Symmetric Group on 4 Letters/Subgroups

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Subgroups of the Symmetric Group on $4$ Letters

Let $S_4$ denote the Symmetric Group on $4$ Letters, whose Cayley table is given as:

$\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}$


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

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


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Examples

Even Permutations

The subset of the Symmetric Group on $4$ Letters $S_4$ which consists of all the even permutations of $S_4$ forms a subgroup of $S_4$.


From Alternating Group is Set of Even Permutations, this is by definition the alternating group on $4$ letters $A_4$

Its Cayley table can be presented as follows:

$\begin{array}{c|cccc|cccc|cccc}

\circ & e & t & u & v & a & b & c & d & p & q & r & s \\ \hline e & e & t & u & v & a & b & c & d & p & q & r & s \\ t & t & e & v & u & b & a & d & c & q & p & s & r \\ u & u & v & e & t & c & d & a & b & r & s & p & q \\ v & v & u & t & e & d & c & b & a & s & r & q & p \\ \hline a & a & c & d & b & p & r & s & q & e & u & v & t \\ b & b & d & c & a & q & s & r & p & t & v & u & e \\ c & c & a & b & d & r & p & q & s & u & e & t & v \\ d & d & b & a & c & s & q & p & r & v & t & e & u \\ \hline p & p & s & q & r & e & v & t & u & a & d & b & c \\ q & q & r & p & s & t & u & e & v & b & c & a & d \\ r & r & q & s & p & u & t & v & e & c & b & d & a \\ s & s & p & r & q & v & e & u & t & d & a & c & b \\ \end{array}$


As $A_4$ has index $2$, it is normal in $S_4$ from Subgroup of Index 2 is Normal.

Hence the quotient group $S_4 / A_4$ is cyclic of order $2$.


Products of Disjoint Transpositions

Let $H$ be the subset of the Symmetric Group on $4$ Letters $S_4$ which consists of the $3$ products of disjoint transpositions of $S_4$, and the identity:

$V := \set {e, \tuple {1 2} \tuple {3 4}, \tuple {1 3} \tuple {2 4}, \tuple {1 4} \tuple {2 3} }$

Then $V$ forms a subgroup of $S_4$.


The Cayley table of $V$ can be presented as:

$\begin{array}{c|cccc}

\circ & e & \tuple {1 2} \tuple {3 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} \\ \hline 

                       e &                         e & \tuple {1 2} \tuple {3 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} \\

\tuple {1 2} \tuple {3 4} & \tuple {1 2} \tuple {3 4} & e & \tuple {1 4} \tuple {2 3} & \tuple {1 3} \tuple {2 4} \\ \tuple {1 3} \tuple {2 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} & e & \tuple {1 2} \tuple {3 4} \\ \tuple {1 4} \tuple {2 3} & \tuple {1 4} \tuple {2 3} & \tuple {1 3} \tuple {2 4} & \tuple {1 2} \tuple {3 4} & e \\ \end{array}$


This is the Klein $4$-group.

$V$ is normal in $S_4$.

The quotient group $S_4 / V$ is the Symmetric Group on $3$ Letters $S_3$.

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