Power Set of Group under Induced Operation is Semigroup/Proof 1

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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.


We need to prove closure and associativity.


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 $\struct {\powerset G, \circ_\PP}$ is closed.



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


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