{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:QXEVAAW3VT3ZJURUTHHHCVXZLC","short_pith_number":"pith:QXEVAAW3","schema_version":"1.0","canonical_sha256":"85c95002dbacf794d23499ce7156f958ad79c61df35f57242a67a44ba2033cdc","source":{"kind":"arxiv","id":"1701.05947","version":1},"attestation_state":"computed","paper":{"title":"On the Symmetry of Images of Word Maps in Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Meng-Che Ho, William Cocke","submitted_at":"2017-01-20T23:24:15Z","abstract_excerpt":"Word maps in a group, an analogue of polynomials in groups, are defined by substitution of formal words. Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We explore sufficient conditions on a group that ensure that the image of all word maps on G are closed under inversion. We then show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inve"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1701.05947","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-20T23:24:15Z","cross_cats_sorted":[],"title_canon_sha256":"3b8103db9e5692861a7c2d3d5b52b993f47121e38fd5fce58d992d82f3b772f4","abstract_canon_sha256":"508308d50c912b4dcd9df0c1c0da09631cf327ed168ee562bc65340d05136de5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:19.771735Z","signature_b64":"3wxBkX9aR5q8ZpuKzsAa23kDEKiTkL4fsBNuRtgz6rTACd0XqWt0uEKS9P0eprhNVAAmv4SOdy+8ST1oRB56BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85c95002dbacf794d23499ce7156f958ad79c61df35f57242a67a44ba2033cdc","last_reissued_at":"2026-05-18T00:52:19.771139Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:19.771139Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Symmetry of Images of Word Maps in Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Meng-Che Ho, William Cocke","submitted_at":"2017-01-20T23:24:15Z","abstract_excerpt":"Word maps in a group, an analogue of polynomials in groups, are defined by substitution of formal words. Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We explore sufficient conditions on a group that ensure that the image of all word maps on G are closed under inversion. We then show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05947","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1701.05947","created_at":"2026-05-18T00:52:19.771246+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05947v1","created_at":"2026-05-18T00:52:19.771246+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05947","created_at":"2026-05-18T00:52:19.771246+00:00"},{"alias_kind":"pith_short_12","alias_value":"QXEVAAW3VT3Z","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_16","alias_value":"QXEVAAW3VT3ZJURU","created_at":"2026-05-18T12:31:39.905425+00:00"},{"alias_kind":"pith_short_8","alias_value":"QXEVAAW3","created_at":"2026-05-18T12:31:39.905425+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC","json":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC.json","graph_json":"https://pith.science/api/pith-number/QXEVAAW3VT3ZJURUTHHHCVXZLC/graph.json","events_json":"https://pith.science/api/pith-number/QXEVAAW3VT3ZJURUTHHHCVXZLC/events.json","paper":"https://pith.science/paper/QXEVAAW3"},"agent_actions":{"view_html":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC","download_json":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC.json","view_paper":"https://pith.science/paper/QXEVAAW3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05947&json=true","fetch_graph":"https://pith.science/api/pith-number/QXEVAAW3VT3ZJURUTHHHCVXZLC/graph.json","fetch_events":"https://pith.science/api/pith-number/QXEVAAW3VT3ZJURUTHHHCVXZLC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC/action/storage_attestation","attest_author":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC/action/author_attestation","sign_citation":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC/action/citation_signature","submit_replication":"https://pith.science/pith/QXEVAAW3VT3ZJURUTHHHCVXZLC/action/replication_record"}},"created_at":"2026-05-18T00:52:19.771246+00:00","updated_at":"2026-05-18T00:52:19.771246+00:00"}