5. Show that the intersection of two normal subgroups is a normal subgroup.

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Let $$ N_1 $$ and $$ N_2 $$ be normal subgroups of a group $$ G $$. By definition, for any $$ g \in G $$ and $$ n_1 \in N_1 $$, we have $$ g n_1 g^{-1} \in N_1 $$. Similarly, for any $$ g \in G $$ and $$ n_2 \in N_2 $$, we have $$ g n_2 g^{-1} \in N_2 $$.
The intersection of the two normal subgroups is defined as $$ N = N_1 \cap N_2 $$. We need to show that $$ N $$ is also a normal subgroup of $$ G $$.
To show that $$ N $$ is normal in $$ G $$, we need to prove that for any $$ g \in G $$ and $$ n \in N $$, the element $$ g n g^{-1} $$ is also in $$ N $$. Since $$ n \in N $$, it follows that $$ n \in N_1 $$ and $$ n \in N_2 $$.
Since $$ N_1 $$ is normal, we have $$ g n g^{-1} \in N_1 $$. Similarly, since $$ N_2 $$ is normal, we have $$ g n g^{-1} \in N_2 $$.
Since $$ g n g^{-1} $$ is in both $$ N_1 $$ and $$ N_2 $$, it follows that $$ g n g^{-1} \in N_1 \cap N_2 = N $$. Thus, $$ N $$ is normal in $$ G $$.
Therefore, the intersection of two normal subgroups $$ N_1 $$ and $$ N_2 $$ is a normal subgroup of $$ G $$.

Show that the intersection of two normal subgroups is a normal subgroup.

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