The Cayley table of a group can be used to find the properties of that group and its elements. The entry in the row of element \(a\) and the column of element \(b\) is the product \(ab\) - note that in general, we cannot assume \(ab = ba\).
Using the Applet
This applet can be used to construct a table for a binary operation on a finite set (up to 10 elements), and determine whether or not the table is the Cayley table of an abstract group. If a group is formed, some information about the group elements will be displayed below.
After selecting the number of elements in the set, a blank Cayley table will be generated. You can rename the group elements, or use the default names. The rows and columns of the table will be labelled with these elements in the order they appear below. Elements must have distinct names. You can still change the element names while constructing your table.
To fill in the table, choose an element by clicking on it in the row or column heading, then click an empty entry of the table to place that element. After the entire table is filled, your table will be tested. If a group is formed, some information about the group will be returned, including which element is acting as an identity, its center, and the orders and inverses of each element in the group. The applet will also test if the group is cyclic or Abelian.
If the table does not represent a group, the applet will indicate which of the group axioms are not satisfied.
About this Applet