Cayley table

Definition

Let $S = \{a_1, \ldots, a_n\}$ be a finite set with $n$ elements, listed in a fixed order, and let $\ast$ be a binary operation on $S$. The Cayley table of $(S, \ast)$ is the $n \times n$ table whose rows and columns are labelled $a_1, \ldots, a_n$. The entry in row $i$ and column $j$ is $a_i \ast a_j$.

Remarks

The operation is commutative exactly when the table is symmetric about its main diagonal. In the Cayley table of a finite group, each element occurs exactly one time in each row and one time in each column. Thus the Cayley table of a finite group is a Latin square.

Explanations

L1 human

A Cayley table is like the multiplication table from school, but for any finite set with an operation. For the integers modulo $2$ under addition, the table has the rows $0, 1$ and $1, 0$. You can read many properties from the table, for example if the operation is commutative.