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Homework Help: Classify this group

  1. Mar 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that ZxZ/<(a,b)> is isomorphic to Z if gcd(a,b)=1.

    3. The attempt at a solution

    I thought I had an idea but apparently I don't.

    I reasoned this geometrically. For ZxZ/<(1,a)> (for all a in Z) can be graphed as a line hitting points (k,a*k) in the x-y plane. If we shift the line covered by <(1,a)> along the y-axis (e.g. use cosets (0,y)+<(1,a)>, where y is an integer), we can hit all points in ZxZ. Hence, ZxZ/<(1,a)> is isomorphic to Z.

    But for ZxZ/<(2,3)>, the story changes, we skip all the (1,y) values if we shift by (0,y)+<(2,3)>. Hence, this leads me to conclude that we also need to include the possible cosets of (1,y)+<(2,a)>, making ZxZ/<(2,3)> isomorphic to Z2xZ. But apparently this is wrong.

    Can anyone shed a light on this? Thank you.
  2. jcsd
  3. Mar 19, 2009 #2
    any ideas?
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