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In 1953, Hall and Paige proved that a finite group admits a complete mapping only if its Sylow-2 subgroup is trivial or non-cyclic. They conjectured that this condition is also sufficient. We prove that it is sufficient to check the conjecture for the 26 sporadic simple groups and the Tits group."},"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":"1010.1323","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-10-07T04:03:11Z","cross_cats_sorted":[],"title_canon_sha256":"2041d011973bc6980b53fdd7a60c7d616739989c85da98aca158a8a303a50666","abstract_canon_sha256":"11e57b3ec556ade1a38451505e414d06e2b8ec8681fea7f915d62785fd9f066d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:41.414608Z","signature_b64":"uvWZk+76F3d0/cWFsg/YHGJpzgXRPJU1jumPdHyIROosh8JuoTS3D0iaAhXBQemDa4hTl5QQAqyQbj2fTIvrCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"522697b6b864aafa10a0b6011ee8ca34e8a00e5c69e5dbacd4fe1809b303275e","last_reissued_at":"2026-05-18T04:39:41.414138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:41.414138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reduction of the Hall-Paige conjecture to sporadic simple groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Stewart Wilcox","submitted_at":"2010-10-07T04:03:11Z","abstract_excerpt":"A complete mapping of a group $G$ is a permutation $\\phi:G\\rightarrow G$ such that $g\\mapsto g\\phi(g)$ is also a permutation. 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