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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
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.
