State definitions of the following:
Group action
Orbit
Stabilizer
Fixed points
Solution
Let \(G\) be a group acting on a set \(X\).
A group action is a map \[ G\times X\to X,\qquad (g,x)\mapsto g\cdot x, \] such that \[ e\cdot x=x, \qquad (gh)\cdot x=g\cdot(h\cdot x). \]
The orbit of \(x\in X\) is \[ Gx=\{g\cdot x:g\in G\}. \]
The stabilizer of \(x\) is \[ G_x=\{g\in G:g\cdot x=x\}. \]
For \(g\in G\), its fixed-point set is \[ X^g=\{x\in X:g\cdot x=x\}. \] The fixed points of the whole action are \[ X^G=\{x\in X:g\cdot x=x\text{ for every }g\in G\}. \]