Brenna Gimenez

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 U23.

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 U23. Since multiplication is associative under the integers, we can see that multiplication is also associative for U23.

Identity:
We need to show that for every element, a, in U23 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.

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