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Cyclic Groups and Subgroups

  1. Oct 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Find all of the subgroups of Z3 x Z3

    2. Relevant equations
    Z3 x Z3 is isomorphic to Z9

    3. The attempt at a solution
    x = (0,1,2,3,4,5,6,7,8)
    <x0> or just <0> = {0}
    <1> = {identity}
    <2> = {0,2,4,6} also wasn't sure if I did this one correctly x o x for x2
    <3> = {0,3,6}
    and so on until I got
    <8> = {0,8}
    <9> = {0}

    I feel like I might be completely wrong but is this even a cyclic group, and do I need to approach finding the subgroups differently? Thanks
  2. jcsd
  3. Oct 11, 2010 #2


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    Z3xZ3 is not cyclic. Any element of Z3xZ3 has order 3. And even if it were, your subgroups of Z9 have problems.
  4. Oct 11, 2010 #3


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    If you're going to compute the subgroup of Z9 generated by 2, you can't stop at 8. We start with 0, 2, 4, 6, 8, and then 2+8=10=1 mod 9, 1+2=3, then you get 5 and 7 so 2 generates the whole group.

    Of course like Dick said those aren't the subgroups you're looking for anyway
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