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Say you want to find all subgroups of <math>Z_n</math>. The corrolary states that, for each positive divisor k of n, the set <math>\langle n/k \rangle</math> is the unique subgroup of <math>Z_n</math> of order k. It also states that these subgroups are the only ones <math>Z_n</math> has. | Say you want to find all subgroups of <math>Z_n</math>. The corrolary states that, for each positive divisor k of n, the set <math>\langle n/k \rangle</math> is the unique subgroup of <math>Z_n</math> of order k. It also states that these subgroups are the only ones <math>Z_n</math> has. | ||
Hence, to enumerate the subgroups, just find all the positive integer divisors of n (in this case 20), and use them to generate the subgroups. | Hence, to enumerate the subgroups, just find all the positive integer divisors of n (in this case 20), and use them to generate the subgroups. | ||
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| + | ---- | ||
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| + | Confusion... | ||
| + | So, does it mean generator = subgroup? I mean,... like for the example above, 1,2,4,5,10,20 are the generators and <1>,<2>,<4>.... are the subgroup??? Correct me if I'm wrong... Thanks | ||
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| + | --[[User:Mmohamad|Mmohamad]] 21:07, 21 September 2008 (UTC) | ||
Revision as of 16:07, 21 September 2008
I do not understand how to tell what a generator of a subgroup is? I think that the subgroups of Z20 are (1,2,4,5,10,20), but that also might not be right. Anyways I could use a little explanation please.
Check the back of the book. Theres a selected answer/hint section. It gives some good information about the problem. The subgroups are given by (1,2,4,5,10,20), which are the generators. So I think you are on the right track. Hope that helps.
There is a corrolary to the Fundamental Theorem of Cyclic Groups on page 79 of the textbook that is really useful for this problem. Say you want to find all subgroups of $ Z_n $. The corrolary states that, for each positive divisor k of n, the set $ \langle n/k \rangle $ is the unique subgroup of $ Z_n $ of order k. It also states that these subgroups are the only ones $ Z_n $ has. Hence, to enumerate the subgroups, just find all the positive integer divisors of n (in this case 20), and use them to generate the subgroups.
Confusion... So, does it mean generator = subgroup? I mean,... like for the example above, 1,2,4,5,10,20 are the generators and <1>,<2>,<4>.... are the subgroup??? Correct me if I'm wrong... Thanks
--Mmohamad 21:07, 21 September 2008 (UTC)
