Cayley Table/Examples

Examples of Cayley Tables

Set of Self-Maps on Doubleton

Let $S$ be the set of self-maps on the doubleton $D = \set {a, b}$.

Let these be enumerated:

$\qquad \epsilon := \begin{pmatrix} a & b \\ a & b \end{pmatrix} \quad \alpha := \begin{pmatrix} a & b \\ b & a \end{pmatrix} \quad \beta := \begin{pmatrix} a & b \\ a & a \end{pmatrix} \quad \gamma := \begin{pmatrix} a & b \\ b & b \end{pmatrix}$

Let $\struct {S, \circ}$ be the semigroup of self-maps under composition of mappings.

The Cayley table of $\struct {S, \circ}$ can be written:

$\qquad \begin{array}{c|cccc} \circ & \epsilon & \alpha & \beta & \gamma \\ \hline \epsilon & \epsilon & \alpha & \beta & \gamma \\ \alpha & \alpha & \epsilon & \gamma & \beta \\ \beta & \beta & \beta & \beta & \beta \\ \gamma & \gamma & \gamma & \gamma & \gamma \\ \end{array}$

Cyclic Group of Order $4$

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

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

Symmetric Group on $3$ Letters

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

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

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:

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