Sylow $p$-subgroup of a direct product is product of Sylow $p$-subgroups of factors

However, the following is true. Let $G=HK$, $H$, $K$ subgroups and let $p$ a prime dividing the order of $G$. Then there exists a $P \in Syl_p(G)$ such that $P=(P\cap H)(P \cap K)$, with $P \cap H \in Syl_p(H)$ and $P \cap K \in Syl_p(K)$.

Proof Let us first find a Sylow $p$-subgroup $P$ of $G$ such that $P\cap H$ is a Sylow $p$-subgroup of $H$ and $P\cap K$ is a Sylow $p$-subgroup of $K$. Let $Q$ be a Sylow $p$-subgroup of $H$ and let $R$ be a Sylow $p$-subgroup of $K$. Choose a Sylow $p$-subgroup $S$ of $G$ such that $Q\subseteq S$. By Sylow theory, there is a $g\in G$ such that $R\subseteq S^g$. In particular, $S\cap H=Q$ and $S^g\cap K=R$.
But $g=hk$ for some $h\in H$ and $k\in K$. Then $S^g\cap K=R=S^{hk} \cap K=(S^h \cap K)^k$, hence $R^{k^{-1}}=S^h \cap K$ and this is a Sylow $p$-subgroup of $K$, being a conjugate of $R$. On the other hand, $S^h \cap H=(S \cap H)^h=Q^h \in Syl_p(H)$, since it is a conjugate of $Q$. So $P=S^h$ is the Sylow $p$-subgroup we were looking for.

Finally we use a counting argument to show that indeed $(P \cap H)(P \cap K)=P$. Observe that $$|(P \cap H)(P \cap K)|=\frac{|P \cap H| \cdot |P \cap K|}{|P \cap H \cap K|}=\frac{|H|_p \cdot |K|_p}{|P \cap H \cap K|}$$ where the $p$-subscript denotes the largest $p$-power dividing a positive integer (which is understood to be $1$ if the integer in question is not divisible by $p$).

Since $P \cap H \cap K$ is a $p$-subgroup of $H \cap K$, note that $|P \cap H \cap K| \leq |H \cap K|_p$. Combining this: $$|(P \cap H)(P \cap K)| \geq \frac{|H|_p \cdot |K|_p}{|H \cap K|_p}=[\frac{|H| \cdot |K|}{|H \cap K|}]_p=|G|_p=|P|$$ since $G=HK$ and $P \in Syl_p(G)$. As a set $(P \cap H)(P \cap K) \subseteq P$, so we conclude $P=(P \cap H)(P \cap K)$.$\square$

Realized that we (with all the comments above) did not answer the OP’s original question fully. So assume $G=MN$, $M \unlhd G$, $N \unlhd G$, with $M \cap N=1$, that is, $G$ is an (internal) direct product of $M$ and $N$. We use the well-known fact that for any $K \unlhd G$, and $P \in Syl_p(G)$, $P \cap K \in Syl_p(K)$, and all Sylow $p$-subgroups of $K$ arise in this way.

The key thing to show here is that for every $P \in Syl_p(G)$ we have $P=(P \cap M)(P \cap N)$. Note that this implies that $P \cong (P \cap M) \times (P \cap N)$.

Observe that $|G|=|MN|=|M|\dot|N|$ and $|(P \cap M)(P \cap N)|=|P \cap M| \dot |P \cap N|$ (remember $P \cap M \cap N \subseteq M \cap N = 1$).

So $|G|$=$ \color{darkblue}{\frac{|M|}{|P \cap M|}\cdot\frac{|N|}{|P \cap N|}}\cdot|(P \cap M)(P \cap N)|$, where the darkblue numbers are not divisible by $p$. It follows that $|(P\cap M)(P \cap N)|$ $|$ is divisible by $|P|$, and hence $P=(P \cap M)(P \cap N)$. The line of proof is similar to my other answer here on this page.

Note that if one would start from an external direct product, the proof would be somewhat simpler, by noting that the direct product of the Sylow $p$-subgroups of each of the factors, gives a Sylow $p$-subgroup of the whole group.

I believe that we can prove everything using the fact that Sylow $p$-subgroups are maximal $p$-subgroups (by definition).

Let $G = H \times K$ and let $P$ be a Sylow $p$-subgroup of $G$. Consider the projections onto each coordinate of the direct product:

\begin{align} \pi_1 : H \times K &\rightarrow H, \quad \pi_2 : H \times K \rightarrow K \\ (h,k) &\mapsto h, \quad (h,k) \mapsto k \end{align}

Notice that $P \leq \pi_1(P) \times \pi_2(P) \leq G$. But $\pi_1(P) \times \pi_2(P)$ is a $p$-group (see note below) and so since $P$ is a Sylow $p$-subgroup (and thus is maximal), we must have $P = \pi_1(P) \times \pi_2(P)$.

Now let's show that $\pi_1(P)$ is a Sylow $p$-subgroup of $H$ and $\pi_2(P)$ is a Sylow $p$-subgroup of $K$.

Without loss of generality, suppose $\pi_1(P)$ is not a Sylow $p$-subgroup of $H$. Since $\pi_1(P)$ is a $p$-subgroup (see note below), then in particular our assumption implies that it is not maximal in $H$. Then there exists a $p$-subgroup $Q \leq H$ with $|\pi_1(P)| < |Q|$. But then $Q \times \pi_2(P)$ is a $p$-subgroup of $G$ with

\begin{align} |P| = |\pi_1(P) \times \pi_2(P)| < |Q \times \pi_2(P)| \end{align}

But this is impossible, because $P$ is maximal.

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Note: I have used the following general property of elements of direct products:

\begin{align} |(h,k)| = \text{lcm}(|h|,|k|) \end{align}

So in particular, if $|(h,k)| = p^n$ for some $n$, then we know that both $h$ and $k$ must have order a power of $p$.