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I've got an old exam question that I can't figure out how to solve. If anyone could let me know what theorems and lemma's I might find useful, please let me know.

Let G be a group of order $p^m$ for some prime $p$ and integer $m\in \mathbb{N}$. Suppose $G$ acts on a finite set $\Omega$. Define $\Delta$ := {$\alpha \in \Omega$|$\alpha$$^g$ = $\alpha$, for all g $\in$ $G$}. Show that |$\Delta$| $\equiv$ |$\Omega$| (mod $p$).

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Every point in $\Delta$ is in its own orbit. All of the other points are in orbits of size larger than $1$. What can we say about the divisibility of the size of an orbit?

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