Algebraically compact group
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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 group is complete in the
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.
