{
  "abstract": "Let $G$ and $H$ be torsion-free nilpotent groups of arbitrary cardinality, without any uniform upper bound imposed on their cardinality, torsion-free rank, number of generators, or nilpotency class. We prove that if the classical Ore division rings of fractions of their rational group algebras are isomorphic as rings, namely \\[ D_{\\mathbb{Q}}(G) := \\operatorname{Frac}_{\\mathrm{Ore}} \\mathbb{Q}[G] \\cong \\operatorname{Frac}_{\\mathrm{Ore}} \\mathbb{Q}[H] =: D_{\\mathbb{Q}}(H), \\] then $G$ and $H$ are isomorphic as groups ($G \\cong H$). This result extends directly to $k$-algebra isomorphisms $D_k(G) \\cong_k D_k(H)$ over any base field $k$ of characteristic zero, as well as to arbitrary torsion-free hypercentral groups (and, more generally, to torsion-free groups satisfying bi-orderability, the Ore condition, and nontrivial intersection of normal subgroups with the center). A key tool in our proof is a weight-preserving isomorphism theorem for rational function fields of free abelian groups: let $P$ and $Q$ be free abelian groups of arbitrary rank equipped with surjective homomorphisms onto a common abelian group $B$; if their rational function fields over a field $k$ containing an algebraically closed subfield of characteristic zero admit a $B$-weight-preserving field isomorphism $k(P) \\cong_B k(Q)$, then there exists a weight-preserving group isomorphism $P \\cong_B Q$. By combining the Malcev--Neumann power series division ring, lowest-degree coset extraction over central localizations, the lowest-term rational coefficient homomorphism, and an embedding into a torsion-free abelianized auxiliary function field, we reduce the division ring isomorphism to a weight-preserving rational function field isomorphism between the quotients $G/L$ and $H/L$, and explicitly construct a group isomorphism $G \\cong H$ via a central modification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34116",
  "author_total": 2,
  "authors": [
    {
      "name": "Jinyu Lin"
    },
    {
      "name": "Xiaodong Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/39cbez4nh8xy9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34116",
  "fulltext_url": null,
  "id": "39cbez4nh8xy9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34116",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7223980642",
  "pdf_url": "https://arxiv.org/pdf/2610.34116",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/39cbez4nh8xy9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34116",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34116"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Reconstruction of the Torsion Free Nilpotent Group from Its Division Ring of the Rational Group Algebra",
  "title_text": "Reconstruction of the Torsion Free Nilpotent Group from Its Division Ring of the Rational Group Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/39cbez4nh8xy9",
  "venue": "arXiv",
  "versions": []
}