Power Structure of Group is Monoid

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Theorem

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

Let $\struct {\powerset G, \circ_\PP}$ be the power structure of $\struct {G, \circ}$.


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 Structure of Monoid is Monoid.

$\blacksquare$