# Symmetric Group on 3 Letters

## Group Example

Let $S_3$ denote the set of permutations on $3$ letters.

The symmetric group on $3$ letters is the algebraic structure:

$\struct {S_3, \circ}$

where $\circ$ denotes composition of mappings.

It is usually denoted, when the context is clear, without the operator: $S_3$.

### Cycle Notation

It can be expressed in the form of permutations given in cycle notation as follows:

 $\ds e$ $:=$ $\ds \text { the identity mapping}$ $\ds p$ $:=$ $\ds \tuple {1 2 3}$ $\ds q$ $:=$ $\ds \tuple {1 3 2}$

 $\ds r$ $:=$ $\ds \tuple {2 3}$ $\ds s$ $:=$ $\ds \tuple {1 3}$ $\ds t$ $:=$ $\ds \tuple {1 2}$

### Cayley Table

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

### Group Presentation

Its group presentation is:

$S_3 := \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$

Hence:

$\begin{array}{c|cccccc} & e & a & a^2 & b & a b & a^2 b \\ \hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$

## Order of Elements

The orders of the various elements of $S_3$ are:

 $\ds e: \,$ $\ds$  $\ds$ Order $1$ $\ds \tuple {123}: \,$ $\ds \tuple {123}^2$ $=$ $\ds \tuple {132}$ $\ds \tuple {123} \tuple {132}$ $=$ $\ds e$ hence Order $3$ $\ds \tuple {132}: \,$ $\ds \tuple {132}^2$ $=$ $\ds \tuple {123}$ $\ds \tuple {132} \tuple {123}$ $=$ $\ds e$ hence Order $3$ $\ds \tuple {12}: \,$ $\ds \tuple {12}^2$ $=$ $\ds e$ hence Order $2$ $\ds \tuple {13}: \,$ $\ds \tuple {13}^2$ $=$ $\ds e$ hence Order $2$ $\ds \tuple {23}: \,$ $\ds \tuple {23}^2$ $=$ $\ds e$ hence Order $2$

## Subgroups

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

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

## Normal Subgroups

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:

$S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$

## Generators

Let:

 $\ds G_1$ $=$ $\ds \set {\tuple {123}, \tuple {12} }$ $\ds G_2$ $=$ $\ds \set {\tuple {13}, \tuple {23} }$

Then:

 $\ds S_3$ $=$ $\ds \gen {G_1}$ $\ds$ $=$ $\ds \gen {G_2}$

where $\gen G$ denotes the group generated by a subset $G$ of $S_3$.

## Centralizers

The centralizers of each element of $S_3$ are given by:

 $\ds \map {C_{S_3} } e$ $=$ $\ds S_3$ $\ds \map {C_{S_3} } {123}$ $=$ $\ds \set {e, \tuple {123}, \tuple {132} }$ $\ds \map {C_{S_3} } {132}$ $=$ $\ds \set {e, \tuple {123}, \tuple {132} }$ $\ds \map {C_{S_3} } {12}$ $=$ $\ds \set {e, \tuple {12} }$ $\ds \map {C_{S_3} } {23}$ $=$ $\ds \set {e, \tuple {23} }$ $\ds \map {C_{S_3} } {13}$ $=$ $\ds \set {e, \tuple {13} }$

## Normalizers of Subgroups

The normalizers of each subgroup of $S_3$ are given by:

 $\ds \map {N_{S_3} } {\set e}$ $=$ $\ds S_3$ $\ds \map {N_{S_3} } {\set {e, \tuple {123}, \tuple {132} } }$ $=$ $\ds S_3$ $\ds \map {N_{S_3} } {\set {e, \tuple {12} } }$ $=$ $\ds \set {e, \tuple {12} }$ $\ds \map {N_{S_3} } {\set {e, \tuple {13} } }$ $=$ $\ds \set {e, \tuple {13} }$ $\ds \map {N_{S_3} } {\set {e, \tuple {23} } }$ $=$ $\ds \set {e, \tuple {23} }$ $\ds \map {N_{S_3} } {S_3}$ $=$ $\ds S_3$

## Center

The center of $S_3$ is given by:

$\map Z {S_3} = \set e$

## Conjugacy Classes

The conjugacy classes of $S_3$ are:

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