The homomorphism theorem of group theory ↩

shorthands: {}
aliases: [The first isomorphism theorem of group theory]
created: 2021-12-17 18:41:11
modified: 2022-01-10 04:13:04

1. (the image of is a subgroup of )
2. (the kernel of is a normal subgroup of )
3. (where denotes quotient group)

Proof

1. Is a subgroup of ?

• has a neutral element, its
• Associativity is given because , whose elements are associative
• , so it is closed to multiplication, which means it forms a group itself

QED

2. Is a normal subgroup of ?

• Since , it has a neutral element
• , so associativity is also given
• , but since , all , so it is closed as well
• Is it normal? , but , so any is also part of the kernel is a normal subgroup

3. Is and isomorphic?

The quotient is: and the image: .

Consider the following function: , (where is a coset of and is an element of (it can be any element since it fully represents it coset)).

a) Is a homomorphism?

b) Is injective?

Is this true?

Or equivalently:

Take a look:

We reached the original claim, QED

c) Is surjective?

In other words: is ?

Look at them in set notation:

And

So we see that they are the same. QED