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

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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 $H / K$

if and only if:

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


Proof

From Power Structure of Group 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 $H / K$.


Then the identity element of $\struct {\LL, \circ_\PP}$ is $e K = K$, which is a subgroup of $\struct {G, \circ}$.

$\Box$


Necessary Condition

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

Denote this subgroup by $\struct {E, \circ}$.


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 $H / K$.

We claim that:

$H = \bigcup \LL$

and:

$K = E$


To show that $\struct {\bigcup \LL, \circ}$ is a subgroup of $G$, we will use the Two-Step Subgroup Test.

We have by hypothesis that $\bigcup \LL$ is non-empty.


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Let $a, b \in \bigcup \LL$ be arbitrary.

Then:

$\exists A, B \in \LL: a \in A \land b \in B$


Since $\struct {\LL, \circ_\PP}$ is a group, by Group Axiom $\text G 0$: Closure:

$A \circ_\PP B \in \LL$


By definition of $\circ_\PP$:

$a \circ b \in A \circ_\PP B$


Hence:

$a \circ b \in \bigcup \LL$

showing that $\bigcup \LL$ is closed.


Since $\struct {\LL, \circ_\PP}$ is a group, by Group Axiom $\text G 3$: Existence of Inverse Element:

$\exists A^{-1} \in \LL: A \circ_\PP A^{-1} = E$


Let $a' \in A^{-1}$ be arbitrary.

We have:

$a \circ a' \in E$


Since $\struct {E, \circ}$ is a group, by Group Axiom $\text G 3$: Existence of Inverse Element again:

$\paren {a \circ a'}^{-1} \in E$


By Group Axiom $\text G 2$: Existence of Identity Element:

\(\ds e\) \(=\) \(\ds \paren{a \circ a' } \circ \paren {a \circ a'}^{-1}\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds a \circ \paren {a' \circ \paren {a \circ a'}^{-1} }\) Group Axiom $\text G 1$: Associativity

By Group Product Identity therefore Inverses:

$a' \circ \paren {a \circ a'}^{-1}$ is an inverse of $a$.


Moreover:

$a' \circ \paren {a \circ a'}^{-1} \in A^{-1} \circ_\PP E = A^{-1} \subseteq \bigcup \LL$

Therefore, every element of $\bigcup \LL$ has an inverse in $\bigcup \LL$.


By the Two-Step Subgroup Test:

$\struct {\bigcup \LL, \circ}$ is a subgroup of $G$.


Now we will show that $E$ is a normal subgroup of $\bigcup \LL$.


We have:

$a \circ E \circ a^{-1} \subseteq A \circ_\PP E \circ_\PP A^{-1} = A \circ_\PP A^{-1} = E$
$a^{-1} \circ E \circ a \subseteq A^{-1} \circ_\PP E \circ_\PP A = A^{-1} \circ_\PP A = E$


and thus $E$ is by definition a normal subgroup.


Finally, we need to show the equality:

$\paren {\bigcup \LL} / E = \LL$


We do this by showing that each set in $\LL$ is a (left) coset of $E$ in $\bigcup \LL$.


That is:

$A = a \circ E$


We have:

$a \circ E \subseteq A \circ_\PP E = A$


Let $x \in A$ be arbitrary.


We have:

$a^{-1} \circ x \in A^{-1} \circ_\PP A = E$


Hence:

\(\ds x\) \(=\) \(\ds e \circ x\) Group Axiom $\text G 2$: Existence of Identity Element
\(\ds \) \(=\) \(\ds \paren {a \circ a^{-1} } \circ x\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds a \circ \paren {a^{-1} \circ x}\) Group Axiom $\text G 1$: Associativity
\(\ds \) \(\in\) \(\ds a \circ E\) as shown above

This shows that:

$A \subseteq a \circ E$

Thus, by definition of set equality:

$A = a \circ E$

Hence the result.


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