Power Set of Group under Induced Operation is Semigroup

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Theorem

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

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


Then $\struct {\powerset G, \circ_\mathcal P}$ is a semigroup.


Proof 1

We need to prove closure and associativity.


Closure

Let $\struct {G, \circ}$ be a group, and let $A, B \subseteq G$.

\(\displaystyle \) \(\) \(\displaystyle \forall a \in A, b \in B: a \circ b \in G\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle A \circ B \subseteq G\) Definition of Subset Product
\(\displaystyle \) \(\leadsto\) \(\displaystyle A \circ B \in \powerset G\) Definition of Power Set

Thus $\struct {\powerset G, \circ_\mathcal P}$ is closed.

$\Box$


Associativity

It follows from Subset Product within Semigroup is Associative that $\circ_\mathcal P$ is associative in $\struct {\powerset G, \circ_\mathcal P}$.

$\Box$


Thus $\struct {\powerset G, \circ_\mathcal P}$ is a semigroup.

$\blacksquare$


Proof 2

By definition a group is also a semigroup.

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

$\blacksquare$