= Solution
Let $Q$ act by left multiplication on the set $G/P$ of left cosets. Every orbit has size a power of $p$, while
$$
|G/P|
$$
is not divisible by $p$. Therefore at least one orbit has size one. If $gP$ is fixed, then $qgP=gP$ for every $q\in Q$, so $g^{-1}Qg\leq P$, equivalently
$$
Q\leq gPg^{-1}.
$$
This is <Sylow containment from a coset fixed point>.
The remaining <Sylow theorems> state that Sylow $p$-subgroups exist, that any two are conjugate, and that their number $n_p$ satisfies
$$
\boxed{n_p\equiv1\pmod p,
\qquad
n_p\mid |G|/|P|.}
$$
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