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 reduced part of the group is Hausdorff and 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:

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