Table of contents |
2 Image and Kernel 3 Examples 4 The category of groups 5 Isomorphisms, Endomorphisms and Automorphisms 6 Homomorphisms of abelian groups |
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)
Older notations for the homomorphism h(x) may be x_{h}, though this may be confused as an index or a general subscript.
Image and Kernel
We define the kernel of h to be
and the image of h to be- im(h) = { h(u) : u in G }.
Examples
- Consider the cyclic group Z_{3} = {0, 1, 2} and the group of integers Z with addition. The map h : Z -> Z_{3} 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.
- Given any two groups G and H, the map h : G -> H which sends every element of G to the identity element of H is a homomorphism; its kernel is all of G.
- Given any group G, the identity map id : G -> G with id(u) = u for all u in G is a group homomorphism.
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, forms 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 of G. 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 Z_{2}.
Homomorphisms of abelian groups
If G and H are abelian (i.e. commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by
- (h + k)(\u) = h(u) + k(u) for all u in G.
- (h + k) o f = (h o f) + (k o f) and g o (h + k) = (g o h) + (g o k).