{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:SRML6QNEDW4AECAYMKUUMC4Q2Y","short_pith_number":"pith:SRML6QNE","schema_version":"1.0","canonical_sha256":"9458bf41a41db802081862a9460b90d612d1522a63deee90c2f5f21cf7605e24","source":{"kind":"arxiv","id":"1703.06979","version":1},"attestation_state":"computed","paper":{"title":"Difference sets disjoint from a subgroup","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Courtney Hoagland, Seth Poulsen, Stephen P. Humphries","submitted_at":"2017-03-20T21:46:24Z","abstract_excerpt":"We study finite groups $G$ having a subgroup $H$ and $D \\subset G \\setminus H$ such that the multiset $\\{ xy^{-1}:x,y \\in D\\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\\it difference set}). We show that $H$ has to be normal, that $|G|=|H|^2$, and that $|D \\cap Hg|=|H|/2$ for all $g \\notin H$. We show that $H$ is contained in every normal subgroup of prime index, and other properties. We give a $2$-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets."},"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":"1703.06979","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-03-20T21:46:24Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d9f1aac99fddc93680006538e312c51268b8cae724c0227eff88f15ba457a978","abstract_canon_sha256":"d1bcd74fad28440a69da9c19b3f8ec215412b96bf17c23ec0951e8369913df6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:11.700250Z","signature_b64":"40sZGShg1HX5wMjzRk5qul+p85UC6SWlCMo3XEHLrEw3AR70Jzx57OrTPH6RVLlV3Mgs0LUhame5hSAwcptrCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9458bf41a41db802081862a9460b90d612d1522a63deee90c2f5f21cf7605e24","last_reissued_at":"2026-05-18T00:48:11.699510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:11.699510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Difference sets disjoint from a subgroup","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Courtney Hoagland, Seth Poulsen, Stephen P. Humphries","submitted_at":"2017-03-20T21:46:24Z","abstract_excerpt":"We study finite groups $G$ having a subgroup $H$ and $D \\subset G \\setminus H$ such that the multiset $\\{ xy^{-1}:x,y \\in D\\}$ has every non-identity element occur the same number of times (such a $D$ is called a {\\it difference set}). We show that $H$ has to be normal, that $|G|=|H|^2$, and that $|D \\cap Hg|=|H|/2$ for all $g \\notin H$. We show that $H$ is contained in every normal subgroup of prime index, and other properties. We give a $2$-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06979","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":"1703.06979","created_at":"2026-05-18T00:48:11.699635+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.06979v1","created_at":"2026-05-18T00:48:11.699635+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.06979","created_at":"2026-05-18T00:48:11.699635+00:00"},{"alias_kind":"pith_short_12","alias_value":"SRML6QNEDW4A","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"SRML6QNEDW4AECAY","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"SRML6QNE","created_at":"2026-05-18T12:31:43.269735+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/SRML6QNEDW4AECAYMKUUMC4Q2Y","json":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y.json","graph_json":"https://pith.science/api/pith-number/SRML6QNEDW4AECAYMKUUMC4Q2Y/graph.json","events_json":"https://pith.science/api/pith-number/SRML6QNEDW4AECAYMKUUMC4Q2Y/events.json","paper":"https://pith.science/paper/SRML6QNE"},"agent_actions":{"view_html":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y","download_json":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y.json","view_paper":"https://pith.science/paper/SRML6QNE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.06979&json=true","fetch_graph":"https://pith.science/api/pith-number/SRML6QNEDW4AECAYMKUUMC4Q2Y/graph.json","fetch_events":"https://pith.science/api/pith-number/SRML6QNEDW4AECAYMKUUMC4Q2Y/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y/action/storage_attestation","attest_author":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y/action/author_attestation","sign_citation":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y/action/citation_signature","submit_replication":"https://pith.science/pith/SRML6QNEDW4AECAYMKUUMC4Q2Y/action/replication_record"}},"created_at":"2026-05-18T00:48:11.699635+00:00","updated_at":"2026-05-18T00:48:11.699635+00:00"}