# Symmetric Group on 3 Letters/Subgroups

## Subgroups of the Symmetric Group on $3$ Letters

Let $S_3$ denote the Symmetric Group on $3$ Letters, whose Cayley table is given as:

$\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$

The subsets of $S_3$ which form subgroups of $S_3$ are:

 $\displaystyle$  $\displaystyle S_3$ $\displaystyle$  $\displaystyle \set e$ $\displaystyle$  $\displaystyle \set {e, \tuple {123}, \tuple {132} }$ $\displaystyle$  $\displaystyle \set {e, \tuple {12} }$ $\displaystyle$  $\displaystyle \set {e, \tuple {13} }$ $\displaystyle$  $\displaystyle \set {e, \tuple {23} }$

## Examples

### Non-Subgroup

Consider the subset $H$ of $S_3$:

$H = \set {e, \tuple {12}, \tuple {13}, \tuple {23} }$

Then $H$ is not a subgroup of $S_3$.