# 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:

$\begin {array} {c|cc} S_3 / H & H & K \\ \hline H & H & K \\ K & K & H \end {array}$