# Symmetric Group on 4 Letters

## Group Example

Let $S_4$ denote the set of permutations on $4$ letters.

The symmetric group on $4$ letters is the algebraic structure:

$\struct {S_4, \circ}$

where $\circ$ denotes composition of mappings.

It is usually denoted, when the context is clear, without the operator: $S_4$.

### Cycle Notation

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

 $\ds e$ $:=$ $\ds \text { the identity mapping}$ $\ds t_{12}$ $:=$ $\ds \tuple {1 2}$ $\ds t_{23}$ $:=$ $\ds \tuple {2 3}$ $\ds r_{132}$ $:=$ $\ds \tuple {1 3 2}$ $\ds r_{123}$ $:=$ $\ds \tuple {1 2 3}$ $\ds t_{13}$ $:=$ $\ds \tuple {1 3}$

 $\ds t_{34}$ $:=$ $\ds \tuple {3 4}$ $\ds v_a$ $:=$ $\ds \tuple {1 2} \tuple {3 4}$ $\ds r_{243}$ $:=$ $\ds \tuple {2 4 3}$ $\ds f_{1432}$ $:=$ $\ds \tuple {1 4 3 2}$ $\ds f_{1243}$ $:=$ $\ds \tuple {1 2 4 3}$ $\ds r_{143}$ $:=$ $\ds \tuple {1 4 3}$

 $\ds r_{234}$ $:=$ $\ds \tuple {2 3 4}$ $\ds f_{1342}$ $:=$ $\ds \tuple {1 3 4 2}$ $\ds t_{24}$ $:=$ $\ds \tuple {2 4}$ $\ds r_{142}$ $:=$ $\ds \tuple {1 4 2}$ $\ds v_b$ $:=$ $\ds \tuple {1 3} \tuple {2 4}$ $\ds f_{1423}$ $:=$ $\ds \tuple {1 4 2 3}$

 $\ds f_{1234}$ $:=$ $\ds \tuple {1 2 3 4}$ $\ds r_{134}$ $:=$ $\ds \tuple {1 3 4}$ $\ds r_{124}$ $:=$ $\ds \tuple {1 2 4}$ $\ds t_{14}$ $:=$ $\ds \tuple {1 4}$ $\ds f_{1324}$ $:=$ $\ds \tuple {1 3 2 4}$ $\ds v_c$ $:=$ $\ds \tuple {1 4} \tuple {2 3}$

### Cayley Table

The Cayley table of $S_4$ can be written:

$\begin{array}{c|cccccc|cccccc|cccccc|cccccc} \circ & e & t_{12} & t_{23} & r_{132} & r_{123} & t_{13} & t_{34} & v_a & r_{243} & f_{1432} & f_{1243} & r_{143} & r_{234} & f_{1342} & t_{24} & r_{142} & v_b & f_{1423} & f_{1234} & r_{134} & r_{124} & t_{14} & f_{1324} & v_c \\ \hline e & e & t_{12} & t_{23} & r_{132} & r_{123} & t_{13} & t_{34} & v_a & r_{243} & f_{1432} & f_{1243} & r_{143} & r_{234} & f_{1342} & t_{24} & r_{142} & v_b & f_{1423} & f_{1234} & r_{134} & r_{124} & t_{14} & f_{1324} & v_c \\ t_{12} & t_{12} & e & r_{132} & t_{23} & t_{13} & r_{123} & v_a & t_{34} & f_{1432} & r_{243} & r_{143} & f_{1243} & f_{1342} & r_{234} & r_{142} & t_{24} & f_{1423} & v_b & r_{134} & f_{1234} & t_{14} & r_{124} & v_c & f_{1324} \\ t_{23} & t_{23} & r_{123} & e & t_{13} & t_{12} & r_{132} & r_{243} & f_{1243} & t_{34} & r_{143} & v_a & f_{1432} & t_{24} & v_b & r_{234} & f_{1423} & f_{1342} & r_{142} & r_{124} & f_{1324} & f_{1234} & v_c & r_{134} & t_{14} \\ r_{132} & r_{132} & t_{13} & t_{12} & r_{123} & e & t_{23} & f_{1432} & r_{143} & v_a & f_{1243} & t_{34} & r_{243} & r_{142} & f_{1423} & f_{1342} & v_b & r_{234} & t_{24} & t_{14} & v_c & r_{134} & f_{1324} & f_{1234} & r_{124} \\ r_{123} & r_{123} & t_{23} & t_{13} & e & r_{132} & t_{12} & f_{1243} & r_{243} & r_{143} & t_{34} & f_{1432} & v_a & v_b & t_{24} & f_{1423} & r_{234} & r_{142} & f_{1342} & f_{1324} & r_{124} & v_c & f_{1234} & t_{14} & r_{134} \\ t_{13} & t_{13} & r_{132} & r_{123} & t_{12} & t_{23} & e & r_{143} & f_{1432} & f_{1243} & v_a & r_{243} & t_{34} & f_{1423} & r_{142} & v_b & f_{1342} & t_{24} & r_{234} & v_c & t_{14} & f_{1324} & r_{134} & r_{124} & f_{1234} \\ \hline t_{34} & t_{34} & v_a & r_{234} & f_{1342} & f_{1234} & r_{134} & e & t_{12} & t_{24} & r_{142} & r_{124} & t_{14} & t_{23} & r_{132} & r_{243} & f_{1432} & f_{1324} & v_c & r_{123} & t_{13} & f_{1243} & r_{143} & v_b & f_{1423} \\ v_a & v_a & t_{34} & f_{1342} & r_{234} & r_{134} & f_{1234} & t_{12} & e & r_{142} & t_{24} & t_{14} & r_{124} & r_{132} & t_{23} & f_{1432} & r_{243} & v_c & f_{1324} & t_{13} & r_{123} & r_{143} & f_{1243} & f_{1423} & v_b \\ r_{243} & r_{243} & f_{1243} & t_{24} & v_b & r_{124} & f_{1324} & t_{23} & r_{123} & r_{234} & f_{1423} & f_{1234} & v_c & e & t_{13} & t_{34} & r_{143} & r_{134} & t_{14} & t_{12} & r_{132} & v_a & f_{1432} & f_{1342} & r_{142} \\ f_{1432} & f_{1432} & r_{143} & r_{142} & f_{1423} & t_{14} & v_c & r_{132} & t_{13} & f_{1342} & v_b & r_{134} & f_{1324} & t_{12} & r_{123} & v_a & f_{1243} & f_{1234} & r_{124} & e & t_{23} & t_{34} & r_{243} & r_{234} & t_{24} \\ f_{1243} & f_{1243} & r_{243} & v_b & t_{24} & f_{1324} & r_{124} & r_{123} & t_{23} & f_{1423} & r_{234} & v_c & f_{1234} & t_{13} & e & r_{143} & t_{34} & t_{14} & r_{134} & r_{132} & t_{12} & f_{1432} & v_a & r_{142} & f_{1342} \\ r_{143} & r_{143} & f_{1432} & f_{1423} & r_{142} & v_c & t_{14} & t_{13} & r_{132} & v_b & f_{1342} & f_{1324} & r_{134} & r_{123} & t_{12} & f_{1243} & v_a & r_{124} & f_{1234} & t_{23} & e & r_{243} & t_{34} & t_{24} & r_{234} \\ \hline r_{234} & r_{234} & f_{1234} & t_{34} & r_{134} & v_a & f_{1342} & t_{24} & r_{124} & e & t_{14} & t_{12} & r_{142} & r_{243} & f_{1324} & t_{23} & v_c & r_{132} & f_{1432} & f_{1243} & v_b & r_{123} & f_{1423} & t_{13} & r_{143} \\ f_{1342} & f_{1342} & r_{134} & v_a & f_{1234} & t_{34} & r_{234} & r_{142} & t_{14} & t_{12} & r_{124} & e & t_{24} & f_{1432} & v_c & r_{132} & f_{1324} & t_{23} & r_{243} & r_{143} & f_{1423} & t_{13} & v_b & r_{123} & f_{1243} \\ t_{24} & t_{24} & r_{124} & r_{243} & f_{1324} & f_{1243} & v_b & r_{234} & f_{1234} & t_{23} & v_c & r_{123} & f_{1423} & t_{34} & r_{134} & e & t_{14} & t_{13} & r_{143} & v_a & f_{1342} & t_{12} & r_{142} & r_{132} & f_{1432} \\ r_{142} & r_{142} & t_{14} & f_{1432} & v_c & r_{143} & f_{1423} & f_{1342} & r_{134} & r_{132} & f_{1324} & t_{13} & v_b & v_a & f_{1234} & t_{12} & r_{124} & r_{123} & f_{1243} & t_{34} & r_{234} & e & t_{24} & t_{23} & r_{243} \\ v_b & v_b & f_{1324} & f_{1243} & r_{124} & r_{243} & t_{24} & f_{1423} & v_c & r_{123} & f_{1234} & t_{23} & r_{234} & r_{143} & t_{14} & t_{13} & r_{134} & e & t_{34} & f_{1432} & r_{142} & r_{132} & f_{1342} & t_{12} & v_a \\ f_{1423} & f_{1423} & v_c & r_{143} & t_{14} & f_{1432} & r_{142} & v_b & f_{1324} & t_{13} & r_{134} & r_{132} & f_{1342} & f_{1243} & r_{124} & r_{123} & f_{1234} & t_{12} & v_a & r_{243} & t_{24} & t_{23} & r_{234} & e & t_{34} \\ \hline f_{1234} & f_{1234} & r_{234} & r_{134} & t_{34} & f_{1342} & v_a & r_{124} & t_{24} & t_{14} & e & r_{142} & t_{12} & f_{1324} & r_{243} & v_c & t_{23} & f_{1432} & r_{132} & v_b & f_{1243} & f_{1423} & r_{123} & r_{143} & t_{13} \\ r_{134} & r_{134} & f_{1342} & f_{1234} & v_a & r_{234} & t_{34} & t_{14} & r_{142} & r_{124} & t_{12} & t_{24} & e & v_c & f_{1432} & f_{1324} & r_{132} & r_{243} & t_{23} & f_{1423} & r_{143} & v_b & t_{13} & f_{1243} & r_{123} \\ r_{124} & r_{124} & t_{24} & f_{1324} & r_{243} & v_b & f_{1243} & f_{1234} & r_{234} & v_c & t_{23} & f_{1423} & r_{123} & r_{134} & t_{34} & t_{14} & e & r_{143} & t_{13} & f_{1342} & v_a & r_{142} & t_{12} & f_{1432} & r_{132} \\ t_{14} & t_{14} & r_{142} & v_c & f_{1432} & f_{1423} & r_{143} & r_{134} & f_{1342} & f_{1324} & r_{132} & v_b & t_{13} & f_{1234} & v_a & r_{124} & t_{12} & f_{1243} & r_{123} & r_{234} & t_{34} & t_{24} & e & r_{243} & t_{23} \\ f_{1324} & f_{1324} & v_b & r_{124} & f_{1243} & t_{24} & r_{243} & v_c & f_{1423} & f_{1234} & r_{123} & r_{234} & t_{23} & t_{14} & r_{143} & r_{134} & t_{13} & t_{34} & e & r_{142} & f_{1432} & f_{1342} & r_{132} & v_a & t_{12} \\ v_c & v_c & f_{1423} & t_{14} & r_{143} & r_{142} & f_{1432} & f_{1324} & v_b & r_{134} & t_{13} & f_{1342} & r_{132} & r_{124} & f_{1243} & f_{1234} & r_{123} & v_a & t_{12} & t_{24} & r_{243} & r_{234} & t_{23} & t_{34} & e \\ \end{array}$

## Subgroups

The subsets of $S_4$ which form subgroups of $S_4$ are:

 $\ds$  $\ds S_4$ $\ds$  $\ds \set e$ $\ds$  $\ds \set {e, \tuple {12} \tuple {34}, \tuple {13} \tuple {24}, \tuple {14} \tuple {23} }$ $\ds$  $\ds \set {e, \tuple {123}, \tuple {132}, \tuple {124}, \tuple {142}, \tuple {134}, \tuple {143}, \tuple {234}, \tuple {243}, \tuple {12} \tuple {34}, \tuple {13} \tuple {24}, \tuple {14} \tuple {23} }$

## Normalizers

Let $\alpha$ denote the permutation in $S_4$ given in cycle notation as $\tuple {1234}$.

The normalizer of $S = \set {\alpha, \alpha^{-1} }$ in $S_4$ is given by:

$\map {N_{S_4} } S = \set {e, \alpha, \alpha^2, \alpha^3, \beta, \alpha \beta, \alpha^2 \beta, \alpha^3 \beta}$

where $\beta$ denotes the permutation in $S_4$ given in cycle notation as $\tuple {24}$.

$\blacksquare$

## Conjugacy Classes

The conjugacy classes of $S_4$ are:

 $\ds$  $\ds \set e$ $\ds$  $\ds \set {\tuple {12}, \tuple {13}, \tuple {14}, \tuple {23}, \tuple {24}, \tuple {34} }$ $\ds$  $\ds \set {\tuple {12} \tuple {34}, \tuple {13} \tuple {24}, \tuple {14} \tuple {23} }$ $\ds$  $\ds \set {\tuple {123}, \tuple {124}, \tuple {134}, \tuple {134}, \tuple {132}, \tuple {142}, \tuple {143}, \tuple {243} }$ $\ds$  $\ds \set {\tuple {1234}, \tuple {1243}, \tuple {1324}, \tuple {1342}, \tuple {1423}, \tuple {1432} }$