A group homomorphism, also called a group morphism, is a map 
between two groups such
that the group operation is preserved: 
for all 
, where the product
on the left-hand side is in 
and on the right-hand side in 
.
As a result, a group homomorphism maps the identity element in 
to the identity element in 
: 
.
Note that a homomorphism must preserve the inverse map because 
, so 
.
In particular, the image of 
is a subgroup of 
and the group kernel, i.e.,
is a subgroup of 
. The kernel is actually a
normal subgroup, as is the preimage
of any normal subgroup of 
. Hence, any (nontrivial) homomorphism from a simple
group must be injective.
