A short exact sequence of groups 
, 
, and 
is given by two group homomorphisms 
and 
and is written
 |
(1)
|
Because it is an exact sequence, 
is injective, and 
is surjective. Moreover,
the group kernel of 
is the image of 
. Hence, the group 
can be considered as a normal
subgroup of 
,
and 
is isomorphic to 
.
A short exact sequence is said to split if there is a group homomorphism 
such that 
is the identity
map on 
.
Then 
is a semidirect product of 
and 
, with the action induced by
conjugation by 
. For Abelian groups
or modules, splitting instead gives the direct
sum of 
and 
.
The notion of a short exact sequence also makes sense for modules and sheaves. 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.
