Set of Subgroups of Abelian Group form Subsemigroup of Power Structure

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

Let $\struct {\powerset G, \circ_\PP}$ denote the power structure of $\struct {G, \circ}$.

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
From Power Structure of Semigroup is Semigroup:
 * $\struct {\powerset S, \circ_\PP}$ is a semigroup.

Let $A$ and $B$ be arbitrary subgroups of $G$.

We need to show that $A \circ_\PP B$ is also a subgroup of $G$.

Let $x$ and $y$ be arbitrary elements of $A \circ_\PP B$.

Then:

Hence the result from the One-Step Subgroup Test.