Maths - Fundamental Homomorphism Theorem

The fundamental homomorphism theorem states that. If φ: G→H is a homomorphism, then
Im(φ)≡G / Ker(φ)

Example: D3 to C3

Lets look at a morphism from D3 to C3:

c3 to d3

So, in this case,

So if we divide {1,1}, {r,1}, {r²,1},{1,m}, {r,m}, {r²,m} by {1,1},{1,m} we get
{1,N}, {r,N}, {r²,N}
where N stands for the larger circles in the diagram.

We can see that the group of big circles is isomorphic to C3 and so Im(φ)≡G / Ker(φ) as required.

Now lets look at a morphism, in the other direction, from C3 to D3:

c3 to d3

So, in this case,

So this is relatively trivial as it is saying that 1,r,r² is isomorphic to {1,1}, {r,1}, {r²,1}

Futher Study

 


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