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This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian.\n  Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$.\n  We further study the next unresolved case not excluded by this congruence, namely $p=97$."},"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":"2608.03003","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-08-04T01:36:02Z","cross_cats_sorted":[],"title_canon_sha256":"45fcf7ecd29aa91d866cc650f466d625f54f4990286f22e97ad841c3b108d0a8","abstract_canon_sha256":"082dfc2a551159a5d04fedcf4191346f94d037c0e9b3533e71fc9f31d6ecebfc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-05T00:44:22.622714Z","signature_b64":"1cDYew2Ekke164mcOu5U0WktsR50YLh0dQExCQbgBXsdr2STJyKCWPwHed3ARKQgKXEBK3W8QVtR6sOSqYcmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e206dc5f8a91e7bfa58b4cfa2fb2c51a58ec302ac21a5fec249cc4b965eb3695","last_reissued_at":"2026-08-05T00:44:22.620298Z","signature_status":"signed_v1","first_computed_at":"2026-08-05T00:44:22.620298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The solution to Kourovka problem 21.88","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Basile Beyer de Ryke","submitted_at":"2026-08-04T01:36:02Z","abstract_excerpt":"We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. 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