Show that if \(G \curvearrowright X\) is a group action, then the stabilizer \(G_x = \operatorname{Stab}_G(x)\) of any point \(x \in X\) is a subgroup of \(G\).
Solution
By definition, \[ G_x=\{g\in G:g\cdot x=x\}. \] The identity lies in \(G_x\) because \(e\cdot x=x\). If \(g,h\in G_x\), then \[ (gh^{-1})\cdot x =g\cdot(h^{-1}\cdot x) =g\cdot x =x, \] because \(h\cdot x=x\) implies \(h^{-1}\cdot x=x\). Thus \(gh^{-1}\in G_x\). By the subgroup criterion, \(G_x\le G\).