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 \[ D_{\mathbb{Q}}(G) := \operatorname{Frac}_{\mathrm{Ore}} \mathbb{Q}[G] \cong \operatorname{Frac}_{\mathrm{Ore}} \mathbb{Q}[H] =: D_{\mathbb{Q}}(H), \] 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.