Condition for Subgroup of Power Set of Group to be Quotient Group

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

Let $\circ_\PP$ be the operation induced by $\circ$ on $\powerset G$, the power set of $G$.

Let $\struct {\LL, \circ_\PP}$ be a subgroup of the algebraic structure $\struct {\powerset G, \circ_\PP}$.

Then:
 * there exists a subgroup $H$ of $G$
 * and a normal subgroup $K$ of $H$
 * such that $\struct {\LL, \circ_\PP}$ is the quotient group $H / K$


 * the identity element of $\struct {\LL, \circ_\PP}$ is a subgroup of $\struct {G, \circ}$.
 * the identity element of $\struct {\LL, \circ_\PP}$ is a subgroup of $\struct {G, \circ}$.

Proof
From Power Set of Group under Induced Operation is Semigroup, we have that $\struct {\powerset G, \circ_\PP}$ is a semigroup.

Sufficient Condition
Let:
 * there exist a subgroup $H$ of $G$
 * and a normal subgroup $K$ of $H$
 * such that $\struct {\LL, \circ_\PP}$ is the quotient group $H / K$.

It is to be shown that the identity element of $\struct {\LL, \circ_\PP}$ is a subgroup of $\struct {G, \circ}$.

Necessary Condition
Let the identity element of $\struct {\LL, \circ_\PP}$ be a subgroup of $\struct {G, \circ}$.

It is to be shown that:
 * there exists a subgroup $H$ of $G$
 * and a normal subgroup $K$ of $H$
 * such that $\struct {\LL, \circ_\PP}$ is the quotient group $H / K$.