Cayley table

Named after the 19th-century British mathematician Arthur Cayley, a Cayley table 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:

× 1 −1
1 1−1
−1 −11
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