Power Set of Group under Induced Operation is Monoid

Theorem
Let $\struct {G, \circ}$ be a group with identity $e$.

Let $\struct {\powerset G, \circ_\PP}$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\powerset G$ by $\circ$.

Then $\struct {\powerset G, \circ_\PP}$ is a monoid with identity $\set e$.

Proof
By definition of a group, $\struct {G, \circ}$ is a monoid.

The result follows from Power Set of Monoid under Induced Operation is Monoid.