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Let $X$ and $Y$ be sets. Under what conditions is there a group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in $Y$?

Equivalently, what conditions need to be satisfied by $X$ and $Y$ for it to be possible to realise $X$ and $Y$ as simultaneous coset spaces $G/H_1$ and $G/H_2$ of $G$ with $H_1\leq H_2$?

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    $\begingroup$ A necessary and sufficient condition is that $X,Y$ are both non-empty and that there is a set $Z$ such that $X$ and $Y\times Z$ are in bijection. In other words, this is true iff $X,Y$ are non-empty, $|X|\ge |Y|$, and, if $X,Y$ are both finite, $|Y|$ divides $|X|$. This does not seem to be of research level. $\endgroup$
    – YCor
    Commented Dec 21, 2015 at 22:35
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    $\begingroup$ Meta discussion: meta.mathoverflow.net/questions/2658/research-level-mathematics $\endgroup$
    – Todd Trimble
    Commented Dec 23, 2015 at 1:45

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