Group Homomorphism

Definition :-
 a)   Homomorphism onto :-   A mapping    f   from a group  G  onto  G'  is called homomorphism of   G'  , if an only if   a , b  is element of   G  .
            
                                     f ( a , b ) = f (a) . f(b)

  In this case group  G'  is said to be homomorphic image of group   G  or else the group  G  is said to be homomorphic to group   G'  .

b)   Homomorphism into :-  A mapping  f  from a group  G  into a grroup  G'  is called homomorphism of  g  if and only if   a,  b  is element of  G   .
     
                                    f ( a , b ) = f (a) f(b)   .

In this case   G"  will not be said to be homomorphic image of  G  which will be   f(G)  to be a subgroup of  G'   .

c)   Endomorphism :-   A homomorphism of a group into itself is called an Endomorphism .

Types of homomorphic maps

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 transformation and multiplication with −1; it is isomorphic to Z/2Z.
An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function.

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 m copies of Z/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. 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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