Order of Double Coset

I am working on a homework problem (so don't just give me the answer) from Herstein's Topics in Algebra, which goes as follows:

If $G$ is a finite group, show that the number of elements in the double coset $AxB$ is $$\dfrac{o(A)o(B)}{o(A\cap xBx^{-1})}$$

It makes sense to me, but my attempts at a proof seem to fall short of undeniability. I have been trying to show that $o(A \cap x B x^{-1})$ equals the number of duplicate terms in the list of all possible products of an element from $A$ and an element from $xB$.

Suppose $a_0 \in A \cap xBx^{-1}$. Then $\exists b_0 \in B : a_0 = x b_0 x^{-1}$. In the list of products of the form $axb, a \in A, b \in B$, any term involving $a_0$ will be of the form $a_0 x b = x b_0 x^{-1} x b = x b_0 b$ for some $b \in B$. But then this list is just the left coset $xB$, which is already accounted for in the product list of $AxB$ by setting $a=e$.

This is where I run into trouble. I can't seem to crystallize this argument. Am I going about this the right way? And if you see how I should extend this argument, can you nudge me in the right direction?

Thanks Math.SE.

Solution 1:

Edit: I didn't notice that "don't just give me the answer" in the first version of my answer. The following are some hints.

Firstly, the formula $$o(HK)=\frac{o(H)o(K)}{o(H\cap K)}$$ holds for subgroups $H$ and $K$ of $G$, but may not hold for subsets $H$ and $K$ of $G$.

Secondly, note that

$$o(AxB)=o(AxBx^{-1}).$$

Solution 2:

This is a very nice formula:

$$|HK|=\frac{|H||K|}{|H\cap K|}.$$

There are a couple of beautiful proofs, suggested intrinsically by its form, that I will hint at:

• Coset spaces. You probably know the formula $|G/H|=|G|/|H|$; in other words you know how to interpret a ratio of two group orders as the size of a coset space when the denominator group is contained in the numerator group. There are four quantities appearing in the desired formula, arranged in the form $A=BC/D$. Can you rearrange this so that the equation involves expressions of the form $|X|/|Y|$? Now interpret the expressions as spaces of either left or right cosets. Exhibit an explicit bijection between these two coset spaces.
• Group actions. If $G$ acts on a set $X$, then the orbit-stabilizer formula asserts that the size of an orbit $Gx$ is equal to the index of $x$'s stabilizer in $G$. You know $|H||K|=|H\times K|$. Can you think of a group action on a set which yields the desired formula from orbit-stabilizer?