Power Structure of Group is Semigroup/Proof 1

Proof
We need to prove closure and associativity.

Closure
Let $\struct {G, \circ}$ be a group, and let $A, B \subseteq G$.

Thus $\struct {\powerset G, \circ_\mathcal P}$ is closed.

Associativity
It follows from Subset Product within Associative Structure is Associative that $\circ_\mathcal P$ is associative in $\struct {\powerset G, \circ_\mathcal P}$.

Thus $\struct {\powerset G, \circ_\mathcal P}$ is a semigroup.