Symmetric Group on 3 Letters/Normal Subgroups/Cayley table of Quotient Group
Normal Subgroups of the Symmetric Group on 3 Letters
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:
- $\begin{array}{c|cccccc}
\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$
Consider the subgroups of $S_3$:
The subsets of $S_3$ which form subgroups of $S_3$ are:
\(\ds \) | \(\) | \(\ds S_3\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set e\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \tuple {123}, \tuple {132} }\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \tuple {12} }\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \tuple {13} }\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {e, \tuple {23} }\) |
Of those, the normal subgroups in $S_3$ are:
- $S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$
Let $H$ denote the normal subgroup $\set {e, \tuple {123}, \tuple {132} }$.
Let $K$ denote the coset of $H$ in $S_3$.
The Cayley table of the quotient of $S_3$ by $H$ is given as:
- $\begin {array} {c|cc} S_3 / H & H & K \\ \hline H & H & K \\ K & K & H \end {array}$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.5 \ \text{(b)}$