# If $[G:H\cap K]= [G:H][G:K]$ then $G=HK$.

Let $G$ a finite group and $H,K$ subgroups of $G$. Show that if $[G:H\cap K]= [G:H][G:K]$ then $G=HK$.

I proved that $G=HK$ implies $[G:H\cap K]= [G:H][G:K]$ but not the other direction.

Hint: Use the fact that $$|HK|=\frac{|H||K|}{|H\cap K|}.$$