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:

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

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

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:

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

Subgroups

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

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

Normal Subgroups

Consider the subgroups of $S_3$:

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

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

Of those, the normal subgroups in $S_3$ are:

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

Generators

Let:

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

Then:

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

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:

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

Normalizers of Subgroups

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

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

Center

The center of $S_3$ is given by:

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

Conjugacy Classes

The conjugacy classes of $S_3$ are:

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