Hw5 Problem 12
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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 $G$.