An extension of a group 
by a group 
is a group 
with a normal subgroup 
such that 
and 
. This information can be encoded into a short exact sequence
of groups
 |
where 
is injective and 
is surjective.
It should be noted that some authors reverse the roles and say that 
is an extension of 
(Spanier 1994, Mac Lane and Birkhoff 1993).
Given groups 
and 
there are (often) many extensions of 
by 
. Examples include the direct
product of 
and 
and a semidirect product of 
and 
. A function 
such that 
is the identity function
on 
is called a transversal function. A group extension is said to be split if there
is a transversal function which is a homomorphism. A group extension is split iff it is a semidirect product.
The study of group extensions has connections with group cohomology.
