{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Q6MD65L6VNDZQ23TCE3LHK2NT7","short_pith_number":"pith:Q6MD65L6","canonical_record":{"source":{"id":"1506.06450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-22T03:25:16Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"445a318d69b31e78600794a957da68f0a71c026151f5b466a9f835d25ad3e11e","abstract_canon_sha256":"c84878533ce93bfe2d34091fd28541339cb8c890d1d6412c80449b62b4faebf3"},"schema_version":"1.0"},"canonical_sha256":"87983f757eab47986b731136b3ab4d9fdd312bfa9a046e2904670ca6e740a0b3","source":{"kind":"arxiv","id":"1506.06450","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06450","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06450v1","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06450","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"pith_short_12","alias_value":"Q6MD65L6VNDZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q6MD65L6VNDZQ23T","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q6MD65L6","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Q6MD65L6VNDZQ23TCE3LHK2NT7","target":"record","payload":{"canonical_record":{"source":{"id":"1506.06450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-22T03:25:16Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"445a318d69b31e78600794a957da68f0a71c026151f5b466a9f835d25ad3e11e","abstract_canon_sha256":"c84878533ce93bfe2d34091fd28541339cb8c890d1d6412c80449b62b4faebf3"},"schema_version":"1.0"},"canonical_sha256":"87983f757eab47986b731136b3ab4d9fdd312bfa9a046e2904670ca6e740a0b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:45.191067Z","signature_b64":"wBDTk3Bbaqb8RIy3d5Cxu1L0IIdEYC7k6Pk1viCZSE7NcFyKt8/UKlHBzRTvUlX7qsckHTUEyDxLuKabSUrWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87983f757eab47986b731136b3ab4d9fdd312bfa9a046e2904670ca6e740a0b3","last_reissued_at":"2026-05-18T01:41:45.190495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:45.190495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.06450","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":"2026-05-18T01:41:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I4eWGlUflFGpK/vO5VpTahhkdhRTzXC/tVjfpxd2yfErlVQV2axynxXxG3LnE98CQAH2k5BYCKaRgd8eC0yZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:39:58.952892Z"},"content_sha256":"18fb2d7320ccb13bbe85e8276673e5d9402270ae1f4237061a9714a4e23c0e38","schema_version":"1.0","event_id":"sha256:18fb2d7320ccb13bbe85e8276673e5d9402270ae1f4237061a9714a4e23c0e38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Q6MD65L6VNDZQ23TCE3LHK2NT7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Characters of p'-degree and Thompson's character degree theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Nguyen Ngoc Hung","submitted_at":"2015-06-22T03:25:16Z","abstract_excerpt":"A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a significant improvement of this result by considering the average of $p'$-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06450","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":"2026-05-18T01:41:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5PP5Xm3cpryeQHiZpe45npyav8drU/dpqMUXull/uali55dIMC882DE1tG190+23vA+HXs+E76mh62BvfRffBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T03:39:58.953244Z"},"content_sha256":"0bc28712324eae48f06710886be8929aacbfbed3b6b6b77f9db98eab1e50d587","schema_version":"1.0","event_id":"sha256:0bc28712324eae48f06710886be8929aacbfbed3b6b6b77f9db98eab1e50d587"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/bundle.json","state_url":"https://pith.science/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/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":"2026-06-02T03:39:58Z","links":{"resolver":"https://pith.science/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7","bundle":"https://pith.science/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/bundle.json","state":"https://pith.science/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Q6MD65L6VNDZQ23TCE3LHK2NT7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Q6MD65L6VNDZQ23TCE3LHK2NT7","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":"c84878533ce93bfe2d34091fd28541339cb8c890d1d6412c80449b62b4faebf3","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-22T03:25:16Z","title_canon_sha256":"445a318d69b31e78600794a957da68f0a71c026151f5b466a9f835d25ad3e11e"},"schema_version":"1.0","source":{"id":"1506.06450","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.06450","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"arxiv_version","alias_value":"1506.06450v1","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.06450","created_at":"2026-05-18T01:41:45Z"},{"alias_kind":"pith_short_12","alias_value":"Q6MD65L6VNDZ","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q6MD65L6VNDZQ23T","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q6MD65L6","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:0bc28712324eae48f06710886be8929aacbfbed3b6b6b77f9db98eab1e50d587","target":"graph","created_at":"2026-05-18T01:41:45Z","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":"A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a significant improvement of this result by considering the average of $p'$-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.","authors_text":"Nguyen Ngoc Hung","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-22T03:25:16Z","title":"Characters of p'-degree and Thompson's character degree theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.06450","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":"sha256:18fb2d7320ccb13bbe85e8276673e5d9402270ae1f4237061a9714a4e23c0e38","target":"record","created_at":"2026-05-18T01:41:45Z","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":"c84878533ce93bfe2d34091fd28541339cb8c890d1d6412c80449b62b4faebf3","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-06-22T03:25:16Z","title_canon_sha256":"445a318d69b31e78600794a957da68f0a71c026151f5b466a9f835d25ad3e11e"},"schema_version":"1.0","source":{"id":"1506.06450","kind":"arxiv","version":1}},"canonical_sha256":"87983f757eab47986b731136b3ab4d9fdd312bfa9a046e2904670ca6e740a0b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87983f757eab47986b731136b3ab4d9fdd312bfa9a046e2904670ca6e740a0b3","first_computed_at":"2026-05-18T01:41:45.190495Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:45.190495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wBDTk3Bbaqb8RIy3d5Cxu1L0IIdEYC7k6Pk1viCZSE7NcFyKt8/UKlHBzRTvUlX7qsckHTUEyDxLuKabSUrWCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:45.191067Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.06450","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:18fb2d7320ccb13bbe85e8276673e5d9402270ae1f4237061a9714a4e23c0e38","sha256:0bc28712324eae48f06710886be8929aacbfbed3b6b6b77f9db98eab1e50d587"],"state_sha256":"3705a1ac1428c48b8ed73e4756f8927e017b406e07f96a631fcee37df8961f3d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1Piv8ReVCmo4SgxALB+JM5/ZfGIAyx/tIDfO8Ae3H2XKMMFxTpF7AkPFq+rKVQVPYRzQfHFggSWQ9H5TL0cgDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T03:39:58.955471Z","bundle_sha256":"d69dcdc984cda92c783017c874c725489319187b010734889637f26de739d060"}}