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Suppose that $G$ is a finitely generated group, $X$ is a $G$-set, and $x \in X$ is a point. Are there any sorts of conditions on $X$ and $G$ that would let me conclude that $\operatorname{Stab}(x)$ is finitely generated as well?

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  • $\begingroup$ Note that for any subgroup $H$ you can take $X=G/H$, so what you are asking is for a characterization of all finitely generated subgroups of a given group. $\endgroup$ – user1688 Jul 14 '16 at 8:08
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"Stabilizer of $x$" is not any more specific than "subgroup of $G$" (consider the regular action of $G$ on $G/H$). In other words, you are asking for conditions on $G$ such that its subgroups are finitely generated.

This is false, in general. It's true in nilpotent (and virtually nilpotent) groups. It's false in solvable groups (consider the lamplighter).

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Lior Silberman is absolutely correct, but I wanted to add one other natural and simple condition in which you can conclude that the stabilizer of $x$ is finitely generated, namely, when the orbit of $x$ is finite. Equivalently, subgroups of finite index in finitely generated groups are finitely generated. This follows from Schreier generators (see "Schreier's lemma" or "Schreier's subgroup lemma").

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