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Group homomorphism

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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 eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that h(u-1) = h(u)-1. Hence one can say that h "is compatible with the group structure".

Image and Kernel

We define the kernel of h to be

ker(h) = { u in G : h(u) = eH }
and the image of h to be
im(h) = { h(u) : u in G }.
The kernel is a normal subgroup of G (in fact, h(g-1 u g) = h(g)-1 eH h(g) = h(g)-1 h(g) = eH) and the image is a subgroup of H. The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {eG}.

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.

  • 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 Z2.

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.
The commutativity of H is needed to prove that h + k is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H,L), then
(h + k) o f = (h o f) + (k o f)   and    g o (h + k) = (g o h) + (g o k).
This shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sum of two copies of Z2 (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in Z2. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.



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