I don't quite understand what order is? I was thinking that it had to do with the number of elements in the set.
Yeah I believe that's right. In class we were given the definitions:
The order of a group G is the number of elements of G, written |G|
The order of an element $ g \in G $ is the smallest positive number k such that g*...*g = 1. (k = # of g's being multiplied).
So for example:
$ ord(7+15\mathbb{Z}) $.
So to find the order of $ (7+15\mathbb{Z}) $, we have to find the numbers of elements in this group.
To do this, we take:
7 mod 15 = 7
7^2 mod 15 = 4
7^3 mod 15 = 13
7^4 mod 15 = 1
Tip: Instead of taking 7^3 mod 15, you can just take 7*4 = 28 mod 15, since:
(7^1 * 7^2) mod 15 = (7 * 4) mod 15 = (7^3) mod 15. Similarly, instead of finding 7^4 mod 15, you can just take (4 * 4) mod 15. This saves a lot of calculator hassle.
Since 7^4 mod 15 is 1, we can stop because we know our cycle is complete. Therefore, we can see our group has 4 elements: (7,4,13,1).
Hence, $ ord(7+15\mathbb{Z}) = 4 $
Hope the example helps you understand the definition of order. -Jesse
