Group homomorphism

Definition

Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that

h(u * v) = h(u) · h(v)

From this property, one can deduce that h maps the identity element e_G of G to the identity element e_H of H, and hence the map "is compatible with the group structure".

Image and Kernel

We define the kernel of h to be

ker(h) = { u in G : h(u) = e_H }

and the image of h to be

im(h) = { h(u) : u in G }.

The kernel is a subgroup of G and the image is a subgroup of H. The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {e_G}.

Examples

• Consider the cyclic group Z₃ = {0, 1, 2} and the group of integers Z with addition. The map h: Z -> Z₃ with h(u) = u modulo 3 is a group homorphism (see modular arithmetic). It is surjective and its kernel consists of all integers which are divisible by 3.

• The exponential map yields a group homorphism from the group of real numbers R with addition to the group of non-zero real numbers R^* with multiplication. The kernel is {0} and the image consists of the positive real numbers.

• The exponential map also yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C^* with multiplication. This map is surjective and has the kernel { 2πki : k in Z }, as can be seen from Euler's formula.

The category of groups

If h : G -> H and k : H -> K are group homomorphisms, then so is k o h : G -> K. This shows that the class of all groups, together with group homomorphisms as morphisms, form a category.

Isomorphisms, Endomorphisms and Automorphisms

If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes.

If h: G -> G is a group homomorphism, we call it an endomorphism. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity and multiplication with -1; it is isomorphic to Z₂.