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P\\\"oschel (1974) that given a prime $p\\ge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $k\\ge 0$ is an integer."},"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":"1111.5216","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-22T15:15:19Z","cross_cats_sorted":[],"title_canon_sha256":"7274548d0e44f0aec76f8a8e7e4d2e7f7839fb846142fbdec9420b26e6ca8907","abstract_canon_sha256":"849f087861d4ec432d3201fbb0e53e509b5de6550548bea1f3be1c971eac3683"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:51.584608Z","signature_b64":"85WvTv84rVyAwlomNEq6hlH2zcMb8RegAyG8hSxNf17UrCVbjrHyLsbdzFxgb88sZemH0JqOiu0jYlsJwRNEAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97e631059a76940654bbc4fd110c92a2210d22a4a312e8dc28c0f31bb7234a0e","last_reissued_at":"2026-05-18T02:16:51.583939Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:51.583939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterization of cyclic Schur groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilya Ponomarenko, Istv\\'an Kov\\'acs, Sergei Evdokimov","submitted_at":"2011-11-22T15:15:19Z","abstract_excerpt":"A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\\\"oschel (1974) that given a prime $p\\ge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $k\\ge 0$ is an integer."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5216","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":"1111.5216","created_at":"2026-05-18T02:16:51.584042+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.5216v2","created_at":"2026-05-18T02:16:51.584042+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5216","created_at":"2026-05-18T02:16:51.584042+00:00"},{"alias_kind":"pith_short_12","alias_value":"S7TDCBM2O2KA","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"S7TDCBM2O2KAMVF3","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"S7TDCBM2","created_at":"2026-05-18T12:26:41.206345+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/S7TDCBM2O2KAMVF3YT6RCDESUI","json":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI.json","graph_json":"https://pith.science/api/pith-number/S7TDCBM2O2KAMVF3YT6RCDESUI/graph.json","events_json":"https://pith.science/api/pith-number/S7TDCBM2O2KAMVF3YT6RCDESUI/events.json","paper":"https://pith.science/paper/S7TDCBM2"},"agent_actions":{"view_html":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI","download_json":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI.json","view_paper":"https://pith.science/paper/S7TDCBM2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.5216&json=true","fetch_graph":"https://pith.science/api/pith-number/S7TDCBM2O2KAMVF3YT6RCDESUI/graph.json","fetch_events":"https://pith.science/api/pith-number/S7TDCBM2O2KAMVF3YT6RCDESUI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI/action/storage_attestation","attest_author":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI/action/author_attestation","sign_citation":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI/action/citation_signature","submit_replication":"https://pith.science/pith/S7TDCBM2O2KAMVF3YT6RCDESUI/action/replication_record"}},"created_at":"2026-05-18T02:16:51.584042+00:00","updated_at":"2026-05-18T02:16:51.584042+00:00"}