# Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection

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## Theorem

Let $\struct {G, \circ}$ be a group.

Let $\HH$ be the set of all normal subgroups of $\struct {G, \circ}$.

Then the algebraic structure $\struct {\HH, \cap}$ is a subsemigroup of the algebraic structure $\struct {\powerset G, \cap}$.

## Proof

From Power Set with Intersection is Commutative Monoid, we have that $\struct {\powerset G, \cap}$ is *a fortiori* a semigroup.

Note that $\HH \subseteq \powerset G$.

Let $H_1$ and $H_2$ be normal subgroups of $\struct {G, \circ}$.

From Intersection of Normal Subgroups is Normal, $H_1 \cap H_2$ is also a normal subgroup of $\struct {G, \circ}$.

Hence $\struct {\HH, \cap}$ is closed.

By Subsemigroup Closure Test, $\struct {\HH, \cap}$ is a subsemigroup of $\struct {\powerset G, \cap}$.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.11$