Two ways to prove f is Isomorphism:

1) By definition:

f is Homomorphism + f bijective (= surjective + injective)

2) f is homomorphism + f has inverse map

Note: The kernel of a map (homomorphism) is the **Ideal** of a ring.

**Two ways to construct an Ideal**:

1) use Kernel of the map

2) by the generators of the map.

**Two ways to prove Injective:**

1) By definition of Injective Map:

f(x) = f(y)

prove x= y

2) By Kernel of homomorphism:

If f is homomorphism

Prove Ker f = {0}

Note: Lemma:

Proof Isomorphism 4 Steps:

1. Define function f：S -> T

Dom(f) = S

2. Show f is 1 to 1(injective)

3. Show f is onto (surjective)

4. Show f(a*b) = f(a). f(b)

Example:

Let T = even Z

Prove (Z,+) and (T,+) isomorphic

Proof:

1. Define f: Z -> T by

f(a) = 2a

2. f(a)=f(b)

2a=2b

a=b

=> f **injective**

3. Suppose b is any even Z

then a= b/2 ∈ Z and

f(a)=f(b/2)=2(b/2)=b

=> f **onto**

4. f(a+b) =2(a+b) =2a+2b =f(a)+f(b)

Hence (Z,+) ≌ (T,+)

Isomorphism (≌)

1. To prove G not isomorphic to H:

=> Prove |G| ≠ |H|

2. Isomorphism class = Equivalence Class

B’cos “≌” is an Equivalence relation.

3. Properties of Isomorphism (≌):

i) match e

ii) match inverse

iii) match power