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Homework Help: Finding subgroups of Factor/ Quotient Groups

  1. Jan 22, 2013 #1
    1. The problem statement, all variables and given/known data

    Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there?

    3. The attempt at a solution

    I don't even know where to start with this question. If someone could just point me in the right direction that would be great.

    Thank you.
  2. jcsd
  3. Jan 22, 2013 #2
    I have made an attempt, if someone could let me know if it is correct or not, that would be much appreciated!

    The elements of Z/9Z are {0,1,2,3,4,5,6,7,8} with operation modulo9.

    The elements (1,2,4,5,7,8} have order 9 and generate the whole group.
    {0,3,6} has order 3.

    Therefore there are two subgroups.
  4. Jan 22, 2013 #3


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    {0} is also a subgroup.
  5. Jan 22, 2013 #4
    It looks like you're good for ##Z/9Z##.

    ##Z/3Z \times Z/3Z## is {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)} - which is not cyclic.
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