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?