# Quotient Subgroup of Semigroup Induced on Power Set

## 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 $\struct {\powerset G, \circ_\mathcal P}$, where:

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

## Proof

Follows directly from:

$\blacksquare$