I have been trying to prove that if $H, K$ are finite subgroups of $G$ then $|HK|=\dfrac{|H| |K|}{|H \cap K|}$. I saw the proofs in Herstein and Gallian textbook and they are essentially the same. However, it confuses me. The claim is that any given element $hk$ appears $|H \cap K|$ times as a product in the list of $HK$ and what follows is my attempt of the proving the same but in a little different way and I wonder if it is correct.
We define the set $\mathcal{R}(hk)=\{ (h',k') \in H\times K \, | \, hk=h'k' \}$. Now, we define the map $f\colon H\cap K \to \mathcal{R}(hk)$ by $f(x)=(hx, x^{-1}k)$. Clearly, $f$ is well defined and we will show that $f$ is bijective.
Let $x,y \in H\cap K$ and suppose that $f(x)=f(y)$. Then $(hx, x^{-1}k)=(hy, y^{-1}k)$ and hence $x=y$.
Let $(h' , k') \in \mathcal{R}(hk)$. Then $hk=h'k'$ and let $x = h^{-1} h' = kk'^{-1}$. Clearly, $x\in H \cap K$ and we have that $h'=hx$ and $k'=x^{-1}k$. Thus, $f(x)=(h',k')$. This shows $f$ is surjective.
Hence, we have that $|\mathcal{R}(hk)|=|H \cap K|$. This proves our claim.
I wonder if this is okay. Are Herstein and Gallian doing the same thing?
Here's the proof from Herstein:
