Group homomorphism

A group homomorphism is a transformation made from one group to another. This change preserves all group axioms and operations.

Let G and H be two groups with group operations + and *. Then a function f between the two groups is a homomorphism if f(x+y)=f(x)*f(y) for every pair of elements x and y in the group G.

Examples

As an example, let the first group be the integers Z. Let the second group be the even numbers 2Z. The group operations are both addition. Consider the function that multiplies numbers by two f(x)=2x. Then this function is a homomorphism. This is because adding two numbers before multiplying by two is the same as multipying by two then adding: f(x+y) = 2(x+y) = 2x + 2y = f(x)+f(y). In fact, f is a group isomorphism.

The even numbers is a normal subgroup of the integers. Similar examples can be created by considering the homomorphism f(x)=nx between the group of integers Z and a normal subgroup nZ.