A short exact sequence of groups
 |
(1)
|
is called split if it essentially presents 
as the direct sum of the groups 
and 
.
More precisely, one can construct a commutative diagram as diagrammed above, where 
is the injection of the first summand 
and 
is the projection onto the second summand 
, and the vertical maps are isomorphisms.
Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since 
and 
are non isomorphic finite groups. Note that
this is also a short exact sequence of 
-modules: this shows that being split is a distinguished property
of short exact sequences also in the category of modules. In fact, it is related
to particular classes of modules.
Given a module 
over a unit ring 
, all short exact sequences
 |
(2)
|
are split iff 
is projective, and all
short exact sequences
 |
(3)
|
is A short exact sequence of vector spaces is always split.
