Quotient Group is Subgroup of Power Structure of Group
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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 $11.4$