Power Structure of Group is Semigroup/Proof 1

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

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

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

Proof
We need to prove closure and associativity.

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

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

Associativity
It follows from Subset Product of Associative is Associative that $\circ_\mathcal P$ is associative in $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$.

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