{"paper":{"title":"The profinite genus of the groups $\\mathbb{Z}^n\\rtimes C_{p^2}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A formula gives the profinite genus of every group of the form Z^n semidirect product C_{p squared}.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Anderson Porto, John MacQuarrie, Marlon Estanislau","submitted_at":"2025-11-27T17:49:49Z","abstract_excerpt":"A formula is given for the profinite genus of groups of the form $\\mathbb{Z}^n \\rtimes C_{p^2}$, completing the calculation of the size of the genus of semidirect products of the form $\\mathbb{Z}^n \\rtimes G$ where $G$ is a finite $p$-group of finite integral representation type."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A formula is given for the profinite genus of groups of the form Z^n ⋊ C_{p²}, completing the calculation of the size of the genus of semidirect products of the form Z^n ⋊ G where G is a finite p-group of finite integral representation type.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The calculation assumes that the groups are exactly of the stated semidirect product form and that the prior results on finite integral representation type for the p-groups remain valid without additional restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A formula is given for the profinite genus of Z^n ⋊ C_{p²}, completing the genus size calculation for Z^n ⋊ G where G is any finite p-group of finite integral representation type.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A formula gives the profinite genus of every group of the form Z^n semidirect product C_{p squared}.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"770b8da43904fd124d04f2b13fbc477b81035382605ad2d664ecdf082ab89a54"},"source":{"id":"2511.22658","kind":"arxiv","version":2},"verdict":{"id":"5c983ad0-dc96-4483-9cff-7e2ac8da8da6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T04:39:51.482546Z","strongest_claim":"A formula is given for the profinite genus of groups of the form Z^n ⋊ C_{p²}, completing the calculation of the size of the genus of semidirect products of the form Z^n ⋊ G where G is a finite p-group of finite integral representation type.","one_line_summary":"A formula is given for the profinite genus of Z^n ⋊ C_{p²}, completing the genus size calculation for Z^n ⋊ G where G is any finite p-group of finite integral representation type.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The calculation assumes that the groups are exactly of the stated semidirect product form and that the prior results on finite integral representation type for the p-groups remain valid without additional restrictions.","pith_extraction_headline":"A formula gives the profinite genus of every group of the form Z^n semidirect product C_{p squared}."},"references":{"count":18,"sample":[{"doi":"","year":1974,"title":"G. Baumslag. Residually finite groups with the same finite images.Compositio Mathematica, 29(3):249–252, 1974","work_id":"f864ac6f-000d-4b64-8b8e-6ac8bda069c9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"C. W. Curtis and I. Reiner.Methods of Representation Theory - with applications to finite groups and orders, volume I. John Wiley & Sons, New York, 1990","work_id":"da62559b-5e87-4dd4-a2f1-ee6c12d53d1b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"V . R. de Bessa, F. Grunewald, and P . A. Zalesskii. Genus for virtually surface groups and pullbacks.Manuscripta Mathematica, 145:221–233, 2014","work_id":"b6e4f7f0-1011-430b-9e51-3b96f2c211d1","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"V . R. de Bessa, A. L. P . Porto, and P . A. Zalesskii. The profinite completion of accessible groups.Monatsh. 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