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These are presented \\[G(m,n,k) = \\left\\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \\right\\rangle \\quad (m,n,k\\in\\mathbb{Z}^+)\\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\\,\\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \\emph{right} and \\emph{left commutation semigroups of} $G,$ denoted ${\\rm P}(G)$ and $\\Lambda (G),$ are the semigroups of mappings generated by $\\rho (g):G \\to G$ and $\\lambda (g):G \\to G$ defi"},"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":"1807.10389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-07-26T22:36:13Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"55193aa25cdd4aa55a23cd94d8019f00e3078316c829a697010e8a90f9a6ef6d","abstract_canon_sha256":"2b075aae49b0edc98ed8c06678c79aae51f3f6844c2bf8e0d4afb499daf4153c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:40.752639Z","signature_b64":"7OB59QimvoxCJqtVaHeWePK0WV0qd1w7PVe6tu12v4PWhKAG3jQtlWYOs6Z54zsGRNU7QS4pS7H4vURp/0A7AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a52ac134c1270a746564ade43ea81a39189d4c3e0a1e6360e962545057a2fd0","last_reissued_at":"2026-05-18T00:09:40.752031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:40.752031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutation Semigroups of Finite Metacyclic Groups with Trivial Centre","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Charles C. Edmunds, Darien DeWolf","submitted_at":"2018-07-26T22:36:13Z","abstract_excerpt":"We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \\[G(m,n,k) = \\left\\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \\right\\rangle \\quad (m,n,k\\in\\mathbb{Z}^+)\\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\\,\\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \\emph{right} and \\emph{left commutation semigroups of} $G,$ denoted ${\\rm P}(G)$ and $\\Lambda (G),$ are the semigroups of mappings generated by $\\rho (g):G \\to G$ and $\\lambda (g):G \\to G$ defi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10389","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":"1807.10389","created_at":"2026-05-18T00:09:40.752113+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.10389v1","created_at":"2026-05-18T00:09:40.752113+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10389","created_at":"2026-05-18T00:09:40.752113+00:00"},{"alias_kind":"pith_short_12","alias_value":"HJJKYE2MCJYK","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HJJKYE2MCJYKORSW","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HJJKYE2M","created_at":"2026-05-18T12:32:28.185984+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/HJJKYE2MCJYKORSWJLPEH2UBUO","json":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO.json","graph_json":"https://pith.science/api/pith-number/HJJKYE2MCJYKORSWJLPEH2UBUO/graph.json","events_json":"https://pith.science/api/pith-number/HJJKYE2MCJYKORSWJLPEH2UBUO/events.json","paper":"https://pith.science/paper/HJJKYE2M"},"agent_actions":{"view_html":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO","download_json":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO.json","view_paper":"https://pith.science/paper/HJJKYE2M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.10389&json=true","fetch_graph":"https://pith.science/api/pith-number/HJJKYE2MCJYKORSWJLPEH2UBUO/graph.json","fetch_events":"https://pith.science/api/pith-number/HJJKYE2MCJYKORSWJLPEH2UBUO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO/action/storage_attestation","attest_author":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO/action/author_attestation","sign_citation":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO/action/citation_signature","submit_replication":"https://pith.science/pith/HJJKYE2MCJYKORSWJLPEH2UBUO/action/replication_record"}},"created_at":"2026-05-18T00:09:40.752113+00:00","updated_at":"2026-05-18T00:09:40.752113+00:00"}