Lagrange Theorem involving double cosets Currently learning about the theorem of Lagrange. I was doing great until they started discussing double cosets. Does Lagrange theorem hold for double cosets? This is the particular question that was asked.
If $G$ is finite and $x,y∈G$, is $|HxK|=|HyK|$ as with single cosets? That is, is there a Lagrange's theorem for double cosets? 
 A: Two ways of arriving at an equivalent counterexample:

$S_3$ has a non-normal subgroup. In fact, it is the smallest group that has a non-normal subgroup. Use that to get a counterexample $H$, $K$ and $x$. In the following, the notation $(a,c,b)$ denotes the cyclic permutation going $a\to c\to b\to a$, which is an element of $S_3$.
$$G=S_3, \\H=K=\{(a,b),e\},\\x=(a,b,c) \\ \implies HK=\{(a,b),e\},HxK=\{(a,b,c),(b,c),(a,c),(a,c,b)\}$$

Let $w=e^{i2\pi/3}$. The set of mappings (with domain and range $\mathbb C$) generated by the set $\{z\mapsto\bar z, z\mapsto wz\}$ equals a group with six elements (that's isomorphic to $D_3$ and $S_3$). Let $H=K=\{z\mapsto z, z\mapsto \bar z\}$. Let $x=(z\mapsto wz)$. Then $HxK=\{z \mapsto wz, z \mapsto w\bar z, z\mapsto \bar wz, z\mapsto \bar w \bar z\}$ while $HK=\{z\mapsto z, z\mapsto \bar z\}$.
A: No. There is no Lagrange's theorem for double cosets. Not only are there relatively easy (counter-) examples, but, also, there are (more complicated) natural, important situations where in fact determination of double cosets is a non-trivial and significant task.
