For an introduction to cayley tables see this page.
Generating cayley tables for hypercomplex numbers see this page.
Cyclic Groups
A group whose elements can be written as e, a, a²… an-1
shifting rows
One possibility would be to start with a row containing all the elements in order, this is the 'identity' row:
| 0 | 1 | 2 | 3 |
Then shift the row to the right (modulo n).
| 1 | 2 | 3 | 0 |
Repeat this until we have done a complete cycle, then put all the rows above each other, the completed table is:
| 0 | 1 | 2 | 3 |
| 1 | 2 | 3 | 0 |
| 2 | 3 | 0 | 1 |
| 3 | 0 | 1 | 2 |
This seems to generate a valid group, I don't know if such an algebra would have a valid application?
