{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:UBIA7OBPQFHF6AGTZ4CZYYTJTD","short_pith_number":"pith:UBIA7OBP","canonical_record":{"source":{"id":"1904.06731","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-04-14T17:52:29Z","cross_cats_sorted":[],"title_canon_sha256":"8e1bcc74157fa7bf3825488c80beaac66e3076cf91454141f574316fa3f7dfcc","abstract_canon_sha256":"a5e9e1d29ee439fdf6ed689ba80ed8ea3bcd152a90db1f4e73f972d8fb905aeb"},"schema_version":"1.0"},"canonical_sha256":"a0500fb82f814e5f00d3cf059c626998d74b3863954ffdcf5df622fbd3dd53b5","source":{"kind":"arxiv","id":"1904.06731","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06731","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06731v1","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06731","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"pith_short_12","alias_value":"UBIA7OBPQFHF","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"UBIA7OBPQFHF6AGT","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"UBIA7OBP","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:UBIA7OBPQFHF6AGTZ4CZYYTJTD","target":"record","payload":{"canonical_record":{"source":{"id":"1904.06731","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-04-14T17:52:29Z","cross_cats_sorted":[],"title_canon_sha256":"8e1bcc74157fa7bf3825488c80beaac66e3076cf91454141f574316fa3f7dfcc","abstract_canon_sha256":"a5e9e1d29ee439fdf6ed689ba80ed8ea3bcd152a90db1f4e73f972d8fb905aeb"},"schema_version":"1.0"},"canonical_sha256":"a0500fb82f814e5f00d3cf059c626998d74b3863954ffdcf5df622fbd3dd53b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:36.371621Z","signature_b64":"5X+Y0QD4oISbox/Re/PJp6EIiiXHpWTE1ZqMS5C29FL24o582ur6XsUzr/8avUhYwkcf/INv0CpDm1f8Jz4MDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0500fb82f814e5f00d3cf059c626998d74b3863954ffdcf5df622fbd3dd53b5","last_reissued_at":"2026-05-17T23:48:36.370961Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:36.370961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.06731","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9yJ10WuUBdtRczQ7Yo4BtSj5HFh5SwLnYcEf3i00nxgf6JEe/DPgCXDB+rASi/vgVgsaqqY/rXidytMAhI4JAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:27:47.354751Z"},"content_sha256":"27918678704c9eb31c39f7c48064324e50dbdc37f70361ec9d9598de2579552c","schema_version":"1.0","event_id":"sha256:27918678704c9eb31c39f7c48064324e50dbdc37f70361ec9d9598de2579552c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:UBIA7OBPQFHF6AGTZ4CZYYTJTD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a lattice characterization of finite soluble $PST$-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alexander N. Skiba, Zhang Chi","submitted_at":"2019-04-14T17:52:29Z","abstract_excerpt":"Let $\\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\\cal L}_{\\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\\in \\mathfrak{F}$. A chief factor $H/K$ of $G$ is $\\mathfrak{F}$-central in $G$ if $(H/K)\\rtimes (G/C_{G}(H/K)) \\in\\mathfrak{F}$.\n  We study the structure of $G$ under the hypothesis that every chief factor\n  of $G$ between $A_{G}$ and $A^{G}$ is $\\mathfrak{F}$-central in $G$ for every subgroup $A\\in {\\cal L}_{\\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06731","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TDyGp/FSLnwxB1XhGU7gM0pgC3S2nYo/UcV0zigtnorvr93gZltXeSw2/YK1Irao0D6eleHOSARa4/pVvm3ZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:27:47.355097Z"},"content_sha256":"cb46502ccc08b3d9998625aaf8b33bab2ec5df0a6a88b4afedf491982bc2bc55","schema_version":"1.0","event_id":"sha256:cb46502ccc08b3d9998625aaf8b33bab2ec5df0a6a88b4afedf491982bc2bc55"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/bundle.json","state_url":"https://pith.science/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/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":"2026-06-02T12:27:47Z","links":{"resolver":"https://pith.science/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD","bundle":"https://pith.science/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/bundle.json","state":"https://pith.science/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UBIA7OBPQFHF6AGTZ4CZYYTJTD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:UBIA7OBPQFHF6AGTZ4CZYYTJTD","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":"a5e9e1d29ee439fdf6ed689ba80ed8ea3bcd152a90db1f4e73f972d8fb905aeb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-04-14T17:52:29Z","title_canon_sha256":"8e1bcc74157fa7bf3825488c80beaac66e3076cf91454141f574316fa3f7dfcc"},"schema_version":"1.0","source":{"id":"1904.06731","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.06731","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"arxiv_version","alias_value":"1904.06731v1","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.06731","created_at":"2026-05-17T23:48:36Z"},{"alias_kind":"pith_short_12","alias_value":"UBIA7OBPQFHF","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"UBIA7OBPQFHF6AGT","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"UBIA7OBP","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:cb46502ccc08b3d9998625aaf8b33bab2ec5df0a6a88b4afedf491982bc2bc55","target":"graph","created_at":"2026-05-17T23:48:36Z","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 $\\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\\cal L}_{\\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\\in \\mathfrak{F}$. A chief factor $H/K$ of $G$ is $\\mathfrak{F}$-central in $G$ if $(H/K)\\rtimes (G/C_{G}(H/K)) \\in\\mathfrak{F}$.\n  We study the structure of $G$ under the hypothesis that every chief factor\n  of $G$ between $A_{G}$ and $A^{G}$ is $\\mathfrak{F}$-central in $G$ for every subgroup $A\\in {\\cal L}_{\\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\\leq ","authors_text":"Alexander N. Skiba, Zhang Chi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-04-14T17:52:29Z","title":"On a lattice characterization of finite soluble $PST$-groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.06731","kind":"arxiv","version":1},"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":"sha256:27918678704c9eb31c39f7c48064324e50dbdc37f70361ec9d9598de2579552c","target":"record","created_at":"2026-05-17T23:48:36Z","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":"a5e9e1d29ee439fdf6ed689ba80ed8ea3bcd152a90db1f4e73f972d8fb905aeb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-04-14T17:52:29Z","title_canon_sha256":"8e1bcc74157fa7bf3825488c80beaac66e3076cf91454141f574316fa3f7dfcc"},"schema_version":"1.0","source":{"id":"1904.06731","kind":"arxiv","version":1}},"canonical_sha256":"a0500fb82f814e5f00d3cf059c626998d74b3863954ffdcf5df622fbd3dd53b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a0500fb82f814e5f00d3cf059c626998d74b3863954ffdcf5df622fbd3dd53b5","first_computed_at":"2026-05-17T23:48:36.370961Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:36.370961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5X+Y0QD4oISbox/Re/PJp6EIiiXHpWTE1ZqMS5C29FL24o582ur6XsUzr/8avUhYwkcf/INv0CpDm1f8Jz4MDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:36.371621Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.06731","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27918678704c9eb31c39f7c48064324e50dbdc37f70361ec9d9598de2579552c","sha256:cb46502ccc08b3d9998625aaf8b33bab2ec5df0a6a88b4afedf491982bc2bc55"],"state_sha256":"c4b6ef8f7557dc16738e372d10ceebc4c4cd556bba08a502918baa0209a8b674"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1j+/ZBszi2Hv+FSIoezaN66/FzGEn4SadEKBe6N4I5hU4c2iwVo1cKaIVrjK92S+nNXLcfYkI4GuCsDMronOBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:27:47.357134Z","bundle_sha256":"678afe4f4304f3af0b2f215b8fed9d35cbb05a041641a9712bda33e3692307d9"}}