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.