{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:LLT4J5IY5K5DSSSETORVXWLV6X","short_pith_number":"pith:LLT4J5IY","schema_version":"1.0","canonical_sha256":"5ae7c4f518eaba394a449ba35bd975f5c18a87d8afb1fde9d051964b88964e48","source":{"kind":"arxiv","id":"1310.3647","version":2},"attestation_state":"computed","paper":{"title":"Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Caroline Lassueur, Shigeo Koshitani","submitted_at":"2013-10-14T12:23:35Z","abstract_excerpt":"We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the group structure of $T(G)$ is known for the cases where a Sylow $p$-subgroup $P$ of $G$ is cyclic, semi-dihedral and generalized quaternion. We investigate $T(G)$, and more accurately, its torsion subgroup $TT(G)$ for the case where $P$ is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involut"},"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":"1310.3647","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-10-14T12:23:35Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"068f0a718360eddfe4f99b5fdebff01185d5514c92e6b45eaf401fb416d94e68","abstract_canon_sha256":"a7311f370d9c074623634aee0a535a0ee143a678d408fe047d9e8be4cc416e4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:47.427585Z","signature_b64":"xdQvElmEH1I0jrNhmYXwyf0GldqjHKIjiuzvIKewtgqP0auwlnbAU8ffJI7aQqk9/1XKGyZ6KbizAZdFYKqkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ae7c4f518eaba394a449ba35bd975f5c18a87d8afb1fde9d051964b88964e48","last_reissued_at":"2026-05-18T02:40:47.426944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:47.426944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Caroline Lassueur, Shigeo Koshitani","submitted_at":"2013-10-14T12:23:35Z","abstract_excerpt":"We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the group structure of $T(G)$ is known for the cases where a Sylow $p$-subgroup $P$ of $G$ is cyclic, semi-dihedral and generalized quaternion. We investigate $T(G)$, and more accurately, its torsion subgroup $TT(G)$ for the case where $P$ is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.3647","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":"1310.3647","created_at":"2026-05-18T02:40:47.427032+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.3647v2","created_at":"2026-05-18T02:40:47.427032+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.3647","created_at":"2026-05-18T02:40:47.427032+00:00"},{"alias_kind":"pith_short_12","alias_value":"LLT4J5IY5K5D","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"LLT4J5IY5K5DSSSE","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"LLT4J5IY","created_at":"2026-05-18T12:27:51.066281+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/LLT4J5IY5K5DSSSETORVXWLV6X","json":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X.json","graph_json":"https://pith.science/api/pith-number/LLT4J5IY5K5DSSSETORVXWLV6X/graph.json","events_json":"https://pith.science/api/pith-number/LLT4J5IY5K5DSSSETORVXWLV6X/events.json","paper":"https://pith.science/paper/LLT4J5IY"},"agent_actions":{"view_html":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X","download_json":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X.json","view_paper":"https://pith.science/paper/LLT4J5IY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.3647&json=true","fetch_graph":"https://pith.science/api/pith-number/LLT4J5IY5K5DSSSETORVXWLV6X/graph.json","fetch_events":"https://pith.science/api/pith-number/LLT4J5IY5K5DSSSETORVXWLV6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X/action/storage_attestation","attest_author":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X/action/author_attestation","sign_citation":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X/action/citation_signature","submit_replication":"https://pith.science/pith/LLT4J5IY5K5DSSSETORVXWLV6X/action/replication_record"}},"created_at":"2026-05-18T02:40:47.427032+00:00","updated_at":"2026-05-18T02:40:47.427032+00:00"}