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 Z3\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.

×\times001122
00
11
22

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:

×\times001122
00000000
11001122
22002211

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

×\times0\color{#D61F06}01\color{#20A900}12\color{#3D99F6}2
0\color{#D61F06}00\color{#D61F06}00\color{#D61F06}00\color{#D61F06}0
1\color{#20A900}10\color{#D61F06}01\color{#20A900}12\color{#3D99F6}2
2\color{#3D99F6}20\color{#D61F06}02\color{#3D99F6}21\color{#20A900}1

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

0\color{#D61F06}00\color{#D61F06}00\color{#D61F06}0
0\color{#D61F06}01\color{#20A900}12\color{#3D99F6}2
0\color{#D61F06}02\color{#3D99F6}21\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 Z17\Bbb Z_{17}

Note by Ali Caglayan
5 years, 1 month ago

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