$|\{ x\in X: g.x=x \space\space\space \forall g\in G \}| = |X|\space mod \space p$

Question

Let $G$ be a p-group. $|G|=p^n$ for some n.

Let X be a finite set so that $\,p\nmid |X|\,$,

G acts upon X.

Denote $A:= \{ x\in X: g.x=x \space\space\space \forall g\in G \}$

I am trying to show $|A| = |X|\space mod \space p$

Answer 1

Hint:

$$|X|=\sum_{x\in X}'\left|\mathcal Orb(x)\right|=|A|+\sum_{x\in X\setminus A}'\left|\mathcal Orb(x)\right|$$

Where $\;\sum'\;$ means sum over disjoint orbits.

Now just remember that

$$\left|\mathcal Orb(x)\right|=[G: G_x]\;,\;\;G_x:=\{g\in G\;:\;gx=x\} =\text{the isotropy or stabilizer group of}\;x$$