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Encyclopedia > 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 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".


Older notations for the homomorphism h(x) may be xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right.


In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

Contents

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 Z/3Z = {0, 1, 2} and the group of integers Z with addition. The map h : Z -> Z/3Z 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/2Z.


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 Z/2Z (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in Z/2Z. 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.


  Results from FactBites:
 
Group homomorphism - Wikipedia, the free encyclopedia (838 words)
For example, a homomorphism of topological groups is often required to be continuous.
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.
For example, the endomorphism ring of the abelian group consisting of the direct sum of two copies of Z/2Z (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in Z/2Z.
  More results at FactBites »

 
 

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