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As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to \\(\\Triv(C_p)\\) for some prime \\(p\\), and every such skew brace admits an ideal of prime index. In particular, \\[ B*B\\neq B \\qquad\\text{and}\\qquad \\partial(B) \\neq B. \\] We also show that"},"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":"2607.19955","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.GR","submitted_at":"2026-07-22T09:30:15Z","cross_cats_sorted":[],"title_canon_sha256":"3437c9dacbc48b5b72f768eb5b97623697de47ef5d8d45beca07ac9c9ad2e047","abstract_canon_sha256":"bd1dc4cf07e49126cf7feaa905936ab260d9cc231011f3f2cf781e83795a1267"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:24:51.953963Z","signature_b64":"FGIc0o4vxNwCIwJ+BD1q3zaUvuQ66xRJssB1Xvgl/vSp5kTNujwM9xCpS6ehp6jbJ715uUl4VTFWTBVn7mwfDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9db2b4e328037bc1042aa98dc788fe168989cf3ef6f2a6e9df5b2d43e7da9095","last_reissued_at":"2026-07-23T01:24:51.953138Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:24:51.953138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ideals and Solvability in Skew Braces","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"G\\\"ulin Ercan, Marco Damele","submitted_at":"2026-07-22T09:30:15Z","abstract_excerpt":"We investigate how nilpotency assumptions on the multiplicative group of a finite skew brace constrain its ideal structure and solvability. Our first main result shows that, if \\(B=(B,+,\\cdot)\\) is finite and \\((B,\\cdot)\\) is nilpotent, then the additive Fitting subgroup \\(F(B,+)\\) is a non-zero ideal of \\(B\\). As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to \\(\\Triv(C_p)\\) for some prime \\(p\\), and every such skew brace admits an ideal of prime index. In particular, \\[ B*B\\neq B \\qquad\\text{and}\\qquad \\partial(B) \\neq B. \\] We also show that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19955","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/2607.19955/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":"2607.19955","created_at":"2026-07-23T01:24:51.953566+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.19955v1","created_at":"2026-07-23T01:24:51.953566+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.19955","created_at":"2026-07-23T01:24:51.953566+00:00"},{"alias_kind":"pith_short_12","alias_value":"TWZLJYZIAN54","created_at":"2026-07-23T01:24:51.953566+00:00"},{"alias_kind":"pith_short_16","alias_value":"TWZLJYZIAN54CBBK","created_at":"2026-07-23T01:24:51.953566+00:00"},{"alias_kind":"pith_short_8","alias_value":"TWZLJYZI","created_at":"2026-07-23T01:24:51.953566+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/TWZLJYZIAN54CBBKVGG4PCH6C2","json":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2.json","graph_json":"https://pith.science/api/pith-number/TWZLJYZIAN54CBBKVGG4PCH6C2/graph.json","events_json":"https://pith.science/api/pith-number/TWZLJYZIAN54CBBKVGG4PCH6C2/events.json","paper":"https://pith.science/paper/TWZLJYZI"},"agent_actions":{"view_html":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2","download_json":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2.json","view_paper":"https://pith.science/paper/TWZLJYZI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.19955&json=true","fetch_graph":"https://pith.science/api/pith-number/TWZLJYZIAN54CBBKVGG4PCH6C2/graph.json","fetch_events":"https://pith.science/api/pith-number/TWZLJYZIAN54CBBKVGG4PCH6C2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2/action/storage_attestation","attest_author":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2/action/author_attestation","sign_citation":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2/action/citation_signature","submit_replication":"https://pith.science/pith/TWZLJYZIAN54CBBKVGG4PCH6C2/action/replication_record"}},"created_at":"2026-07-23T01:24:51.953566+00:00","updated_at":"2026-07-23T01:24:51.953566+00:00"}