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 $\Bbb Z_{17}$

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2^{34}

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
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`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$