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
