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How could this group possibly have elements of this order?

  1. Dec 8, 2011 #1
    1. The problem statement, all variables and given/known data
    resulto3.jpg

    It seems to me that every element of Z_45 has order 1, 3, 5, 9, 15, or 45. It seems impossible to have an element of order 2 by Lagrange's Theorem.

    Is there another way of looking at this problem?
     
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  3. Dec 8, 2011 #2

    micromass

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    For example, (30,0,0,0) has order 2.

     
  4. Dec 8, 2011 #3
    I see! Do you think a counting argument would be acceptable? For example there are two elements of order 2 in Z_60, (0 and 30), one element of order 2 in Z_45 (0), two elements of order 2 in Z_12 (0 and 6) and two elements of order 2 in Z_36 (0 and 18)

    So it seems to me, all of the elements of order 2 in A would be:

    (0, 0, 0, 0)
    (0, 0, 0, 18)
    (0, 0, 6, 0)
    (0, 0, 6, 18)
    (30, 0, 0, 0)
    (30, 0, 6, 0)
    (30, 0, 0, 18)
    (30, 0, 6, 18)

    So 8 elements, correct?
     
  5. Dec 8, 2011 #4

    micromass

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    (0,0,0,0) does not have order 2.
     
  6. Dec 8, 2011 #5
    Oh thank you, you are correct. Is my argument mathematically correct where I just list out all of the possible elements of order 2?

    Also, do you know of a better way to look at the number of subgroups of index 2 in A?
     
  7. Dec 8, 2011 #6

    micromass

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    Yes, the rest of what you did is absolutely correct!!!!
     
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