Homomorphism

Homomorphism is a function between groups that preserves the group operations.

Suppose $f:G \rightarrow H$ is a function and $G,H$ are groups. Then, to be a homomorphism $f$ has to satisfy: $$f(xy)=f(x)f(y).$$ Here on the left the operation in in $G$ and on the right in $H$.

Properties.

• The identity element is preserved:

$$f(e_A) = e_B.$$

• Inverses of elements are preserved:

$$f(a^{-1}) = [f(a)]^{-1}.$$

If a homomorphism is an injective function (i.e. one-to-one), then we say that it is a monomorphism. If a homomorphism is an surjective function (i.e. onto), then we say that it is an epimorphism. If a homomorphism has an inverse, then we say that it is an isomorphism.

In linear algebra the analogue is linear operator. One can define similar natural concepts of homomorphisms for other algebraic structures, such as ring homomorphisms, etc.