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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, 1 month ago

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