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tamintl
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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
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