Definition:Cayley Table

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.

Some sources call this 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.

Examples

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

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

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

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

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

Non-Abelian Groups

When depicting an abelian group, it is clear there is no ambiguity as to where to place the elements. As $x y = y x$, the table is symmetrical about the major axis.

However, when the group $G$ being depicted is non-abelian, by definition there are entries $x, y \in G$ such that $x y \ne y x$.

The convention is that the first element of a pair goes down the column at the left, while the second element goes across the top.

This can be seen in the second of the above tables, where, for example, $r p = t$ and $p r = s$.

Source of Name

This entry was named for Arthur Cayley.

Sources

• W.E. Deskins: Abstract Algebra (1964): $\S 1.4$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.1$: Example $58$