Let a group \(G\) with identity \(e\) act on a set \(X\), and let \(x\in X\). The stabilizer of \(x\) is \[ G_x \coloneqq \theset{g\in G \st g\cdot x = x}. \]
\(G_x\) is a subgroup of \(G\): \(e\cdot x=x\); if \(g,h\in G_x\), then \((gh)\cdot x=g\cdot(h\cdot x)=x\); and if \(g\in G_x\), then \(g^{-1}\cdot x=g^{-1}\cdot(g\cdot x)=x\).
For a group action $G \actson X$, what is $G_x$?
What is the stabilizer of a point under a group action?