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The second isomorphism theorem [wiki] in group theory is as follows:

Let $G$ be a group. $H \triangleleft G, K \le G$. Then:

  • $HK \le G$,
  • $(H \cap K) \triangleleft K$, and
  • $K/(H \cap K) \cong HK/H \le G/H$.

The third isomorphism theorem [wiki] is as follows:

Let $G$ be a group. $K \triangleleft G$. Then:

  • If $K \le H \le G$, $(H/K \triangleleft G/K) \iff H \triangleleft G$, and
  • If $K \le H \triangleleft G$, $(G/K)/(H/K) \cong G/H$.

The two isomorphism theorems seem to be quite different. However, I notice that the nature homomorphism $g \mapsto gN (g \in G, N \triangleleft G)$ plays a key role in both of their proofs. Specifically, for the third isomorphism theorem, it is $\sigma_3: g \in G \mapsto gK \in G/K$ while for the second one, it is $\sigma_2: k \in K \mapsto kH \in G/H$ where $K$ is a subgroup of $G$.

It seems that they are both talking about something about the subgroups and quotient groups under the natural homomorphism. However, I am not able to figure it out. Could anybody make it more clear?

Specifically, what is the relationship between these two isomorphism theorems?
Can they be stated in a unified, more general framework?

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The homomorphism you mention is indeed important because, if $N\triangleleft G$ then the projection $$p:G\to G/N$$ of $G$ onto its quotient $G/N$ is "universal among morphisms killing $N$":

Suppose $N\triangleleft G$ and let $\phi:G\to L$ be a group homomorphism with $\phi (N)=1$. Then there is a unique morphism $f:G/N\to L$ such that $$\phi=f\circ p$$

This is the basic result from which all the properties of quotient groups derive.

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Could you please elaborate more on your idea that "This is the basic result from which all the properties of quotient groups derive", for instance, by explaining the second and third isomorphism theorems in your "framework"? – hengxin Apr 23 '14 at 11:07

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