Reconstruction of the Torsion Free Nilpotent Group from Its Division Ring of the Rational Group Algebra
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 Dℚ(G) := FracOre ℚ[G] ≅ FracOre ℚ[H] =: Dℚ(H), then G and H are isomorphic as groups (G ≅ H). This result extends directly to k-algebra isomorphisms Dk(G) ≅k Dk(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) ≅B k(Q), then there exists a weight-preserving group isomorphism P ≅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 ≅ H via a central modification.
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BibTeX
@misc{lin2026reconstruction,
title = {{Reconstruction of the Torsion Free Nilpotent Group from Its Division Ring of the Rational Group Algebra}},
author = {Jinyu Lin and Xiaodong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34116},
eprint = {2610.34116},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34116},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reconstruction of the Torsion Free Nilpotent Group from Its Division Ring of the Rational Group Algebra
AU - Jinyu Lin
AU - Xiaodong Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34116
UR - https://arxiv.org/abs/2610.34116
AB - 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_ℚ(G) := Frac_Ore ℚ[G] ≅ Frac_Ore ℚ[H] =: D_ℚ(H), then G and H are isomorphic as groups (G ≅ H). This result extends directly to k-algebra isomorphisms D_k(G) ≅_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) ≅_B k(Q), then there exists a weight-preserving group isomorphism P ≅_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 ≅ H via a central modification.
ER -