The set of integers less than 23 and relatively prime to 23 with multiplication modulo 23.

This is the set {1, 2, 3, 4, 5, …., 22}, and is also known as U_{23}.

x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 |

3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 1 | 4 | 7 | 10 | 13 | 16 | 19 | 22 | 2 | 5 | 8 | 11 | 14 | 17 | 20 |

4 | 4 | 8 | 12 | 16 | 20 | 1 | 5 | 9 | 13 | 17 | 21 | 2 | 6 | 10 | 14 | 18 | 22 | 3 | 7 | 11 | 15 | 19 |

5 | 5 | 10 | 15 | 20 | 2 | 7 | 12 | 17 | 22 | 4 | 9 | 14 | 19 | 1 | 6 | 11 | 16 | 21 | 3 | 8 | 13 | 18 |

6 | 6 | 12 | 18 | 1 | 7 | 13 | 19 | 2 | 8 | 14 | 20 | 3 | 9 | 15 | 21 | 4 | 10 | 16 | 22 | 5 | 11 | 17 |

7 | 7 | 14 | 21 | 5 | 12 | 19 | 3 | 10 | 17 | 1 | 8 | 15 | 22 | 6 | 13 | 20 | 4 | 11 | 18 | 2 | 9 | 16 |

8 | 8 | 16 | 1 | 9 | 17 | 2 | 10 | 18 | 3 | 11 | 19 | 4 | 12 | 20 | 5 | 13 | 21 | 6 | 14 | 22 | 7 | 15 |

9 | 9 | 18 | 4 | 13 | 22 | 8 | 17 | 3 | 12 | 21 | 7 | 16 | 2 | 11 | 20 | 6 | 15 | 1 | 10 | 19 | 5 | 14 |

10 | 10 | 20 | 7 | 17 | 4 | 14 | 1 | 11 | 21 | 8 | 18 | 5 | 15 | 2 | 12 | 22 | 9 | 19 | 6 | 16 | 3 | 13 |

11 | 11 | 22 | 10 | 21 | 9 | 20 | 8 | 19 | 7 | 18 | 6 | 17 | 5 | 16 | 4 | 15 | 3 | 14 | 2 | 13 | 1 | 12 |

12 | 12 | 1 | 13 | 2 | 14 | 3 | 15 | 4 | 16 | 5 | 17 | 6 | 18 | 7 | 19 | 8 | 20 | 9 | 21 | 10 | 22 | 11 |

13 | 13 | 3 | 16 | 6 | 19 | 9 | 22 | 12 | 2 | 15 | 5 | 18 | 8 | 21 | 11 | 1 | 14 | 4 | 17 | 7 | 20 | 10 |

14 | 14 | 5 | 19 | 10 | 1 | 15 | 6 | 20 | 11 | 2 | 16 | 7 | 21 | 12 | 3 | 17 | 8 | 22 | 13 | 4 | 18 | 9 |

15 | 15 | 7 | 22 | 14 | 6 | 21 | 13 | 5 | 20 | 12 | 4 | 19 | 11 | 3 | 18 | 10 | 2 | 17 | 9 | 1 | 16 | 8 |

16 | 16 | 9 | 2 | 18 | 11 | 4 | 20 | 13 | 6 | 22 | 15 | 8 | 1 | 17 | 10 | 3 | 19 | 12 | 5 | 21 | 14 | 7 |

17 | 17 | 11 | 5 | 22 | 16 | 10 | 4 | 21 | 15 | 9 | 3 | 20 | 14 | 8 | 2 | 19 | 13 | 7 | 1 | 18 | 12 | 6 |

18 | 18 | 13 | 8 | 3 | 21 | 16 | 11 | 6 | 1 | 19 | 14 | 9 | 4 | 22 | 17 | 12 | 7 | 2 | 20 | 15 | 10 | 5 |

19 | 19 | 15 | 11 | 7 | 3 | 22 | 18 | 14 | 10 | 6 | 2 | 21 | 17 | 13 | 9 | 5 | 1 | 20 | 16 | 12 | 8 | 4 |

20 | 20 | 17 | 14 | 11 | 8 | 5 | 2 | 22 | 19 | 16 | 13 | 10 | 7 | 4 | 1 | 21 | 18 | 15 | 12 | 9 | 6 | 3 |

21 | 21 | 19 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 | 1 | 22 | 20 | 18 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 |

22 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |

**Associativity:**

We need to show that (ab)c=a(bc) for all a,b,c in U_{23}. Since multiplication is associative under the integers, we can see that multiplication is also associative for U_{23}.

**Identity:**

We need to show that for every element, a, in U_{23} there exists an identity element that satisfies: ea = ae =a.

Letting e=1, we can see from the table above that 1 is the identity for each element from 1….22.

**Inverse:**

We need to show that for every element in the set, there exists an inverse, b, such that ab = ba =e. As we can see from the table, there is only one 1 in each row. The corresponding entries who have 1 as their product are inverses.

For example: 3(8) = 8(3) = 1 and 13(16) = 16(13) = 1

Since all three properties hold, we have proved that this is a group.