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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?

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Consider an example such as the symmetric group G on 3 points X = { 1, 2, 3 }.

Take x = 1, then Gx = { (), (2,3) } has 3 cosets Gxg in G, namely $G_x () = \{ (), (2,3) \}, ~ G_x(1,2,3) = \{ (1,2,3), (1,2) \}, \text{ and } G_x(1,3,2) = \{ (1,3,2), (1,3) \},$ but it has only two orbits on X, namely {1} and {2,3}.

Of course, Gx has only 2 double cosets,

$G_x () G_x = \{ (), (2,3) \}, \text{ and } G_x (1,2,3) G_x = \{ (1,2,3), (1,2), (1,3), (1,3,2) \}$

You may be thinking of $[G:G_x] = |X|$ is the size of the transitive G-set X, not the number of orbits of Gx on X.

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    Ah ok so the cosets $G_x(1,2,3)$ and $G_x(1,3,2)$ both have an element that send the point $1$ to $2$ and they both have an element that sends the point $1$ to $3$. So the "range" of where the point $1$ is sent by each coset is $\{2,3\}$. Now this is the orbit of point $2$ or $3$ under $G_x$. But this property does not chaacterize a unique coset.2012-01-18
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As you say, the rank of $X$ (or perhaps more usually, the rank of the permutation group $G$), is the number of orbits that $G_x$ has on $X.$ But $|X| = [G:G_x]$ by transitivity, so the only way that the rank can be equal to $[G:G_x]$ is if $G_x$ acts trivially on $X.$ This happens if and only if $G_{x} \lhd G.$ In the case of a faithful action, this forces $G_x = 1,$ so that $X$ is a regular $G$-set.