Group of order 81 acting on a set of order 98

Question

A group P acts on a set Ω. We know that|P| = 81 and |Ω| = 98. Let $Ω_0$ be the set of elements of Ω that are fixed by every element of P. In other words, $Ω_0$ ={α ∈ Ω | α · g = α for all g ∈ P}. Show |$Ω_0$| = 3k + 2 for some integer k with 0 ≤ k ≤ 32.

Answer 1

This is a direct application of a theorem about $p$-groups: if $X$ is a finite set with an action of a $p$-group $G$, then the order of $X$ is equivalent modulo $p$ to the fixed points of the action. I will leave it to you to get the desired consequence.

And while we're at it, let's remember why this is true. The size of each orbit of the action must divide the order of $G$, and so it is either a power of $p$ or $1$. Note that the orbits of size one are exactly the fixed points of the action. Thus $$ \#X = \sum_{\text{orbits } O \text{ in } X} \# O \equiv \sum_{\text{orbits of size one in } X} 1 \equiv (\text{number of fixed points}) \pmod p. $$