What is the relationship between the second isomorphism theorem and the third one in group theory?

Question

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?

Answer 1

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.