1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show ZXZ/<1,1> is an infinite cyclic group.

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

    Show ZXZ/<1,1> is an infinite cyclic group.

    2. Relevant equations



    3. The attempt at a solution
    <1,1> = {...(-1,-1), (0,0), (1,1),...}

    implies ZXZ/<1,1> = {<1,0>+<1,1>, <0,1>+<1,1>} which is isomorphic to ZXZ.

    But ZXZ is not cyclic, is my description of the factor group wrong , is it not isomorphic to ZXZ, or am I missing other points?
     
  2. jcsd
  3. Jan 11, 2009 #2
    what characterize an infinite cyclic group?

    1. generated by a single element
    2. infinite order

    can you put an element canonically into a certain form?
    so say you have (m, n), what can I write it as?
     
  4. Jan 11, 2009 #3
    if (m,n) is an element of one of the left cosets of the factor group then,
    if m=n (m,n) is in <1,1>. if m/=n then (m,n) = (m,n) + <1,1>. (because (0,0) is
    generated by <1,1>). I'm not sure what this implies though.
     
  5. Jan 11, 2009 #4
    so actually <1,0> + <1,1> generated all elements of ZXZ. Is it acceptable to say ZXZ/<1,1> is isomorphic to Z+0Z which is isomorphic to Z? (Which is generated by <1> or <-1>)
     
  6. Jan 11, 2009 #5
    <1,1> is just the identity. So, yeah.
     
  7. Jan 11, 2009 #6
    ... this is the additive group, (0,0) is the identity, and I'm using the notation <1,1> to imply that this is the cyclic generator. I hope that clears up any confusion.

    I'm not sure if the conclusion I've come to is correct.

    ZXZ/<1,1> = {Z+<1,1>}
    Clearly Z is infinite and cyclic generated by <1> or <-1>.
    So, ZXZ/<1,1> is a cyclic infinite group.
     
  8. Jan 11, 2009 #7
    well, (0,0)=(1,1) since you are modding out the diagonal set.
    (1,0) is the generator (or using coset notations (1,0)+<(1,1)>). Your conclusion is definitely correct (as in post 4)
     
  9. Jan 11, 2009 #8
    awesome thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?