Power Structure of Group is Semigroup

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

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

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

Proof 1

We need to prove closure and associativity.


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

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


Proof 2

By definition a group is also a semigroup.

The result then follows from Power Structure of Semigroup is Semigroup.