Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup

Theorem
Let $G$ be a group.

Let $P$ be a Sylow $p$-subgroup of $G$.

Let $H$ be a subgroup of $G$.

Then $P \cap H$ is not necessarily a Sylow $p$-subgroup of $H$.

Proof
We note that from Intersection of Subgroups is Subgroup that $P \cap H$ is a subgroup of $G$ and also of $H$.

Let $G$ be the dihedral group $D_3$, given by its group presentation:

By definition of Sylow $p$-subgroup, $\gen a$ is a Sylow $3$-subgroup of $G$.

However, $\gen b$ is also a subgroup of $G$, of order $2$.

But:
 * $\gen b \cap \gen a = e$

and $e$ is not a Sylow $3$-subgroup of $G$.