# Intersection with Normal Subgroup is Normal

## Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$, and let $N$ be a normal subgroup of $G$.

Then $H \cap N$ is a normal subgroup of $H$.

## Proof

By Intersection of Subgroups is Subgroup, $H \cap N$ is a subgroup of $N$.

It remains to be shown that $H$ is normal in $H$.

Because $N \lhd G$:

$\forall n \in N: \forall g \in G: g n g^{-1} \in N$

Let $x \in H \cap N$.

Because $H \le G$ and therefore closed:

$\forall x \in H \cap N: \forall g \in H: g x g^{-1} \in H$

But since $x \in N$ and $N \lhd G$, $g x g^{-1} \in N$.

The result follows.

$\blacksquare$

## Examples

### Subset Product of Normal Subgroup with Intersection

Let $\struct G$ be a group whose identity is $e$.

Let $H_1, H_2$ be subgroups of $G$.

Let:

$N_1 \lhd H_1$
$N_2 \lhd H_2$

where $\lhd$ denotes the relation of being a normal subgroup.

Then:

$N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$

### Subset Product of Intersection with Intersection

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $H_1, H_2$ be subgroups of $G$.

Let:

$N_1 \lhd H_1$
$N_2 \lhd H_2$

where $\lhd$ denotes the relation of being a normal subgroup.

Then:

$\paren {H_1 \cap N_2} \paren {H_2 \cap N_1} \lhd \paren {H_1 \cap H_2}$