Symmetric Group on 4 Letters

From ProofWiki
Jump to navigation Jump to search

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:

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


This needs considerable tedious hard slog to complete it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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:

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


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Symmetric_Group_on_4_Letters&oldid=527997"