Cayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group — such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center — can be discovered from its Cayley table.
A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication:
Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ n = 1". In that paper they were referred to simply as tables, and were merely illustrative — they came to be known as Cayley tables later on, in honour of their creator.
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:
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:
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.
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