# Group homomorphism

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

Formally, therefore, a map ${\displaystyle f:G\rightarrow H}$ is a homomorphism if

${\displaystyle f:1_{G}\mapsto 1_{H};\,}$
${\displaystyle f:x^{-1}\mapsto f(x)^{-1};\,}$
${\displaystyle f:xy\mapsto f(x)f(y).\,}$

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.