Page 82 - DMTH403_ABSTRACT_ALGEBRA

Unit 6: Group Isomorphism

 Ker  = {1 (0, 0)} Notes
  is 1-1.

So we have proved that ! is a homomorphism, which is bijective.
And now let us look at a very useful property of a homomorphism that is surjective.
Theorem 5: Iff : G  G is an onto group homomorphism and S is a subset that generates G ,
1
2
1
then f(S) generates G .
2
Proof: We know that
G = < S > = { x x ...x m m r | m  N, x  S, r  Z for all i). We will show that
1 r
2 r
1
2
1
1
1
G = < f(S) >
2
Let x  G , Since f is surjective, there exists y  G such that f(y) = x. Since y  G , y = x ...x m m r ,
1 r
1
1
1
2
for some m  N, where xi  S and ri  Z,  1  i  m.
1 r
Thus, x = f (y) =  f x ...x m m r 
1
m r
= (f(x )) ... (f(x )) , since f is a homomorphism.
1 r
m
1
 x  < f(S) >. since f(x )  f(S) for every i = 1, 2, ..., r.
1
Thus G = < f(S) >.
2
So far you have seen examples of various kinds of homomorphisms-injective, surjective and
bijective. Let us now look at bijective homomorphism in particular.
6.2 Isomorphisms
Definition: Let G and G be two groups. A homomorphism f : G  G is called an isomorphism
2
1
1
2
if f is 1-1 and onto.
In this case we say that the group G is isomorphic to the group G or G and G are isomorphic.
1
2
2
1
We denote this fact by G  G .
2
1
An isomorphism of a group G onto itself is called an automorphism of G. For example, the
identity’ function IG : G  G : I (x) = x is an automorphism.
G
Note The word ‘isomorphisms’ is derived from the Greek word ‘ISOS’ meaning
‘equal’.
Let us look at another example of an isomorphism.
 a   b  

Example: Consider the set G =   b a  a, b R . 
     
Then G is a group with respect to matrix addition.
 a  b  
Show that f : G  C : f   b a   = a + ib is an isomorphism.
    

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