Hw5 Problem 12

### Problem 12

(+) If $H$ and $K$ are subgroups of $G$, show that $H \bigcap K$ is a subgroup of $G$.

Solution

Closed:
Let $a, b \in H \bigcap K$.
Then $a, b \in H$ and $a, b \in K$.
Now because $H$ and $K$ are subgroups and thus closed, $ab \in H$ and $ab \in K$.
Hence $ab \in H \bigcap K$ and $H \bigcap K$ is closed under the group operation.

Identity:
Since $H$ and $K$ are subgroups $e \in H$ and $e \in K$, so then $e \in H \bigcap K$.

Inverse:
Let $a \in H \bigcap K$. Then because $H$ and $K$ are subgroups $a^{-1} \in H$ and $a^{-1} \in K$, so $a^{-1} \in H \bigcap K$.

Therefore, since all conditions hold, $H \bigcap K$ is a subgroup of $G$.