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We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral (the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that the unique obstacle here is a hypothetical S-ring of rank $5$ associated with"},"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":"1503.02621","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-09T19:02:23Z","cross_cats_sorted":[],"title_canon_sha256":"ec3243a8c2fdbcb0f5af542daedaa35f27b95f92a482848cc01263e9a5d76860","abstract_canon_sha256":"048994c870e7d47e4722fc94d5ca764175609be7085f55aa76ad9270c6a9c01f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:07.985111Z","signature_b64":"aRCE7x75qcO7IA4Ja6Eul/PM5lZ0XERCn2SL5RMmhg4dxROnFzmWT2Noxww7uIhQH5PwpiGmlaQGjIrYMkfdCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8673ab2709cd4475a4a58a38d50acc9191d20b83f243d1819f5dcf1bd865f2e1","last_reissued_at":"2026-05-18T00:42:07.984649Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:07.984649Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Schur 2-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya Ponomarenko, Mikhail Muzychuk","submitted_at":"2015-03-09T19:02:23Z","abstract_excerpt":"A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\\sym(G)$ that contains all right translations. 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Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that the unique obstacle here is a hypothetical S-ring of rank $5$ associated with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02621","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":"1503.02621","created_at":"2026-05-18T00:42:07.984719+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02621v1","created_at":"2026-05-18T00:42:07.984719+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02621","created_at":"2026-05-18T00:42:07.984719+00:00"},{"alias_kind":"pith_short_12","alias_value":"QZZ2WJYJZVCH","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"QZZ2WJYJZVCHLJFF","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"QZZ2WJYJ","created_at":"2026-05-18T12:29:39.896362+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/QZZ2WJYJZVCHLJFFRI4NKCWMSG","json":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG.json","graph_json":"https://pith.science/api/pith-number/QZZ2WJYJZVCHLJFFRI4NKCWMSG/graph.json","events_json":"https://pith.science/api/pith-number/QZZ2WJYJZVCHLJFFRI4NKCWMSG/events.json","paper":"https://pith.science/paper/QZZ2WJYJ"},"agent_actions":{"view_html":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG","download_json":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG.json","view_paper":"https://pith.science/paper/QZZ2WJYJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02621&json=true","fetch_graph":"https://pith.science/api/pith-number/QZZ2WJYJZVCHLJFFRI4NKCWMSG/graph.json","fetch_events":"https://pith.science/api/pith-number/QZZ2WJYJZVCHLJFFRI4NKCWMSG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG/action/storage_attestation","attest_author":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG/action/author_attestation","sign_citation":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG/action/citation_signature","submit_replication":"https://pith.science/pith/QZZ2WJYJZVCHLJFFRI4NKCWMSG/action/replication_record"}},"created_at":"2026-05-18T00:42:07.984719+00:00","updated_at":"2026-05-18T00:42:07.984719+00:00"}