Alternating Group on 4 Letters/Cayley Table

Cayley Table of Alternating Group on $4$ Letters

The Cayley table of the alternating group on $4$ letters can be written:

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

where the expression for $A_4$ in cycle notation is given as:

\(\ds e\)

\(:=\)

\(\ds \text { the identity mapping}\)

\(\ds t\)

\(:=\)

\(\ds \tuple {1 2} \tuple {3 4}\)

\(\ds u\)

\(:=\)

\(\ds \tuple {1 3} \tuple {2 4}\)

\(\ds v\)

\(:=\)

\(\ds \tuple {1 4} \tuple {2 3}\)

\(\ds a\)

\(:=\)

\(\ds \tuple {1 2 3}\)

\(\ds b\)

\(:=\)

\(\ds \tuple {1 3 4}\)

\(\ds c\)

\(:=\)

\(\ds \tuple {2 4 3}\)

\(\ds d\)

\(:=\)

\(\ds \tuple {1 4 2}\)

\(\ds p\)

\(:=\)

\(\ds \tuple {1 3 2}\)

\(\ds q\)

\(:=\)

\(\ds \tuple {2 3 4}\)

\(\ds r\)

\(:=\)

\(\ds \tuple {1 2 4}\)

\(\ds s\)

\(:=\)

\(\ds \tuple {1 4 3}\)

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Tables: $3$. Alternating group $\map A 4$