# Cotorsion group

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In mathematics, in the realm of abelian group theory, an abelian group is said to be **cotorsion** if every extension of it by a torsion-free group splits. If the group is , this is equivalent to asserting that for all torsion-free groups . It suffices to check the condition for being the group of rational numbers.

Some properties of cotorsion groups:

- Any quotient of a cotorsion group is cotorsion.
- A direct product of groups is cotorsion if and only if each factor is.
- Every divisible group or injective group is cotorsion.
- The
**Baer Fomin Theorem**states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group, that is, a group of bounded exponent. - A torsion-free abelian group is cotorsion if and only if it is algebraically compact.
- Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.

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