Symmetric Group on 4 Letters/Cayley Table
Cayley Table of Symmetric Group on $4$ Letters
The Cayley table of the symmetric group on $4$ letters 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}$
where the expression for $S_4$ in cycle notation is given as:
\(\ds e\) | \(:=\) | \(\ds \text { the identity mapping}\) | ||||||||||||
\(\ds t_{12}\) | \(:=\) | \(\ds \tuple {1 2}\) | ||||||||||||
\(\ds t_{23}\) | \(:=\) | \(\ds \tuple {2 3}\) | ||||||||||||
\(\ds r_{132}\) | \(:=\) | \(\ds \tuple {1 3 2}\) | ||||||||||||
\(\ds r_{123}\) | \(:=\) | \(\ds \tuple {1 2 3}\) | ||||||||||||
\(\ds t_{13}\) | \(:=\) | \(\ds \tuple {1 3}\) |
\(\ds t_{34}\) | \(:=\) | \(\ds \tuple {3 4}\) | ||||||||||||
\(\ds v_a\) | \(:=\) | \(\ds \tuple {1 2} \tuple {3 4}\) | ||||||||||||
\(\ds r_{243}\) | \(:=\) | \(\ds \tuple {2 4 3}\) | ||||||||||||
\(\ds f_{1432}\) | \(:=\) | \(\ds \tuple {1 4 3 2}\) | ||||||||||||
\(\ds f_{1243}\) | \(:=\) | \(\ds \tuple {1 2 4 3}\) | ||||||||||||
\(\ds r_{143}\) | \(:=\) | \(\ds \tuple {1 4 3}\) |
\(\ds r_{234}\) | \(:=\) | \(\ds \tuple {2 3 4}\) | ||||||||||||
\(\ds f_{1342}\) | \(:=\) | \(\ds \tuple {1 3 4 2}\) | ||||||||||||
\(\ds t_{24}\) | \(:=\) | \(\ds \tuple {2 4}\) | ||||||||||||
\(\ds r_{142}\) | \(:=\) | \(\ds \tuple {1 4 2}\) | ||||||||||||
\(\ds v_b\) | \(:=\) | \(\ds \tuple {1 3} \tuple {2 4}\) | ||||||||||||
\(\ds f_{1423}\) | \(:=\) | \(\ds \tuple {1 4 2 3}\) |
\(\ds f_{1234}\) | \(:=\) | \(\ds \tuple {1 2 3 4}\) | ||||||||||||
\(\ds r_{134}\) | \(:=\) | \(\ds \tuple {1 3 4}\) | ||||||||||||
\(\ds r_{124}\) | \(:=\) | \(\ds \tuple {1 2 4}\) | ||||||||||||
\(\ds t_{14}\) | \(:=\) | \(\ds \tuple {1 4}\) | ||||||||||||
\(\ds f_{1324}\) | \(:=\) | \(\ds \tuple {1 3 2 4}\) | ||||||||||||
\(\ds v_c\) | \(:=\) | \(\ds \tuple {1 4} \tuple {2 3}\) |