### 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 g…