If G acts on X, show that there must be a fixed point for this action. Please help. Suppose that $G$ is a group of order $p^k$, where $p$ is prime and $k$ is a positive integer. Suppose that $X$ is a finite set and assume that $p$ does not divide the $|X|$. If $G$ acts on $X$, show that there must be a fixed point for this action. i.e show that there is some element $x\in X$ whose stabilizer $G_x=\{g.x;g\in G\}$ is all of $G$.

Put together facts you should know:


*

*A $G$-set $X$ is the disjoint union of its orbits, so $|X|$ is the sum of the sizes of the orbits. 

*The size of an orbit divides $|G|$ by orbit-stabilizer.

*Fixed points correspond to orbits of size $1$.

