Power Structure of Group is Semigroup

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $P = \left({ \mathcal P \left({G}\right), \circ_{\mathcal P} }\right)$ be the algebraic structure consisting of the power set of $G$ and the subset product induced by $\circ$.

Then $P$ is a semigroup

Proof
We need to prove closure and associativity.

Closure
Let $\left({G, \circ}\right)$ be a group, and let $A, B \subseteq G$.

Associativity
This follows from Subset Product of Associative is Associative.

Thus $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$ is a semigroup.