Symmetric Group on 4 Letters/Subgroups/Examples/Even Permutations

Example of Subgroup of Symmetric Group on 4 Letters

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 $2$.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations