Formally, therefore, a map is a homomorphism if
although the first two are in fact consequences of the third.
The kernel of a homomorphism is the set of all elements of the domain that map to the identity element of the codomain. This subset is a normal subgroup, and every normal subgroup is the kernel of some homomorphism.
We say that two groups are isomorphic if there is a bijective homomorphism of one onto the other : the mapping is called an isomorphism. Isomorphic groups have identical structure and are often thought of as just being relabelings of one another.