Rotman's Introduction to to the theory of groups. Exercise 3.45.

Question

Can you give me a hint on the first part of the exercise?

Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ with $\tau x = x$ for all $\tau \in G$.

I'm trying to prove that there's only one orbit on the $G$-set but the only "special feature" that I see is that the order of $G$ is a power of a prime. However, I cannot figure out how to use that in order to solve prove that there's only one orbit.

Answer 1

The question asks you to show that there is an orbit of size $1$, not to show that there is only one orbit. Note that orbits having size not equal to 1 must have size divisible by $p$ by orbit-stabilizer theorem.

Answer 2

Hint: Decompose $X$ into a union of orbits and use the orbit-stabilizer theorem (or prove it), and the fact that any subgroup of a $p$-group has as its index a power of $p$.