Suppose $H$ and $K$ are subgroups of a finite group $G$. Then I read $ |HxK|=|H|[K:x^{-1}Hx\cap K] $
where $x\in G$. How can this equality be derived?
I wanted to prove the equivalent equality $|HxK|/|H|=[K:x^{-1}Hx\cap K]$ by exhibiting a bijection of the cosets of $x^{-1}Hx\cap K$ in $K$ with cosets of $H$ in $HxK$, but I realize $HxK$ need not be a group, nor $H$ a subgroup. Thanks.
Source: This is the third statement of Exercise 5 of Jacobson's Basic Algebra I, page 53.