---
tag: CAAPB5WJSV
kind: concept
name: Cotorsion group
aliases: [cotorsion, cotorsion-module]
sources:
  - wikipedia: en:5739241@r1366628056
    locator: Cotorsion group
    note: adopted
    mention: wikipedia://en/5739241
    sections: [Summary]
same_as:
  wikidata: Q5175462
  wikipedia: Cotorsion group
inventory: [Q5BHSM8Y72]
---

## Summary

In [[abelian|071DDFVG4J]] group theory, an abelian group is said to be **cotorsion** if any extension of it by a torsion-free abelian group splits. If the group is $M$, this says that the Ext-group $\operatorname{Ext}_\mathbb{Z}(F,M)$ is zero for all torsion-free groups $F$. Since any such $F$ embeds into a direct sum of copies of $\mathbb{Q}$, it suffices to check the condition on the group of [[rational numbers|0NCJ5WXN6Z]].