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We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order $2^7 \\cdot 3^2 \\cdot 5 \\cdot 7^2 \\cdot 19^2$ whose integral group ring contains a unit of order $7 \\cdot 19$ which, in the rational 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":"1710.08780","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-10-24T14:04:14Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"9b8b577455364a050e09d9075245727012775d002530cac7f9140bdb114e1bc9","abstract_canon_sha256":"a38ce1d81350f31ff0ec38d67a8719e5b72cff15bcfb17a5c936c526ce442bed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:11.218479Z","signature_b64":"oRyfl4oIEbyaqc4w6Q8NHs/Ea2Ql2GclK+HSihwpNNRupsmKzOuOytDmk94Iin5NiZqDB1NxjVAQq9irUKprCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48996af8f8a208a0e9c0e48249474f5528e334db074b9a51771747b5ee82d318","last_reissued_at":"2026-05-18T00:30:11.217818Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:11.217818Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Counterexample to the First Zassenhaus Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RT"],"primary_cat":"math.RA","authors_text":"Florian Eisele, Leo Margolis","submitted_at":"2017-10-24T14:04:14Z","abstract_excerpt":"Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\\cdot u \\cdot a=\\pm g$ for some $g\\in G$. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order $2^7 \\cdot 3^2 \\cdot 5 \\cdot 7^2 \\cdot 19^2$ whose integral group ring contains a unit of order $7 \\cdot 19$ which, in the rational group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.08780","kind":"arxiv","version":2},"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":"1710.08780","created_at":"2026-05-18T00:30:11.217918+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.08780v2","created_at":"2026-05-18T00:30:11.217918+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.08780","created_at":"2026-05-18T00:30:11.217918+00:00"},{"alias_kind":"pith_short_12","alias_value":"JCMWV6HYUIEK","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_16","alias_value":"JCMWV6HYUIEKB2OA","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_8","alias_value":"JCMWV6HY","created_at":"2026-05-18T12:31:21.493067+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/JCMWV6HYUIEKB2OA4SBESR2PKU","json":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU.json","graph_json":"https://pith.science/api/pith-number/JCMWV6HYUIEKB2OA4SBESR2PKU/graph.json","events_json":"https://pith.science/api/pith-number/JCMWV6HYUIEKB2OA4SBESR2PKU/events.json","paper":"https://pith.science/paper/JCMWV6HY"},"agent_actions":{"view_html":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU","download_json":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU.json","view_paper":"https://pith.science/paper/JCMWV6HY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.08780&json=true","fetch_graph":"https://pith.science/api/pith-number/JCMWV6HYUIEKB2OA4SBESR2PKU/graph.json","fetch_events":"https://pith.science/api/pith-number/JCMWV6HYUIEKB2OA4SBESR2PKU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU/action/storage_attestation","attest_author":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU/action/author_attestation","sign_citation":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU/action/citation_signature","submit_replication":"https://pith.science/pith/JCMWV6HYUIEKB2OA4SBESR2PKU/action/replication_record"}},"created_at":"2026-05-18T00:30:11.217918+00:00","updated_at":"2026-05-18T00:30:11.217918+00:00"}