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:


 * $\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:
 * $c \circ a = e$

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.

Also known as
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.

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.

Non-Commutative Structures
When depicting a commutative structure, 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 structure $S$ being depicted is non-commutative, 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 $p r = s$.