# 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).

 $\times$ $\color{#D61F06}0$ $\color{#20A900}1$ $\color{#3D99F6}2$ $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#20A900}1$ $\color{#D61F06}0$ $\color{#20A900}1$ $\color{#3D99F6}2$ $\color{#3D99F6}2$ $\color{#D61F06}0$ $\color{#3D99F6}2$ $\color{#20A900}1$

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

 $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#D61F06}0$ $\color{#20A900}1$ $\color{#3D99F6}2$ $\color{#D61F06}0$ $\color{#3D99F6}2$ $\color{#20A900}1$

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 A Former Brilliant Member
6 years, 8 months ago

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