Proofs of Isomorphism

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 f^{-1}

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
f: A \mapsto B
Prove Ker f = {0}

Note: Lemma:
\text{f is injective} \iff Ker f = {0}

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)

Let T = even Z
Prove (Z,+) and (T,+) isomorphic
1. Define f: Z -> T by
f(a) = 2a
2. f(a)=f(b)
=> f injective

3. Suppose b is any even Z
then a= b/2 ∈ Z and
=> 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
\phi(e) = e'
ii) match inverse
\phi (g^{-1} )= (\phi (g))^{-1}

iii) match power
\phi(g^{k})= (\phi(g))^{k}

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