{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2LA6G4VBHJB4DT4O7OFXGRPUCF","short_pith_number":"pith:2LA6G4VB","canonical_record":{"source":{"id":"1304.0143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-30T22:21:15Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"a2398664277a360ced08dbb9fe4fddc9fb8b0a0e7db34eb053c94bb29872da7c","abstract_canon_sha256":"1e033bbc4b5a74da9e3a9770a2f6b12cbe84db1c117ab010b460696655cebb70"},"schema_version":"1.0"},"canonical_sha256":"d2c1e372a13a43c1cf8efb8b7345f4114ad54a5c6f7dce8634bef1d852f4a22c","source":{"kind":"arxiv","id":"1304.0143","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.0143","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1304.0143v1","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.0143","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"2LA6G4VBHJB4","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"2LA6G4VBHJB4DT4O","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"2LA6G4VB","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2LA6G4VBHJB4DT4O7OFXGRPUCF","target":"record","payload":{"canonical_record":{"source":{"id":"1304.0143","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-30T22:21:15Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"a2398664277a360ced08dbb9fe4fddc9fb8b0a0e7db34eb053c94bb29872da7c","abstract_canon_sha256":"1e033bbc4b5a74da9e3a9770a2f6b12cbe84db1c117ab010b460696655cebb70"},"schema_version":"1.0"},"canonical_sha256":"d2c1e372a13a43c1cf8efb8b7345f4114ad54a5c6f7dce8634bef1d852f4a22c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:21.074850Z","signature_b64":"AsKY8+xw1hTtjFOXhADEbmM4ja/+X+iNaG/KXIapxBzeTNqgwNwr/Q3+WqVgyKnfw2B5D7ZYJqRF6dKHO8K9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2c1e372a13a43c1cf8efb8b7345f4114ad54a5c6f7dce8634bef1d852f4a22c","last_reissued_at":"2026-05-18T02:28:21.074278Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:21.074278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.0143","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":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CjulHwzIuwM9IwpNC84ayJ45ytUpq8xAl9u50dyaOvjALcsPQAUxbMiZZNw2nmEr7rjaLBr/2pn2gYdaUqJZCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:56:56.834499Z"},"content_sha256":"99e23d900e290883dc624180b7253a7e16a6f0d5a2d2a504afe4782619aab20c","schema_version":"1.0","event_id":"sha256:99e23d900e290883dc624180b7253a7e16a6f0d5a2d2a504afe4782619aab20c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2LA6G4VBHJB4DT4O7OFXGRPUCF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Which alternating and symmetric groups are unit groups?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RT"],"primary_cat":"math.RA","authors_text":"Christopher Davis, Tommy Occhipinti","submitted_at":"2013-03-30T22:21:15Z","abstract_excerpt":"We prove there is no ring with unit group isomorphic to S_n for n \\geq 5 and that there is no ring with unit group isomorphic to A_n for n \\geq 5, n \\neq 8. We give examples of rings with unit groups isomorphic to S_1, S_2, S_3, S_4, A_1, A_2, A_3, A_4, and A_8. We expect our methods to work similarly for other groups with trivial center; in particular, we plan to consider other simple groups in later work."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0143","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":"2026-05-18T02:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y5cZhukvb008s0MJprPkZI0DyCdULKjl4l72DtrlIFxH7s38etvdXQBaptz8covFg/U/INuH9z2PRv4FTHFlBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:56:56.834859Z"},"content_sha256":"4393a6d3c9ecafc6caa6c7cc17d299474e79c1702584069bc87fc678948c4a12","schema_version":"1.0","event_id":"sha256:4393a6d3c9ecafc6caa6c7cc17d299474e79c1702584069bc87fc678948c4a12"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/bundle.json","state_url":"https://pith.science/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/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":"2026-06-22T14:56:56Z","links":{"resolver":"https://pith.science/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF","bundle":"https://pith.science/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/bundle.json","state":"https://pith.science/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2LA6G4VBHJB4DT4O7OFXGRPUCF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2LA6G4VBHJB4DT4O7OFXGRPUCF","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":"1e033bbc4b5a74da9e3a9770a2f6b12cbe84db1c117ab010b460696655cebb70","cross_cats_sorted":["math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-30T22:21:15Z","title_canon_sha256":"a2398664277a360ced08dbb9fe4fddc9fb8b0a0e7db34eb053c94bb29872da7c"},"schema_version":"1.0","source":{"id":"1304.0143","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.0143","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1304.0143v1","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.0143","created_at":"2026-05-18T02:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"2LA6G4VBHJB4","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"2LA6G4VBHJB4DT4O","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"2LA6G4VB","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:4393a6d3c9ecafc6caa6c7cc17d299474e79c1702584069bc87fc678948c4a12","target":"graph","created_at":"2026-05-18T02:28:21Z","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":"We prove there is no ring with unit group isomorphic to S_n for n \\geq 5 and that there is no ring with unit group isomorphic to A_n for n \\geq 5, n \\neq 8. We give examples of rings with unit groups isomorphic to S_1, S_2, S_3, S_4, A_1, A_2, A_3, A_4, and A_8. We expect our methods to work similarly for other groups with trivial center; in particular, we plan to consider other simple groups in later work.","authors_text":"Christopher Davis, Tommy Occhipinti","cross_cats":["math.GR","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-30T22:21:15Z","title":"Which alternating and symmetric groups are unit groups?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0143","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":"sha256:99e23d900e290883dc624180b7253a7e16a6f0d5a2d2a504afe4782619aab20c","target":"record","created_at":"2026-05-18T02:28:21Z","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":"1e033bbc4b5a74da9e3a9770a2f6b12cbe84db1c117ab010b460696655cebb70","cross_cats_sorted":["math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-03-30T22:21:15Z","title_canon_sha256":"a2398664277a360ced08dbb9fe4fddc9fb8b0a0e7db34eb053c94bb29872da7c"},"schema_version":"1.0","source":{"id":"1304.0143","kind":"arxiv","version":1}},"canonical_sha256":"d2c1e372a13a43c1cf8efb8b7345f4114ad54a5c6f7dce8634bef1d852f4a22c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d2c1e372a13a43c1cf8efb8b7345f4114ad54a5c6f7dce8634bef1d852f4a22c","first_computed_at":"2026-05-18T02:28:21.074278Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:21.074278Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AsKY8+xw1hTtjFOXhADEbmM4ja/+X+iNaG/KXIapxBzeTNqgwNwr/Q3+WqVgyKnfw2B5D7ZYJqRF6dKHO8K9Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:21.074850Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.0143","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99e23d900e290883dc624180b7253a7e16a6f0d5a2d2a504afe4782619aab20c","sha256:4393a6d3c9ecafc6caa6c7cc17d299474e79c1702584069bc87fc678948c4a12"],"state_sha256":"7cebdffe18d74a3002eabef8bb516eced69bc4eb2dcb6c293e0d8fd4b6caccc4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8JyoIHFFZ4djcIr+T85aFAraxs9ppSZ8yEAknN04PeQV1ylbD1oXfHp95PPElyOU7Grk+0t07QjFIu+Pum9vDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T14:56:56.837057Z","bundle_sha256":"0484e77b8dcc8355eb3772ef6cf4822d15794b300993276440f38985c37cdf68"}}