7HUN6U5YHX/action/replication_record"}},"created_at":"2026-05-18T04:05:37.365087+00:00","updated_at":"2026-05-18T04:05:37.365087+00:00"}