The intersection of normal subgroups is normal

Let $𝑁, 𝑀 ⊴ 𝐺$ be normal subgroups. Then $𝑁 \cap 𝑀 ⊴ 𝐺$ is a normal subgroup. group

Proof

The intersection of subgroups is a subgroup, so $𝑁 \cap 𝑀$ is a subgroup. Likewise, for any $𝑎 \in 𝑁 \cap 𝑀$ , $𝑔 𝑎 𝑔 - 1 \in 𝑁 \cap 𝑀$ for all $𝑔 \in 𝐺$ , hence $𝑁 \cap 𝑀$ is normal.

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The intersection of normal subgroups is normal

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