Abstract Algebra Challenge  

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Read my article introducing group theory.

 

A normal subgroup is a subgroup H such that for any g ∈ G and h ∈ H, ghg-1 ∈ H, meaning it is closed under an operation called conjugation. We define the quotient group G/H for any normal subgroup H to be the set of all gH for g ∈ G (the set of left cosets), with multiplication aH · bH = (ab)H. Prove this is a group.

Prove that there is a homomorphism between the group G and the quotient group G/H (quite a trivial one, as well). It is true that the kernel (the set of points that map to the identity element) is H in this homomorphism. This proves that any normal subgroup is the kernel of a homomorphism.

That one guy that's always on

 
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