A group homomorphism 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.
