# Power Structure of Group is Monoid

## 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.

$\blacksquare$