{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:PRBX2AUNEKQPMRBBAUMMNIE5TE","short_pith_number":"pith:PRBX2AUN","canonical_record":{"source":{"id":"1310.6518","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-10-24T08:06:17Z","cross_cats_sorted":[],"title_canon_sha256":"e837e21e0c32ec8978334e52823eacf322e95afdcf43f37ae2df2803b2bc9fc1","abstract_canon_sha256":"ad0d42732c2cfcb00d84a2d2dea485cb31f27a9a69a1060b131442ddf9b93598"},"schema_version":"1.0"},"canonical_sha256":"7c437d028d22a0f644210518c6a09d9928164e9599303adf88a136ddc92e75e6","source":{"kind":"arxiv","id":"1310.6518","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6518","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6518v3","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6518","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"PRBX2AUNEKQP","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PRBX2AUNEKQPMRBB","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PRBX2AUN","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:PRBX2AUNEKQPMRBBAUMMNIE5TE","target":"record","payload":{"canonical_record":{"source":{"id":"1310.6518","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-10-24T08:06:17Z","cross_cats_sorted":[],"title_canon_sha256":"e837e21e0c32ec8978334e52823eacf322e95afdcf43f37ae2df2803b2bc9fc1","abstract_canon_sha256":"ad0d42732c2cfcb00d84a2d2dea485cb31f27a9a69a1060b131442ddf9b93598"},"schema_version":"1.0"},"canonical_sha256":"7c437d028d22a0f644210518c6a09d9928164e9599303adf88a136ddc92e75e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:46.597223Z","signature_b64":"Q2b5BVfs8/Vhfohv2hNwJWNuwa2YerDfWnL1E+4Mfh7iugwNZI/hO2OBgjNgJiLHHtaWIDlf03APcwMB7pVwBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c437d028d22a0f644210518c6a09d9928164e9599303adf88a136ddc92e75e6","last_reissued_at":"2026-05-18T03:02:46.596703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:46.596703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.6518","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZeRiX3VQ3nOI1gFhlJe6IOfFSOhbdM1fMKOfAnGfQae43WWv51ztjatBuvBpQZf03NA+O/NOWj57p/TE6+6dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T08:20:52.466503Z"},"content_sha256":"b55e5550930db1bc76b69947403a777cfcc436853605dab3d855abfb7faf9e7d","schema_version":"1.0","event_id":"sha256:b55e5550930db1bc76b69947403a777cfcc436853605dab3d855abfb7faf9e7d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:PRBX2AUNEKQPMRBBAUMMNIE5TE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boosting an analogue of Jordan's theorem for finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexandre Turull, Ignasi Mundet i Riera","submitted_at":"2013-10-24T08:06:17Z","abstract_excerpt":"Let $\\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. Suppose that for any $G\\in\\mathcal C$ whose order is divisible by at most two distinct primes there exists an abelian subgroup $A\\subseteq G$ such that $A$ is generated by at most $d$ elements and $[G : A] \\le M$. We prove that there exists a positive constant $C_0$ such that any $G \\in \\mathcal C$ has an abelian subgroup $A$ satisfying $[G : A] \\le C_0$, and $A$ can be generated by at most $d$ elements. We also prove some related results. Our proofs use the Classificatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6518","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V2z9hjdQ5t+4GorSGz13bAiVcYaYqnO5FQeJ36H2NAkA+UEWClRTMMH3iuzB+Bguq+AKLheJP4CP+vhHQ7i+DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T08:20:52.466847Z"},"content_sha256":"02a31c5b1451e6cdff80c86b5a4959298c1752540a26ce693eb8779950bb9be1","schema_version":"1.0","event_id":"sha256:02a31c5b1451e6cdff80c86b5a4959298c1752540a26ce693eb8779950bb9be1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/bundle.json","state_url":"https://pith.science/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T08:20:52Z","links":{"resolver":"https://pith.science/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE","bundle":"https://pith.science/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/bundle.json","state":"https://pith.science/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PRBX2AUNEKQPMRBBAUMMNIE5TE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PRBX2AUNEKQPMRBBAUMMNIE5TE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ad0d42732c2cfcb00d84a2d2dea485cb31f27a9a69a1060b131442ddf9b93598","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-10-24T08:06:17Z","title_canon_sha256":"e837e21e0c32ec8978334e52823eacf322e95afdcf43f37ae2df2803b2bc9fc1"},"schema_version":"1.0","source":{"id":"1310.6518","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6518","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6518v3","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6518","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"PRBX2AUNEKQP","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PRBX2AUNEKQPMRBB","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PRBX2AUN","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:02a31c5b1451e6cdff80c86b5a4959298c1752540a26ce693eb8779950bb9be1","target":"graph","created_at":"2026-05-18T03:02:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $\\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. Suppose that for any $G\\in\\mathcal C$ whose order is divisible by at most two distinct primes there exists an abelian subgroup $A\\subseteq G$ such that $A$ is generated by at most $d$ elements and $[G : A] \\le M$. We prove that there exists a positive constant $C_0$ such that any $G \\in \\mathcal C$ has an abelian subgroup $A$ satisfying $[G : A] \\le C_0$, and $A$ can be generated by at most $d$ elements. We also prove some related results. Our proofs use the Classificatio","authors_text":"Alexandre Turull, Ignasi Mundet i Riera","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-10-24T08:06:17Z","title":"Boosting an analogue of Jordan's theorem for finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6518","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b55e5550930db1bc76b69947403a777cfcc436853605dab3d855abfb7faf9e7d","target":"record","created_at":"2026-05-18T03:02:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ad0d42732c2cfcb00d84a2d2dea485cb31f27a9a69a1060b131442ddf9b93598","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-10-24T08:06:17Z","title_canon_sha256":"e837e21e0c32ec8978334e52823eacf322e95afdcf43f37ae2df2803b2bc9fc1"},"schema_version":"1.0","source":{"id":"1310.6518","kind":"arxiv","version":3}},"canonical_sha256":"7c437d028d22a0f644210518c6a09d9928164e9599303adf88a136ddc92e75e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c437d028d22a0f644210518c6a09d9928164e9599303adf88a136ddc92e75e6","first_computed_at":"2026-05-18T03:02:46.596703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:46.596703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q2b5BVfs8/Vhfohv2hNwJWNuwa2YerDfWnL1E+4Mfh7iugwNZI/hO2OBgjNgJiLHHtaWIDlf03APcwMB7pVwBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:46.597223Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.6518","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b55e5550930db1bc76b69947403a777cfcc436853605dab3d855abfb7faf9e7d","sha256:02a31c5b1451e6cdff80c86b5a4959298c1752540a26ce693eb8779950bb9be1"],"state_sha256":"7b659e65a58feeff9e742b58c62c136099d9c58d1ddd9cf0aff35a5c6d3f3634"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/0h91q3BXDlZhJJAoT70mTVl6zLKFTiTZxgH2FV+ESxey9DRN2/YHQVB0uozfEGw9b012znHShHvQXIL7OQCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T08:20:52.468680Z","bundle_sha256":"11c17f87da4921d449cf89f9f80421c3aa6fbace94ad0a7ce2449be7ce4d0fb5"}}