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Group Theory Question

  1. Nov 10, 2014 #1
    1. The problem statement, all variables and given/known data
    Determine all the subgroups of (A,x_85) justify.
    where A = {1, 2, 4, 8, 16, 32, 43, 64}.


    3. The attempt at a solution
    To determine all of the subgroups of A, we find the distinct subgroups of A.
    <1> = {1}
    <2> = {1,2,4..} and so on?
    <4> = ...
    ...

    is this true? are there any other possible subgroups, i know i havnt posted my full solution.
     
  2. jcsd
  3. Nov 10, 2014 #2

    Dick

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    Ok, so you're dealing with a group of integers under multiplication mod 85. I think you should fill in the ...'s before anyone can figure out whether you have all of the subgroups.
     
  4. Nov 10, 2014 #3
    <1> = {1}
    <2> = {1,2,4, 8,16,32,43,64}
    <4> = {4,16,64,1}
    <8> = {1,2,4, 8,16,32,43,64}
    <16> = {16,1}

    Them are all the distinct subgroups of the group. for example. <2> = <43> .
     
  5. Nov 10, 2014 #4

    Dick

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    <2> and <8> aren't really distinct, are they? There are only four distinct subgroups. And I'm not sure what you are supposed to supply for justification. In a general group a subgroup might have more than one generator. But do you know what a cyclic group is?
     
  6. Nov 11, 2014 #5
    Yes. A cyclic group is a group with order n which contains an element of order n. Or better still a cyclic group is a group which contains an element that generates the group. 2 and 8 are not distinct! Silly me!

    I can't think of any other subgroups. I think I may have to show that <2> = <8> = <43> and so on?
     
  7. Nov 11, 2014 #6

    Dick

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    There is a theorem that if you have a cyclic group of order n then there is exactly one subgroup for each divisor of n. Since the four divisors of 8 are 1,2,4,8 then once you find four subgroups you know you are done.
     
  8. Nov 11, 2014 #7
    I am aware of this theorem!
    Thank you so much!
     
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