# Finite generation of stabilizers in a $G$-set [closed]

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?

• 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. – user1688 Jul 14 '16 at 8:08

"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.
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").