On abelian subgroups of finitely generated metabelian groups

TL;DR

This paper introduces Hall groups to classify abelian subgroups within finitely generated metabelian groups, providing explicit classifications and counterexamples to a longstanding conjecture.

Contribution

It defines Hall groups and uses them to classify all abelian subgroups in finitely-generated metabelian groups, addressing a conjecture by Baumslag.

Findings

Explicit classification of abelian subgroups in finitely-generated metabelian groups

Negative resolution of Baumslag's conjecture on abelian subgroup cardinality

Introduction of Hall groups as a tool for subgroup analysis

Abstract

In this note we introduce the class of $\mathcal H$-groups (or Hall groups) related to the class of $\mathcal B$-groups defined by Ph. Hall in 1950's. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely-generated metabelian groups. Hall groups allow to give a negative answer to the Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups.