Power Structure of Group is Semigroup
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Theorem
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.
Closure
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.
$\Box$
Associativity
It follows from Subset Product within Semigroup is Associative that $\circ_\PP$ is associative in $\struct {\powerset G, \circ_\PP}$.
$\Box$
Thus $\struct {\powerset G, \circ_\PP}$ is a semigroup.
$\blacksquare$
Proof 2
By definition a group is also a semigroup.
The result then follows from Power Structure of Semigroup is Semigroup.
$\blacksquare$