# Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup

## Theorem

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

Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset G$, the power set of $G$.

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

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

## Proof

From Power Structure of Group is Semigroup, we have that $\struct {\powerset G, \circ_\PP}$ is a semigroup.

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

From Subset Product of Normal Subgroups is Normal, $\struct {\HH, \circ_\PP}$ is closed.

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

$\blacksquare$