show that there is some element x∈X whose stabilizer Gx is all of G where G is a group of order p^k, where p is prime and k is a positive integer

Question

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Suppose that G is a group of order p^k, where p is prime and k is a positive integer.

Answer 1

The idea is fine. $G$ acts on $X$; so we know that $$|X|=\sum |\mathcal O(x)|$$

Now, a orbit $\mathcal O(x)$ is a singleton iff $G$ fixes $x$, so we may write $$\tag 1 |X|=|X_0|+\sum' \mathcal |O(x)|$$ where the sum runs through nonfixed points (if any) and $|X_0|$ is the size of the set of fixed points. By Orbit-Stabilizer, $|\mathcal O(x)|\mid |G|=p^k$; and if $|\mathcal O(x)|>1$ this gives $|\mathcal O(x)|=0\mod p$, so modding out $(1)$ gives $$|X|=|X_0|\mod p$$

In particular, if $|X|\neq 0$ or $|X|\neq 0\mod p$; $|X_0|\neq 0$.