Intersection with Normal Subgroup is Normal/Examples/Subset Product of Normal Subgroup with Intersection

Theorem
Let $\struct G$ be a group whose identity is $e$.

Let $H_1, H_2$ be subgroups of $G$.

Let:
 * $N_1 \lhd H_1$
 * $N_2 \lhd H_2$

where $\lhd$ denotes the relation of being a normal subgroup.

Then:
 * $N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$

Proof
Consider arbitrary $x_n \in N_1, x_h \in H_1 \cap H_2$.

Thus:
 * $x_n x_h \in N_1 \paren {H_1 \cap H_2}$

Note that as $x_h \in H_1 \cap H_2$ it follows that $x_h \in H_1$.

We aim to show that:
 * $x_n x_h N_1 \paren {H_1 \cap N_2} \paren {x_n x_h}^{-1} \subseteq N_1 \paren {H_1 \cap H_2}$

thus demonstrating $N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$ by the Normal Subgroup Test.

We have: