Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup
Jump to navigation
Jump to search
Theorem
Let $\struct {G, \circ}$ be a group.
Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset G$, the power set of $G$.
Let $\HH$ be the set of all normal subgroups of $\struct {G, \circ}$.
Then the algebraic structure $\struct {\HH, \circ_\PP}$ is a subsemigroup of the algebraic structure $\struct {\powerset G, \circ_\PP}$.
Proof
From Power Structure of Group is Semigroup, we have that $\struct {\powerset G, \circ_\PP}$ is a semigroup.
Note that $\HH \subseteq \powerset G$.
From Subset Product of Normal Subgroups is Normal, $\struct {\HH, \circ_\PP}$ is closed.
By Subsemigroup Closure Test, $\struct {\HH, \circ_\PP}$ is a subsemigroup of $\struct {\powerset G, \circ_\PP}$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.11$