$p$-group acting on a finite set

Question

Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$.

Any thoughts? I'm stumped because $G$ is not necessarily finite.

Answer 1

For each $x\in X$, the orbit of $x$ is the set $\mathcal O(x)=\{gx:g\in G\}$. Since $G$ is a $p$-group, $|\mathcal O(x)|$ is a power of $p$. Note that it can not be infinite because $X$ is finite.

Note that the set orbits is a partition in $X$. Since the cardinal of each orbit is a power of $p$, $|X|$ must be also a multiple of $p$, unless some of these orbits have cardinal $p^0=1$.