{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:GIDJXZSDOVVDSMDXDRTAYWSM46","short_pith_number":"pith:GIDJXZSD","canonical_record":{"source":{"id":"1402.0799","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-04T17:19:58Z","cross_cats_sorted":[],"title_canon_sha256":"a98eaa00f2cc7e9fdaca4847d5153a0992f8aa9b4fa9d9b9d0226c7c88e390e8","abstract_canon_sha256":"b735e771f9980ee972e58cf224f1712194a5aa5571a5195e0ceff48e6125b139"},"schema_version":"1.0"},"canonical_sha256":"32069be643756a3930771c660c5a4ce785b258fd16976e0cdb5f968857edcb9a","source":{"kind":"arxiv","id":"1402.0799","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.0799","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"arxiv_version","alias_value":"1402.0799v2","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.0799","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"pith_short_12","alias_value":"GIDJXZSDOVVD","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GIDJXZSDOVVDSMDX","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GIDJXZSD","created_at":"2026-05-18T12:28:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:GIDJXZSDOVVDSMDXDRTAYWSM46","target":"record","payload":{"canonical_record":{"source":{"id":"1402.0799","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-04T17:19:58Z","cross_cats_sorted":[],"title_canon_sha256":"a98eaa00f2cc7e9fdaca4847d5153a0992f8aa9b4fa9d9b9d0226c7c88e390e8","abstract_canon_sha256":"b735e771f9980ee972e58cf224f1712194a5aa5571a5195e0ceff48e6125b139"},"schema_version":"1.0"},"canonical_sha256":"32069be643756a3930771c660c5a4ce785b258fd16976e0cdb5f968857edcb9a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:29.555391Z","signature_b64":"sbVnzLCt51miqLL5tkuBdsdB08EqpdOQPX1QOMQmjuM0+MSbHHVM5ctRAwtm+lFf1z6hQO82MpoaoSaQwQ3zDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32069be643756a3930771c660c5a4ce785b258fd16976e0cdb5f968857edcb9a","last_reissued_at":"2026-05-18T01:01:29.554947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:29.554947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.0799","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z/L9X/+e27AiBucRYIJrec66sHrsh0svzjmpKKjT6Z4hrWf4TfzOXeRvKYFNqU1+Ri8pXeEtTujpGNKojWSsAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:14:44.118944Z"},"content_sha256":"f0511e1b99f17d814a0dd0820c06c5f35f66dadbecf5519932efab0d27a701b1","schema_version":"1.0","event_id":"sha256:f0511e1b99f17d814a0dd0820c06c5f35f66dadbecf5519932efab0d27a701b1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:GIDJXZSDOVVDSMDXDRTAYWSM46","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Transversals as generating sets in finitely generated groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Jack Button, Mariano Zeron-Medina Laris, Maurice Chiodo","submitted_at":"2014-02-04T17:19:58Z","abstract_excerpt":"We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$, and that the construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n \\leq3$, there is a simultaneous left-right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank $n$ group $G$ with in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0799","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OJlCI77e9MVpLaWCpuChR3shWwz7Cqy5gjXKyvAFyoV/ML2ObyfoUuEuAgGPvUEDEnOANe/6c3EE3SL4a/fkCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T22:14:44.119284Z"},"content_sha256":"809fe3726d83addeb559b111cf9e2ff92bac1a1570f5ba035e6abdedf87275bf","schema_version":"1.0","event_id":"sha256:809fe3726d83addeb559b111cf9e2ff92bac1a1570f5ba035e6abdedf87275bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/bundle.json","state_url":"https://pith.science/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/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":"2026-06-03T22:14:44Z","links":{"resolver":"https://pith.science/pith/GIDJXZSDOVVDSMDXDRTAYWSM46","bundle":"https://pith.science/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/bundle.json","state":"https://pith.science/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GIDJXZSDOVVDSMDXDRTAYWSM46/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:GIDJXZSDOVVDSMDXDRTAYWSM46","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":"b735e771f9980ee972e58cf224f1712194a5aa5571a5195e0ceff48e6125b139","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-04T17:19:58Z","title_canon_sha256":"a98eaa00f2cc7e9fdaca4847d5153a0992f8aa9b4fa9d9b9d0226c7c88e390e8"},"schema_version":"1.0","source":{"id":"1402.0799","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.0799","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"arxiv_version","alias_value":"1402.0799v2","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.0799","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"pith_short_12","alias_value":"GIDJXZSDOVVD","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_16","alias_value":"GIDJXZSDOVVDSMDX","created_at":"2026-05-18T12:28:30Z"},{"alias_kind":"pith_short_8","alias_value":"GIDJXZSD","created_at":"2026-05-18T12:28:30Z"}],"graph_snapshots":[{"event_id":"sha256:809fe3726d83addeb559b111cf9e2ff92bac1a1570f5ba035e6abdedf87275bf","target":"graph","created_at":"2026-05-18T01:01:29Z","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 explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$, and that the construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n \\leq3$, there is a simultaneous left-right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank $n$ group $G$ with in","authors_text":"Jack Button, Mariano Zeron-Medina Laris, Maurice Chiodo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-04T17:19:58Z","title":"Transversals as generating sets in finitely generated groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0799","kind":"arxiv","version":2},"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:f0511e1b99f17d814a0dd0820c06c5f35f66dadbecf5519932efab0d27a701b1","target":"record","created_at":"2026-05-18T01:01:29Z","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":"b735e771f9980ee972e58cf224f1712194a5aa5571a5195e0ceff48e6125b139","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-04T17:19:58Z","title_canon_sha256":"a98eaa00f2cc7e9fdaca4847d5153a0992f8aa9b4fa9d9b9d0226c7c88e390e8"},"schema_version":"1.0","source":{"id":"1402.0799","kind":"arxiv","version":2}},"canonical_sha256":"32069be643756a3930771c660c5a4ce785b258fd16976e0cdb5f968857edcb9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"32069be643756a3930771c660c5a4ce785b258fd16976e0cdb5f968857edcb9a","first_computed_at":"2026-05-18T01:01:29.554947Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:29.554947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sbVnzLCt51miqLL5tkuBdsdB08EqpdOQPX1QOMQmjuM0+MSbHHVM5ctRAwtm+lFf1z6hQO82MpoaoSaQwQ3zDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:29.555391Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.0799","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0511e1b99f17d814a0dd0820c06c5f35f66dadbecf5519932efab0d27a701b1","sha256:809fe3726d83addeb559b111cf9e2ff92bac1a1570f5ba035e6abdedf87275bf"],"state_sha256":"4b79cb9ebca469040b5a10aa64bb935ca807caa3f8898f8ba34f3be82dc3e2c1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FII4WmOzAxNcYg+p1P91jlb8qnF2z6iSkG0mNrEf8J99PbdPKLSxtaNtMWCxSxeN5HivqZsaNc2P9ilopPC+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T22:14:44.121163Z","bundle_sha256":"7c3a40fdc5da78cf1f71e21070079b6c223f59ac1ef0e73efc9693fa5bbf49fb"}}