In this sections we shall assume that G and H are groups and that
is a homomorphism from G to H.
The Fundamental Theorem of Homomorphisms (also known as the First Isomorphism
Theorem) states that 
. If we denote the natural
homomorphism from G to 
by 
and the isomorphism from
to 
by 
then we have that 
.
That is, for all 
, 
.
One question we have asked is how we can construct a non trivial homomorphism
. The Fundamental theorem provides a different
way to think about the problem. It tells us that if we look at the
normal subgroups of G, we can form all the possible factor groups.
Then if H contains a subgroup isomorphic to that factor group there is
a homomorphism from G to H.
For example, let us denote the elements of 
where r is a rotation of 
, x reflection about the x-axis,
y reflection about the y-axis, d reflection about the diagonal
y=x and a reflection about the diagonal y=-x. The subgroup
is a normal subgroup of 
. The cosets are
N, 
, 
and 
.
Since 
, 
then 
is isomorphic to the Klein-4
group. Thus there is a non trivial homomorphism from to any
group that has the Klein-4 group as a subgroup. As examples of groups
that could act as codomain are the Klein-4 group itself and
.
The group has 10 subgroups. These are:
, 
, 
, 
, 
, 
, 
, 
, 
is isomorphic to the Klein-4 group.
Now 
, 
, is a group of order 2 and must be isomorphic
to the cyclic group of order 2, 
. This tells us that the only other
non-trivial homomorphisms that can be constructed with as domain
are ones to groups that have a subgroup isomorphic to . Examples
of possible codomains are 
for n a positive integer,
and 
. We could also answer the problem, say, of how many
epimorphisms are there from to . For each possible epimorphism
there is a kernel of order 4 in . The "other" coset must map to the
generator of . There are 3 candidates for kernel and so there are
3 epimorphisms from to . All told, there are 4 homomorphisms
from to (the fourth one being the trivial homomorphism).