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 $x\in X$.
"An Introduction to the Theory of Groups" by Joseph Rotman gave one way to characterize the rank. 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 $G$.
My question is why isn't the rank equal to the number of cosets $G_xg$ of $G$ and thus requiring the characterization in terms of double cosets?