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First, when both $\\langle x\\rangle$ and $\\langle z\\rangle$ are normal in~$G$, we obtain a complete classification of such exact products by an explicit system of six polynomial congruences. Second, when the cores $\\langle x\\rangle^G$ and $\\langle z\\rangle^G$ are arbitrary subgroups of $\\langle x\\rangle$"},"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":"2605.24480","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-23T09:06:43Z","cross_cats_sorted":[],"title_canon_sha256":"c87c8159528e0181f463f166782934af108206865409af5b9e88ed53c1db7a52","abstract_canon_sha256":"792e0b3cfa6d32c105fe14d801fec6bb8c76adf89195d4bd945361a7c388a9c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:42.079446Z","signature_b64":"c7BKW/fHRcLX8laAzDJx49pGwuobR/73lOj1iNO7W5Oh4J5y2hhUV5BqHm6W5t6q2x9uMUFER2/AW83L5fwrBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4db5afb29a2caf072ac83d5e32cfd673ee8e26dba16588686939e3df94783ac5","last_reissued_at":"2026-05-26T01:03:42.078639Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:42.078639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Zappa-Sz\\'ep products of two semidihedral groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Riccardo Aragona","submitted_at":"2026-05-23T09:06:43Z","abstract_excerpt":"Let $n, m \\ge 4$. 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Second, when the cores $\\langle x\\rangle^G$ and $\\langle z\\rangle^G$ are arbitrary subgroups of $\\langle x\\rangle$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24480","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.24480/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.24480","created_at":"2026-05-26T01:03:42.078785+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.24480v1","created_at":"2026-05-26T01:03:42.078785+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24480","created_at":"2026-05-26T01:03:42.078785+00:00"},{"alias_kind":"pith_short_12","alias_value":"JW227MU2FSXQ","created_at":"2026-05-26T01:03:42.078785+00:00"},{"alias_kind":"pith_short_16","alias_value":"JW227MU2FSXQOKWI","created_at":"2026-05-26T01:03:42.078785+00:00"},{"alias_kind":"pith_short_8","alias_value":"JW227MU2","created_at":"2026-05-26T01:03:42.078785+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/JW227MU2FSXQOKWIHVPDFT6WOP","json":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP.json","graph_json":"https://pith.science/api/pith-number/JW227MU2FSXQOKWIHVPDFT6WOP/graph.json","events_json":"https://pith.science/api/pith-number/JW227MU2FSXQOKWIHVPDFT6WOP/events.json","paper":"https://pith.science/paper/JW227MU2"},"agent_actions":{"view_html":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP","download_json":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP.json","view_paper":"https://pith.science/paper/JW227MU2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.24480&json=true","fetch_graph":"https://pith.science/api/pith-number/JW227MU2FSXQOKWIHVPDFT6WOP/graph.json","fetch_events":"https://pith.science/api/pith-number/JW227MU2FSXQOKWIHVPDFT6WOP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP/action/storage_attestation","attest_author":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP/action/author_attestation","sign_citation":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP/action/citation_signature","submit_replication":"https://pith.science/pith/JW227MU2FSXQOKWIHVPDFT6WOP/action/replication_record"}},"created_at":"2026-05-26T01:03:42.078785+00:00","updated_at":"2026-05-26T01:03:42.078785+00:00"}