This is the multiplication table for the group. This completely defines the group although many of the table entries can derived from the others so it contains quite a lot of redundancy.
Cayley tables can be used to define a group as explained on this page or as part of a definition for a ring or associated algebra. They can also be used in a slightly modified form for hypercomplex numbers see this page.
Latin Squares
The Cayley Table is a Latin Square. The properties of a group result in the following properties when a Cayley table represents a group
- no repetition in any row or column
- one row and one column will represent the identity element and so will be in order.
So each element of the group appears once in each row and each column.
Proof of this - suppose x appears in a row labeled with 'a' twice:
x = a b
x = a c
then we cancel to get: b=c but we use distinct elements to label the columns.
Example
Here is a Cayley table representing a group with the elements '1','a','b','aba','ba' and 'ab' as follows:
| 1 | a | b | aba | ba | ab | |
| 1 | 1 | a | b | aba | ba | ab |
| a | a | 1 | ab | ba | aba | b |
| b | b | ba | 1 | ab | a | aba |
| aba | aba | ab | ba | 1 | b | a |
| ba | ba | b | aba | a | ab | 1 |
| ab | ab | aba | a | b | 1 | ba |
If we take any element say 'aba' then we can see that it appears in each row and each column only once.
| 1 | a | b | aba | ba | ab | |
| 1 | 1 | a | b | aba | ba | ab |
| a | a | 1 | ab | ba | aba | b |
| b | b | ba | 1 | ab | a | aba |
| aba | aba | ab | ba | 1 | b | a |
| ba | ba | b | aba | a | ab | 1 |
| ab | ab | aba | a | b | 1 | ba |
