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 mapsIf 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 groupsIf 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 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).