# Quotient Subgroup of Semigroup Induced on Power Set

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## Theorem

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

Let $\struct {H, \circ}$ be a normal subgroup of $\struct {G, \circ}$.

Then $\struct {G / H, \circ_H}$ is a subgroup of $\struct {\powerset G, \circ_\PP}$, where:

- $\struct {G / H, \circ_H}$ is the quotient group of $G$ by $H$
- $\struct {\powerset G, \circ_\PP}$ is the semigroup induced by the operation $\circ$ on the power set $\powerset G$ of $G$.

## Proof

Follows directly from:

- Quotient Group is Group
- Cosets of $G$ by $H$ are subsets of $G$ and therefore elements of $\powerset G$
- The operation $\circ_H$ is defined as the subset product of cosets.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.4$