{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:A2RDI7GKDPSZGEDDLO6E3AZ66D","short_pith_number":"pith:A2RDI7GK","schema_version":"1.0","canonical_sha256":"06a2347cca1be59310635bbc4d833ef0e2ddfe95ef02a054d7d977470c5c97e2","source":{"kind":"arxiv","id":"1109.5007","version":1},"attestation_state":"computed","paper":{"title":"Commutativity pattern of finite non-abelian $p$-groups determine their orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"A. Abdollahi, H. Dorbidi, H. Shahverdi, S. Akbari","submitted_at":"2011-09-23T06:48:35Z","abstract_excerpt":"Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $\\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\\setminus Z(G)$ as the vertices of $\\Gamma_G$ and join two distinct vertices $x$ and $y$, whenever $xy\\neq yx$. Here, we prove that \"the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups\"; this means that if $P$ is a finite non-abelian $p$-group such that $\\Gamma_P\\cong \\Gamma_H$ for some group $H$, then $|P|=|H|$."},"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":"1109.5007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-09-23T06:48:35Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"f2f1ab428c2198747a0e3f622b00a93a981e6ac682ef37524a86365503b68880","abstract_canon_sha256":"994cab2d64b6a1ac7989bff23b6e26b0717eadc6e25531e9c612a632a1209a1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:31.731265Z","signature_b64":"+j59V2b75V6VpSWWBv/f5ZEg4Bvs3XArMTwSXkvzl4UdHXBuq3ZZv5R3x+Lfn69UWw3ePxcs6whx3z29NH6mAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06a2347cca1be59310635bbc4d833ef0e2ddfe95ef02a054d7d977470c5c97e2","last_reissued_at":"2026-05-18T04:12:31.730512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:31.730512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutativity pattern of finite non-abelian $p$-groups determine their orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"A. Abdollahi, H. Dorbidi, H. Shahverdi, S. Akbari","submitted_at":"2011-09-23T06:48:35Z","abstract_excerpt":"Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph $\\Gamma_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\\setminus Z(G)$ as the vertices of $\\Gamma_G$ and join two distinct vertices $x$ and $y$, whenever $xy\\neq yx$. Here, we prove that \"the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups\"; this means that if $P$ is a finite non-abelian $p$-group such that $\\Gamma_P\\cong \\Gamma_H$ for some group $H$, then $|P|=|H|$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5007","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":"1109.5007","created_at":"2026-05-18T04:12:31.730637+00:00"},{"alias_kind":"arxiv_version","alias_value":"1109.5007v1","created_at":"2026-05-18T04:12:31.730637+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5007","created_at":"2026-05-18T04:12:31.730637+00:00"},{"alias_kind":"pith_short_12","alias_value":"A2RDI7GKDPSZ","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"A2RDI7GKDPSZGEDD","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"A2RDI7GK","created_at":"2026-05-18T12:26:22.705136+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/A2RDI7GKDPSZGEDDLO6E3AZ66D","json":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D.json","graph_json":"https://pith.science/api/pith-number/A2RDI7GKDPSZGEDDLO6E3AZ66D/graph.json","events_json":"https://pith.science/api/pith-number/A2RDI7GKDPSZGEDDLO6E3AZ66D/events.json","paper":"https://pith.science/paper/A2RDI7GK"},"agent_actions":{"view_html":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D","download_json":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D.json","view_paper":"https://pith.science/paper/A2RDI7GK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1109.5007&json=true","fetch_graph":"https://pith.science/api/pith-number/A2RDI7GKDPSZGEDDLO6E3AZ66D/graph.json","fetch_events":"https://pith.science/api/pith-number/A2RDI7GKDPSZGEDDLO6E3AZ66D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D/action/storage_attestation","attest_author":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D/action/author_attestation","sign_citation":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D/action/citation_signature","submit_replication":"https://pith.science/pith/A2RDI7GKDPSZGEDDLO6E3AZ66D/action/replication_record"}},"created_at":"2026-05-18T04:12:31.730637+00:00","updated_at":"2026-05-18T04:12:31.730637+00:00"}