Hw11 Problem 4

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Problem 4

Show that if $H$ and $N$ are subgroups of a group $G$, and $N$ is normal in $G$, then $H\cap N$ is normal in $H$. Show by an example that $H\cap N$ need not be normal in $G$.

Solution

$H \cap N$ is a subgroup of $G$, and is in $H$, hence $H \cap N$ is a subgroup of $H$. Let $h \in H$ and $x \in H \cap N$, so $x \in N$. We know $N$ is normal in $G$, so by Theorem 14.13, $hxh^{-1} \in N \ \ \ \forall h \in G$ and $x \in N$. Since $H$ is a subgroup (closed and contains inverse) $hxh^{-1} \in H$, since $h \in H$ and $x \in H$. Therefore, $hxh^{-1} \in H \cap N$ and $H \cap N$ is normal in $H$.