A split exact sequence of Abelian groups or modules is a short exact sequence
 |
(1)
|
that presents 
as the direct sum of 
and 
. Equivalently, its surjection 
has a section morphism 
satisfying 
. For general groups,
the corresponding splitting gives a semidirect
product, which need not be a direct product.
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, consider
 |
(2)
|
where the homomorphisms are given by ![alpha([a]_2)=[2a]_4](/images/equations/SplitExactSequence/Inline11.svg)
and ![beta([b]_4)=[b]_2](/images/equations/SplitExactSequence/Inline12.svg)
, with ![[a]_m](/images/equations/SplitExactSequence/Inline13.svg)
denoting the residue class
of 
modulo 
. In particular, 
and 
. This short
exact sequence cannot be split, since 
and 
are not 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
 |
(3)
|
are split iff 
is projective, and all
short exact sequences
 |
(4)
|
is A short exact sequence of vector spaces is always split.
