Rank of a transitive G-set and double cosets This is a follow up question to a previous question: If $X$ is a transitive $G$-set, then the rank of $X$ is the number of $G_x$-orbits of $X$ where $G_x$ is the stabilizer of some $x\in X$. If $X$ is a transitive $G$-set and $x \in X$, then rank $X$ is the number of ($G_x$-$G_x$)-double cosets in $G$. Is there a way to understand what these double cosets that count the rank mean? The subgroup $G_x$ acts on $X$ and so the cosets of $G_x$ have meaning through that. Is there some kind of analogous context to understand double cosets?

Let $X$ be transitive, fix $x\in X$, let $G_x$ be its stabiliser. Then as a $G$-set, $X$ is isomorphic to the set of right cosets $G/G_x$ with $G$ acting on it on the left. The isomorphism is given by sending $\sigma\in G$ to $\sigma(x)$ (check!). So now, $G_x$ acting on $X$ is just the same as $G_x$ acting on $G/G_x$ by left multiplication, and the orbits of this action correspond precisely to the double cosets $G_x\backslash G/G_x$. Namely if $y\in X$ corresponds to $\sigma G_x\in G/G_x$, then the orbit of $y$ corresponds to the double coset $G_x\sigma G_x$. You should write it out, carefully checking all bijections, but if you think about it for long enough it will become trivial.