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
3 years, 10 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> 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"
MathAppears 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} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...