# Group homomorphism

In group theory a **group homomorphism** is a map from one group to another group that preserves the group operations.

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.

An embedding or monomorphism is an injective homomorphism (or, equivalently, one whose kernel consists only of the identity element).

## Isomorphism

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.