Question
If $H$ and $K$ are subgroups of $G$, show that $H \cap K$ is a subgroup of $G$. (Can you see that the same proof shows that the intersection of any number of subgroups of $G$, finite or infinite, is again a subgroup of $G ?)$
Step 1
First, we need to show that $H \cap K$ is non-empty. Since $H$ and $K$ are subgroups of $G$, they both contain the identity element $e$ of $G$. Therefore, $e \in H \cap K$, and $H \cap K$ is non-empty. Show more…
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If $H$ and $K$ are subgroups of a group $G$, prove that $H \cap K$ is a subgroup of $G$. (Remember that $x \in H \cap K \quad$ iff $\quad x \in H \quad$ and $\quad x \in K)$.
SUBGROUPS
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