Algebraically compact group

In mathematics, in the realm of abelian group theory, a group is said to be algebraically compact if it is a direct summand of every abelian group containing it as a pure subgroup.

Equivalent characterizations of algebraic compactness:

• The reduced part of the group is Hausdorff and complete in the ${\displaystyle \mathbb {Z} }$ adic topology.
• The group is pure injective, that is, injective with respect to exact sequences where the embedding is as a pure subgroup.

Relations with other properties:

• A torsion-free group is cotorsion if and only if it is algebraically compact.
• Every injective group is algebraically compact.
• Ulm factors of cotorsion groups are algebraically compact.