Quotient Group is Subgroup of Power Structure of Group

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 power structure of $\struct {G, \circ}$.

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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Theorem $11.4$