# Freely indecomposable almost free groups with free abelianization

## TL;DR

This paper constructs large, freely indecomposable groups with free abelianization at uncountable cardinals within G"odel's universe, advancing understanding of group structures with specific algebraic properties.

## Contribution

It introduces a method to build such groups for uncountable cardinals in G"odel's universe, strengthening previous results by Eklof and Mekler.

## Key findings

- Existence of groups with specified properties at uncountable cardinals
- Construction within G"odel's constructible universe $L$
- Enhancement of earlier theoretical results

## Abstract

For certain uncountable cardinals $\kappa$ we produce a group of cardinality $\kappa$ which is freely indecomposable, strongly $\kappa$-free, and whose abelianization is free abelian of rank $\kappa$. The construction takes place in G\"odel's constructible universe $L$. This strengthens an earlier result of Eklof and Mekler.