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 \(\Bbb Z_{17}\)

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