Group homomorphism
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 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".
Older notations for the homomorphism h(x) may be x_{h}, though this may be confused as an index or a general subscript.
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 normal subgroup of
G (in fact,
h(
g^{-1} u g) =
h(
g)
^{-1} e_{H} h(
g) =
h(
g)
^{-1} h(
g) =
e_{H}) 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}}.
- 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.
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}.
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
Z_{2} (the
Klein four-group) is isomorphic to the ring of 2-by-2
matrices with entries in
Z_{2}. 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.