Isomorphism between $H/H\cap N$ and $HN/N$

Question

Suppose we have a group G under operation +, and let H be a subgroup and N a normal subgroup. I want to prove that $H/H\cap N$ is isomorphic to $HN/N$.

Where, if I am not mistaken:

• $H/H\cap N = \{H\cap N + h | h \in H\}$
• $HN/N = \{N+(h+n)|h \in H, n \in N\}$

Answer 1

Compose the maps $$H\xrightarrow{\text{inclusion}}HN\xrightarrow{\text{quotient}}HN/N$$ Show that the composite map from $H$ to $HN/N$ is surjective. Observe that its kernel is $H\cap N$. Then apply the first isomorphism theorem.

(Incidentally, the fact that $H/(H\cap N)\cong HN/N$ is known as the second isomorphism theorem.)

Answer 2

Hint: Define $\phi:H \to \frac{HN}{N}$ by $\phi(h)=hN$ and show that this is surjective homomorphism whose kernel is $H\cap N$ and therefore from isomorphism theorem $\frac{H}{H\cap N}$ and $\frac{HN}{N}$ are isomorphic.