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An [[Abelian group|Abelian group]]  $  C $
is a cotorsion group if  $  { \mathop{\rm Ext} } ( G,C ) = 0 $
for all [[Torsion-free group|torsion-free]] Abelian groups  $  G $, 
i.e. every extension of  $  C $
by a torsion-free group  $  G $
splits (cf. also [[Extension of a group|Extension of a group]]). For  $  C $
to be a cotorsion group it suffices to assume that  $  { \mathop{\rm Ext} } ( \mathbf Q,C ) = 0 $. 
The importance of cotorsion groups lies in the facts that  $  { \mathop{\rm Ext} } ( B,A ) $
is a cotorsion group for all Abelian groups  $  A $
and  $  B $, 
and that they have several nice features.

Cotorsion groups can also be characterized by their injective property with respect to those exact sequences  $  0 \rightarrow G \rightarrow H \rightarrow K \rightarrow 0 $
of Abelian groups which split when  $  K $
is restricted to its torsion part  $  t ( K ) $(
cf. also [[Exact sequence|Exact sequence]]).

Epimorphic images of cotorsion groups are cotorsion, and so are the extensions of cotorsion groups by cotorsion groups. A direct product of groups is cotorsion if and only if each summand is cotorsion.

Examples of cotorsion groups are: 1) divisible (i.e., injective) Abelian groups, like  $  \mathbf Q $, 
$  \mathbf Z ( p  ^  \infty  ) $(
cf. also [[Divisible group|Divisible group]]); and 2) algebraically compact groups, like finite groups and the additive group of the  $  p $-
adic integers (for any prime  $  p $); 
cf. also [[Compact group|Compact group]]. A torsion Abelian group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group (the Baer–Fomin theorem), and a torsion-free Abelian group is cotorsion exactly if it is algebraically compact. Ulm subgroups of cotorsion groups are cotorsion, and the Ulm factors of cotorsion groups are algebraically compact.

For a reduced cotorsion group  $  C $, 
there is a natural isomorphism  $  { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,C ) \cong C $. 
This fact is relevant in showing that every Abelian group  $  A $
can be imbedded as a subgroup in a cotorsion group  $  C $
such that the cokernel  $  C/A $
is torsion-free and divisible. If  $  A $
is reduced, then  $  C $
can be chosen as  $  { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,A ) $; 
this is the smallest cotorsion group in which  $  A $
can be imbedded in this manner. It is called the cotorsion hull of  $  A $, 
and is unique up to isomorphism over  $  A $.

A cotorsion group is said to be adjusted if it is reduced and contains no non-trivial torsion-free summand. The cotorsion hull of a reduced torsion group is adjusted, and the correspondence  $  T \mapsto { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,T ) $
between the class of reduced torsion groups  $  T $
and the class of adjusted cotorsion groups is a bijection; its inverse is the formation of the torsion part. As a consequence, the classification of reduced torsion groups and that of adjusted cotorsion groups are equivalent problems. The Harrison structure theorem [[#References|[a2]]] states that every cotorsion group  $  C $
is a direct sum of three groups:  $  C = D \oplus G \oplus A $, 
where  $  D $
is a divisible group,  $  G $
is a reduced torsion-free algebraically compact group and  $  A $
is an adjusted cotorsion group. Such a decomposition of  $  C $
is unique up to isomorphism.

Some authors use  "cotorsion"  as  "cotorsion in the above sense + reduced" .

A general reference is [[#References|[a1]]]. See [[#References|[a3]]] for a generalization to cotorsion modules over commutative domains.

See also [[Cotorsion-free group|Cotorsion-free group]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Fuchs,   "Infinite abelian groups" , '''1''' , Acad. Press  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.K. Harrison,   "Infinite abelian groups and homological methods"  ''Ann. of Math.'' , '''69'''  (1959)  pp. 366–391</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Matlis,   "Cotorsion modules" , ''Memoirs'' , '''49''' , Amer. Math. Soc.  (1964)</TD></TR></table>