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 $\map A 4$

Alternating Group on 4 Letters/Cayley Table

Cayley Table of Alternating Group on $4$ Letters

Sources

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