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We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of $u$ and apply this to the study of the Zassenhaus Conjecture. This conjecture states that any unit of finite order in $\\mathbb{Z} G$ is conjugate in the rational group algebra of $G$ to an element in $\\pm G$."},"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":"1706.02483","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-06-08T09:04:06Z","cross_cats_sorted":[],"title_canon_sha256":"5f628595789967ffe31d67dbd2479b7276494347c988faf7171a9b8602b54853","abstract_canon_sha256":"4643baba3fcdeecee2f8027647752e5716c12e314c3d8f73ceaeba3f9e485dbc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:01.070662Z","signature_b64":"/MRa/7a4gU+GTz+MAlzvUTnJDpezDydTEfQNsInSbfiKH7r97e/EOZ4S5VBQTAp7n6q2B/6ZZFBK6iD9Rxu8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37e2e01d5a6c85c75e9664a2d6ca9198daefd54ee79cb8dbb9838fec0b5399ee","last_reissued_at":"2026-05-18T00:32:01.070257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:01.070257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cliff-Weiss Inequalities and the Zassenhaus Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"\\'Angel del R\\'io, Leo Margolis","submitted_at":"2017-06-08T09:04:06Z","abstract_excerpt":"Let $N$ be a nilpotent normal subgroup of the finite group $G$. Assume that $u$ is a unit of finite order in the integral group ring $\\mathbb{Z} G$ of $G$ which maps to the identity under the linear extension of the natural homomorphism $G \\rightarrow G/N$. We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of $u$ and apply this to the study of the Zassenhaus Conjecture. 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