= Solution
Write $|G|=p^r m$ with $p$ prime and $p\nmid m$. The <Sylow theorems> assert that a subgroup of order $p^r$ exists; every <p-group> subgroup is contained in a conjugate of a <Sylow subgroup>; all <Sylow subgroups> are conjugate; and their number $n_p$ satisfies $n_p\equiv1\pmod p$ and $n_p\mid m$.
For the containment and conjugacy assertions, fix a <Sylow subgroup> $P$ and let any <p-group> subgroup $Q$ act by left multiplication on the $m$ left <cosets> $G/P$. By the <orbit-stabilizer theorem>, every orbit has size a power of $p$. Since $p\nmid m$, at least one orbit has size one. If $gP$ is such a fixed point, then $qgP=gP$ for all $q\in Q$, equivalently $g^{-1}Qg\subseteq P$. Thus $Q\subseteq gPg^{-1}$. If $Q$ too has order $p^r$, this inclusion is equality, proving conjugacy. This is <Sylow containment from a coset fixed point>.
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