Let $p$ be a prime number. Then
$$
F_{p}(X)=\frac{X^{p}-1}{X-1}=1+X+\cdots+X^{p-1}
$$
and more generally
$$
F_{p^{m}}(X)=\frac{X^{p^{m}}-1}{X p^{m-1}-1}=1+X^{p^{m-1}}+\cdots+X^{(p-1) p^{m-1}}
$$
in particular, then,
$$
\varphi\left(p^{m}\right)=(p-1) p^{m-1}=p^{m}-p^{m-1}
$$
MATH3962 COURSE NOTES ：
Let $G$ act on $M$ and take $x \in M$. The subgroup
$$
G_{x}={\sigma \in G \mid \sigma x=x}
$$
is called the stabilizer of $x$.
F4. Let $G$ act on $M$ and take $x \in M$. The map
$$
G \rightarrow G x, \quad \sigma \mapsto \sigma x
$$
defines a bijection $i: G / G_{X} \rightarrow G x$. Thus, if $G x$ is finite, so is $G: G_{X}$, and
$$
|G x|=G: G_{X} .
$$
If $G$ is finite, so is $G x$, and
$$
|G x|=\frac{|G|}{\left|G_{x}\right|} ;
$$
in particular then the size of any orbit divides the order of $G$.