Quotient Group is Subgroup of Power Structure of Group

Theorem
Let $\left({G, \circ}\right)$ be a group and let $\left({H, \circ}\right)$ be a normal subgroup of $\left({G, \circ}\right)$.

Then $\left({G / H, \circ_H}\right)$ is a subgroup of $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$, where:
 * $\left({G / H, \circ_H}\right)$ is the quotient group of $G$ by $H$;
 * $\left({\mathcal P \left({G}\right), \circ_\mathcal P}\right)$ is the semigroup induced by the operation $\circ$ on the power set $\mathcal P \left({G}\right)$ of $G$.

Proof
Follows directly from:


 * Quotient Group is a Group
 * Cosets of $G$ by $H$ are subsets of $G$ and therefore elements of $\mathcal P \left({G}\right)$
 * The operation $\circ_H$ is defined as the subset product of cosets.