In advanced group theory. $p$ is a prime and $n$ is a positive integer. $G$ is a group with order $p^n$ and set $X$ is a finite $G$-set. It means that, there exists an action of $G$ on $X$. Id like to show that prime $p$ divides $|X|-|X_G|$
Specifically, i did try to consider that with $|X|-|X_G|=\sum_{x \in X}|Gx|$ (called Burnside Identity) where $Gx$ is an orbit of $x$ under $G$ and $x\in X$, $|G|=|G_x||Gx|$. In deed, i say a main idea about this theorem, im not sure that $|Gx| \geq p$. If there exists a singleton set $Gx$ for some $x\in X$with $|Gx| = 1$ then how can i explain that?
$\{Gx \}_{x\in X}$ is partition of $X$.
Note that $|Gx|=1$ $\iff$ $x\in X_G \iff Gx = \{x\}$
Let $Gx_1, Gx_2 , Gx_3,...,Gx_r, Gx_{r+1} ,... , Gx_k$ be all disjoint orbit and $Gx_1 ,.., Gx_r$ are singleton set. so $X_G = \bigcup_1^rGx_i $.
and $Gx_r , Gx_{r+1} ,..., Gx_k$ have more than one element so that p divides $|Gx_i|$.
$$|X| = \sum_1^k |Gx_i| = \sum_1^r |Gx_i| + \sum_{r+1}^k |Gx_j|=|X_G| + \sum_{r+1}^k|Gx_j|$$
so $$|X|=|X_G| \ mod \ p$$