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I don't quite understand what order is? I was thinking that it had to do with the number of elements in the set. | I don't quite understand what order is? I was thinking that it had to do with the number of elements in the set. | ||
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| + | ---- | ||
| + | Yeah I believe that's right. In class we were given the definitions: | ||
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| + | The ''order of a group'' G is the number of elements of G, written |G| <br> | ||
| + | The ''order of an element'' <math> g \in G </math> is the smallest positive number k such that g*...*g = 1. (k = # of g's being multiplied). | ||
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| + | So for example: <br> | ||
| + | <math>ord(7+15\mathbb{Z})</math>. | ||
| + | |||
| + | <p>So to find the order of <math>(7+15\mathbb{Z})</math>, we have to find the numbers of elements in this group.</p> | ||
| + | To do this, we take: <br> | ||
| + | 7 mod 15 = 7 <br> | ||
| + | 7^2 mod 15 = 4 <br> | ||
| + | 7^3 mod 15 = 13 <br> | ||
| + | 7^4 mod 15 = 1 <br> | ||
| + | |||
| + | <p>Tip: Instead of taking 7^3 mod 15, you can just take 7*4 = 28 mod 15, since: <br> | ||
| + | (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.</p> | ||
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| + | 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). <br> | ||
| + | Hence, <math>ord(7+15\mathbb{Z}) = 4</math> <br> | ||
| + | Hope the example helps you understand the definition of order. -Jesse | ||
Revision as of 14:41, 14 September 2008
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
