Cayley Table/Examples

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Examples of Cayley Tables

Cyclic Group of Order $4$

The Cayley table of the cyclic group of order $4$ can be written:

$\begin{array}{c|cccc} & e & a & b & c \\ \hline e & e & a & b & c \\ a & a & b & c & e \\ b & b & c & e & a \\ c & c & e & a & b \\ \end{array}$


Symmetric Group on $3$ Letters

The Cayley table of the symmetric group on $3$ letters can be written:

$\begin{array}{c|cccccc} \circ & e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p & p & q & e & s & t & r \\ q & q & e & p & t & r & s \\ r & r & t & s & e & q & p \\ s & s & r & t & p & e & q \\ t & t & s & r & q & p & e \\ \end{array}$


Arbitrary Structure of Order 3

A Cayley table does not necessarily describe the structure of a group.

The Cayley table of an algebraic structure of order $3$ can be presented:

$\begin{array}{c|cccc} \circ & a & b & c \\ \hline a & b & c & b \\ b & b & a & c \\ c & a & c & c \\ \end{array}$