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It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \\leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \\neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|"},"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":"1712.05899","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-16T03:53:59Z","cross_cats_sorted":[],"title_canon_sha256":"3670956b8174fde196839dfac7345cf1221751b9ae05f92d6e12a27594b6ed3c","abstract_canon_sha256":"12c8254d589140703110e2aca9e96724c3c741ab9b2f5f2d45b5cb61565481eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:51.583357Z","signature_b64":"VWVG8i5Jj1BceaLlisNtNWTMTYATxEChCoPJ6dyqf+4X2vQwnPaAU1gdBr1ZKXorpY+aZAcn08fmemF3hidoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b956b44252acd69e1911efd7536c7a2e43f385c59a66a63b0c91e3470cd3d8d","last_reissued_at":"2026-05-18T00:27:51.582872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:51.582872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the second-largest Sylow subgroup of a finite simple group of Lie type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alice C. Niemeyer, S.P. Glasby, Tomasz Popiel","submitted_at":"2017-12-16T03:53:59Z","abstract_excerpt":"Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \\leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \\neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.05899","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":"1712.05899","created_at":"2026-05-18T00:27:51.582950+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.05899v1","created_at":"2026-05-18T00:27:51.582950+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.05899","created_at":"2026-05-18T00:27:51.582950+00:00"},{"alias_kind":"pith_short_12","alias_value":"JOKWWRBFFLGW","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_16","alias_value":"JOKWWRBFFLGWTYMR","created_at":"2026-05-18T12:31:24.725408+00:00"},{"alias_kind":"pith_short_8","alias_value":"JOKWWRBF","created_at":"2026-05-18T12:31:24.725408+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/JOKWWRBFFLGWTYMRD36XKNWHUL","json":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL.json","graph_json":"https://pith.science/api/pith-number/JOKWWRBFFLGWTYMRD36XKNWHUL/graph.json","events_json":"https://pith.science/api/pith-number/JOKWWRBFFLGWTYMRD36XKNWHUL/events.json","paper":"https://pith.science/paper/JOKWWRBF"},"agent_actions":{"view_html":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL","download_json":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL.json","view_paper":"https://pith.science/paper/JOKWWRBF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.05899&json=true","fetch_graph":"https://pith.science/api/pith-number/JOKWWRBFFLGWTYMRD36XKNWHUL/graph.json","fetch_events":"https://pith.science/api/pith-number/JOKWWRBFFLGWTYMRD36XKNWHUL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL/action/storage_attestation","attest_author":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL/action/author_attestation","sign_citation":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL/action/citation_signature","submit_replication":"https://pith.science/pith/JOKWWRBFFLGWTYMRD36XKNWHUL/action/replication_record"}},"created_at":"2026-05-18T00:27:51.582950+00:00","updated_at":"2026-05-18T00:27:51.582950+00:00"}