Structure and Layout
Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal to the product ba for all a and b in the group. In order to avoid confusion, the convention is that the factor that labels the row (termed nearer factor by Cayley) comes first, and that the factor that labels the column (or further factor) is second. For example, the intersection of row a and column b is ab and not ba, as in the following example:
| * | a | b | c |
|---|---|---|---|
| a | a2 | ab | ac |
| b | ba | b2 | bc |
| c | ca | cb | c2 |
Cayley originally set up his tables so that the identity element was first, obviating the need for the separate row and column headers featured in the example above. For example, they do not appear in the following table:
| a | b | c |
| b | c | a |
| c | a | b |
In this example, the cyclic group Z3, a is the identity element, and thus appears in the top left corner of the table. It is easy to see, for example, that b2 = c and that cb = a. Despite this, most modern texts — and this article — include the row and column headers for added clarity.
Read more about this topic: Cayley Table
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