Isomorphic Groups

Question

Can Please anybody describe the concept of Isomorphic Groups with an example ? Reference to https://gateoverflow.in/1219/gate2007-21

Comments

Do you want to know only the concept of isomorphic groups or you want to know the proof of

The number of non-isomorphic abelian groups of order p^n is equal to the number of partitions of n.

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I can explain to you the concept of isomorphism but I don't know the proof of the theorem

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Yes Sir Please Explain it no need of Proof :)

Answer 1

To know about Isomorphism you have to know the definitions of Homomorphism,Epimorphism and Monomorphism.

Definition of Homomorphism of groups :

Let (G,*) and (G,o) be two groups and f be a function from G into G1. Then f is called a homomorphism of G into G1 if for all a,b ∈ G,
                     f(a*b) = f(a) o f(b).

Example

Consider the group (Z,+).

Define f:Z →Z by f(n) = 3n for all n∈ Z

Here the function f is from the group (Z,+) to (Z,+)

Let n,m ∈ Z then we get n+m ∈ Z

and we have
f(n+m) = 3(n+m) = 3n + 3m = f(n) + f(m)

Hence the function f is a homomorphism.

Definition of epimorphism and monomorphism of groups

Let (G,*) and (G,o) be two groups and f : G → G1 be a homomorphism of groups then

1) f is called a monomorphism if f is an injective function.

2) f is called an epimorphism if f is an surjective function.

Example of Monomorphism

The example for monomorphism is same as the example given for homomorphism. 

The function f: Z → Z defined by f(n) = 3n is an injective function. So, f is a monomorphism.  

Example of Epimorphism

Consider the function f : Z →{1,-1} by
f(n) = 1, if n is even
        = -1 if n is odd

Here the groups are (Z,+) and  ({-1,1},*)

(where (Z,+) is a group of the set of integers under addition

and ({-1,1},*) is a group of the set {1,-1} under multiplication.)

Check the above function f is a homomorphism.

Since f is surjective f is an epimorphism.

Definition of Isomorphism

Let (G,*) and (G,o) be two groups and f : G → G1 be a homomorphism of groups then
f is called a isomorphism if f is a bijective function.

Example of Isomorphism
Consider the function f: Z → Z by f(x) = x

Now I will show that f is a homomorphism.

Take any two elements x,y belongs to Z

Then x + y  belongs to Z

Hence f(x+y) = x + y = f(x) + f(y)

Hence f is homomorphism.

Since the function f(x) = x is bijective.

f is an isomorphism.

Definition of Isomorphic Groups

Two groups (G,*) and (G1,o) are isomorphic to each other if there exist a homomorphic bijective function from G to G1.

Example for Isomorphic groups 

Every infinite cyclic group is isomorphic to infinite cyclic group of set of integers under addition

https://gateoverflow.in/215185/group-theory