Cayley Tables


A Cayley table is a multiplication table for a finite group. You'll sometimes hear them called group multiplication tables. The Cayley table for a group G helps you with computation: you can easily look up group products and even the inverses of elements. And for very small groups, you can even use Cayley tables to find all possible groups for each order!


How many groups are there of order 3? Let's name our elements { e, x, y } where e is the identity element. If you write the 3x3 Cayley table, you can fill in a lot of elements. Since e is an identity element we know that e*e = e, e*x = x, e*y = y, x*e = x, and y*e = y. This leaves 4 empty squares. There's also the added restriction that every row and every column must contain all the elements in the group - repeats are not allowed. (Play around and try to figure out why this is the case.) Once you try all possibilities you'll find there is only one group with 3 elements. It can take different forms, but they're all the same. The most familiar manifestation of this group would be the integers mod 3 under addition.