×

# Cayley Tables

A Cayley Table is a multiplication table for a group. This sounds simple however some interesting patterns and things come from this.

For example take the integers modulo 3, denoted as $$\Bbb Z_3$$. Let's define multiplication in the usual sense. You will find that this forms a group (you can check this).

Now writing the elements of the group out in Lexicographical order we can form a table.

 $$\times$$ $$0$$ $$1$$ $$2$$ $$0$$ $$1$$ $$2$$

Now we simply fill it in by applying the multiplication to the corresponding elements in the rows and columns in the table. You should have:

 $$\times$$ $$0$$ $$1$$ $$2$$ $$0$$ $$0$$ $$0$$ $$0$$ $$1$$ $$0$$ $$1$$ $$2$$ $$2$$ $$0$$ $$2$$ $$1$$

Now if I colour the elements lexicographically we can see a pattern emerge (more clearly).

 $$\times$$ $$\color{red}0$$ $$\color{green}1$$ $$\color{blue}2$$ $$\color{red}0$$ $$\color{red}0$$ $$\color{red}0$$ $$\color{red}0$$ $$\color{green}1$$ $$\color{red}0$$ $$\color{green}1$$ $$\color{blue}2$$ $$\color{blue}2$$ $$\color{red}0$$ $$\color{blue}2$$ $$\color{green}1$$

We can go further and assume lexicographical order (of some sort) and remove the reference rows and columns.

 $$\color{red}0$$ $$\color{red}0$$ $$\color{red}0$$ $$\color{red}0$$ $$\color{green}1$$ $$\color{blue}2$$ $$\color{red}0$$ $$\color{blue}2$$ $$\color{green}1$$

The interesting thing is, is that we can do this for any group and reveal some of it's symmetries and patterns.

Over the next few days I shall be releasing the coloured Cayley tables for some groups that I have generated. We shall start of with simpler groups and eventually get more complicated. But first:

The Image at the top of the post is the Cayley Table for $$\Bbb Z_{17}$$

Note by Ali Caglayan
2 years, 11 months ago