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Show ZXZ is an infinite cyclic group.

  1. Jan 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Show ZXZ is an infinite cyclic group. Under addition of course.

    2. Relevant equations

    3. The attempt at a solution
    So this obviously is an infinite cyclic group. Z is generated by <1> or <-1>.
    The problem I run into here is I think <(1,1)> will only generate elements of the form (a,a) s.t. a is an element of Z, (think along the diagonal). I do not see a way to generate something like (2,3).
  2. jcsd
  3. Jan 11, 2009 #2
    Wow, sorry I was confused, I see that there can not be any generators of ZXZ.
  4. Jan 11, 2009 #3


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    Science Advisor
    Homework Helper

    There can be generators. You just need more than one to generate the group. E.g. {(1,0), (0,1)} is a set that generates the whole group.
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