Torsion-free Abelian Groups Revisited

TL;DR

This paper revisits the structure of torsion-free abelian groups of finite rank, focusing on automorphism actions and classification of indecomposable components, providing new insights into their decomposition properties.

Contribution

It introduces a classification of finite rank strongly indecomposable torsion-free abelian groups based on automorphism group actions and indecomposable decompositions.

Findings

Automorphism group orbits correspond to indecomposable decompositions.

Existence of a characteristic subgroup of finite index that is a direct sum of strongly indecomposable groups.

Provides a classification framework for finite rank strongly indecomposable torsion-free abelian groups.

Abstract

Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G contains a direct sum of strongly indecomposable groups as a characteristic subgroup of finite index, giving rise to a classification of finite rank strongly indecomposable torsion--free abelian groups.