Power Structure of Group is Semigroup

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

Then its power set $$\mathcal{P} \left({G}\right)$$ is a semigroup $$\left({\mathcal{P} \left({G}\right), \circ_{\mathcal {P}}}\right)$$ with respect to the subset product.

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.