Set of Subgroups of Abelian Group form Subsemigroup of Structure Induced on Power Set

Theorem
Let $\struct {G, \circ}$ be an abelian group.

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

Let $\SS$ be the set of all subgroups of $G$.

Then $\struct {\SS, \circ_\PP}$ is a subsemigroup of $\struct {\powerset G, \circ_\PP}$.

Proof
A fortiori, an abelian group is a commutative semigroup.

Also a fortiori, a subgroup of $G$ is also a subsemigroup of $G$.

The result then follows directly from Set of Subsemigroups of Commutative Semigroup form Subsemigroup of Structure Induced on Power Set.