Quotient Subgroup of Semigroup Induced on Power Set

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

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

Then $\struct {G / N, \circ_N}$ is a subgroup of $\struct {\powerset G, \circ_\PP}$, where:
 * $\struct {G / N, \circ_N}$ is the quotient group of $G$ by $N$
 * $\struct {\powerset G, \circ_\PP}$ 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 $N$ are subsets of $G$ and therefore elements of $\powerset G$
 * The operation $\circ_N$ is defined as the subset product of cosets.