# 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:

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$

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 $G$.

$\blacksquare$