Orders of elements in a group.

[tamintl]

1. Homework Statement
Remember, the set of groups (Gn, *), the group of multiplicatively-invertible elements of Z/n under multiplication. For p a prime, the elements of Gp are all elements of Z/p except 0; for n not a prime, the elements of Gn are all the elements of Z/n except 0 and those (besides 1) which divide n and all multiples of those elements.

a) Consider G15. What are the elements of G15, and what is the order of each elements? What group is G15 isomorphic to?


3. The Attempt at a Solution

For the first part, What are the elements of G15:

I think what the statement at the top is trying to tell me is that it is all of the elements of Z/15 except for 0 and the elements that divide 15 (besides 1) and the elements that are multiples of those elements.

So if that logic is correct I come up with the group: G15 = {1, 2, 4, 7, 8, 11, 13, 14}

Next it asks for the order of each element, this is where I am a little confused.

1 has order ?
2 has order ?
4 has order ?
7 has order ..
8 has order ..
11 has order..
13 has order..
14 has order ..


I have literally no idea how to find the orders of each element. Is there a systematic way?

Please shed some light

Regards tamintl

 

[micromass]

Simply multiply the number with itself, until you get 1 (mod 15). The number of times you had to multiply is the order.

Let me do two examples, I'll let you do the rest:
- 1 has order 1, this is by definition. So let's do something more interesting:
- 2. Multiply by 2, we get 4. Multiply by 2, we get 8. Multiply by 2 we get 16=1 (mod 15). Thus 2⁴=1, and thus 4 is the order.