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**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**/2

**Z**.

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*.

*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*).

*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**/

*n*

**Z**is isomorphic to the ring of m-by-m matrices with entries in

**Z**/

*n*

**Z**. 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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