A finite $p$-group action has total points congruent mod $p$ to its global fixed points.Suggest a slogan
Let \(p\) be a prime, let \(G\) be a finite $p$-group, and let \(G\) act on a finite set \(X\). Let \(\Fix(G)\coloneqq\theset{x\in X\suchthat gx=x\text{ for all }g\in G}\). Then \[ \abs X\equiv\abs{\Fix(G)}\pmod p. \]
The orbits of \(G\) partition \(X\). A point of \(\Fix(G)\) is an orbit of size \(1\). Every other orbit \(Gx\) has size \([G:G_x]>1\) by the orbit-stabilizer theorem, and this index divides \(\abs G\), so it is a positive power of \(p\). Reducing \(\abs X=\abs{\Fix(G)}+\sum\abs{Gx}\), the sum over the orbits of size greater than \(1\), modulo \(p\) gives the congruence.