How could I prove this equation true?

Question

Let $H, K$ be subgroups of a finite group $G$ with $K \subset N_H$. Show that $$\#(HK) = \frac{\#(H)\#(K)}{\#(H \cap K)}$$

What would I need to do to prove this equation true?

Thanks in advance.

Answer 1

If $K\subseteq N_G(H)$, and of course $H\subseteq N_G(H)$, you have $HK\subseteq N_G(H)$, and so $H\lhd HK$.

Simply by counting items in cosets you always have that $|G/N||N|=|G|$ in finite groups, where $N\lhd G$. Rewritten this says $|G/N|=|G|/|N|$

In this case, the second isomorphism theorem says that

$$HK/H\cong K/(H\cap K)$$

Applying the first observation, $|HK|/|H|=|K|/|H\cap K|$.

Multiplying by $|H|$ on both sides, you have your equation.