Intersection with Normal Subgroup is Normal/Examples

Examples of Use of Intersection with Normal Subgroup is Normal

Subset Product of Normal Subgroup with Intersection

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:

$N_1 \paren {H_1 \cap N_2} \lhd N_1 \paren {H_1 \cap H_2}$

Subset Product of Intersection with Intersection

Let $\struct {G, \circ}$ 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:

$\paren {H_1 \cap N_2} \paren {H_2 \cap N_1} \lhd \paren {H_1 \cap H_2}$