Power Set of Group under Induced Operation is Monoid

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Theorem

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

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 $\mathcal P \left({G}\right)$ by $\circ$.


Then $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$ is a monoid with identity $\left\{{e}\right\}$.


Proof

By the definition of a group, $\left({G, \circ}\right)$ is a monoid.

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

$\blacksquare$