[[Normal subgroup]]
# The intersection of normal subgroups is normal

Let $N,M \trianglelefteq G$ be normal subgroups.
Then $N \cap M \trianglelefteq G$ is a normal subgroup. #m/thm/group 

> [!check]- Proof
> [[The intersection of subgroups is a subgroup]], so $N \cap M$ is a subgroup.
> Likewise, for any $a \in N \cap M$, $gag^{-1} \in N \cap M$ for all $g \in G$, hence $N \cap M$ is normal.
> <span class="QED"/>

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