Cardinality of double cosets

Question

We show that the sets of double cosets $K\backslash G/H$ is in bijection with $K\backslash(G/H)$. So $\vert K\backslash(G/H) \vert = \vert G/H \vert /\vert K \vert = (\vert G \vert / \vert H \vert)/\vert K \vert$. But for the example $S_2\backslash S_3/S_2=\{\{Id,(12)\},\{(132),(13),(123),(23)\}\}$, $ S_2 \backslash S_3 /S_2$ is of cardinality 2 but $(\vert S_3 \vert / \vert S_2 \vert)/\vert S_2 \vert=1.5$

Answer 1

The left action of $K$ on $G/H$ is not related to double cosets. For example if $K\subseteq H$ then all orbits of that action are singltons. So there is no bijection between $K\backslash G/H$ and $K\backslash (G/H)$ in general.