Definition:Cayley Table

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Definition

A Cayley table is a technique for describing an algebraic structure (usually a finite group) by putting all the products in a square array:

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


The column down the left hand side denotes the first (leading) operand of the operation.

The row across the top denotes the second (following) operand of the operation.

Thus, in the above Cayley table:

$c \circ a = d$


If desired, the symbol denoting the operation itself can be put in the upper left corner, but this is not essential if there is no ambiguity.


The order in which the rows and columns are placed is immaterial.

However, it is conventional, when representing an algebraic structure with an identity element, to place that element at the head of the first row and column.


Entry

The occurrences in a Cayley table of the elements of the algebraic structure it defines are called the entries of the Cayley table.


Also known as

Some sources refer to a Cayley table as an operation table, but there exists the view that this sounds too much like a piece of hospital apparatus.

Another popular name for this is a multiplication table, but this holdover from grade school terminology may be considered irrelevant to a table where the operation has nothing to do with multiplication as such.

In the field of logic, a truth table in this format is often referred to as matrix form, but note that this terminology clashes with the definition of a matrix in mathematics.


Examples

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:

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


Also see

  • Results about Cayley tables can be found here.


Source of Name

This entry was named for Arthur Cayley.


Historical Note

The concept of the Cayley table was proposed in $1854$ by Arthur Cayley.


Sources

  • 1854: Arthur Cayley: On the theory of groups, as depending on the symbolic equation $\theta^n - 1$ (Phil. Mag. Ser. 4 Vol. 7: pp. 40 – 47)
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