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Homework Help: Weird group isomorphism problem

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the group Z/<(a,b)> is isomorphic to Z if gcd(a,b)=1. Find generators of Z/<(a,b)>.

    2. Relevant information
    Please note that the question is asking for Z/<(a,b)>, not ZxZ/<(a,b)>. I am having trouble understanding the meaning behind <(a,b)> as a subgroup of Z. My professor only gave the hint that <(a,b)> is cyclic and has an analogous case for ZxZ/<(a,b)> if gcd(a,b)=1.

    3. The attempt at a solution
    From my understanding, <(a,b)> should represent the set generated by a and b, ie, <(a,b)>={ n*a+m*b | n and m are integers }. However, for a and b s.t. gcd(a,b)=1, there are integers n and m such that n*a+m*b=1, in which case every element in Z can be generated from this linear combination.

    This is clearly not the case since Z/<(a,b)> would be Z/Z={0}.

    Does anyone have any idea what else <(a,b)> could be in this context

  2. jcsd
  3. Mar 3, 2009 #2
    Normally, the group generated by a and b is written <a,b>. Also, gcd(a,b) is sometimes written (a,b). So maybe Z/<(a,b)> is actually Z/<1>. But that doesn't help either. Where did you find this problem?
  4. Mar 3, 2009 #3
    It came up during class following the problem ZxZ/<(7,37)>. Apparently these two problems share some similar characteristics.
  5. Mar 5, 2009 #4
  6. Mar 5, 2009 #5
    Have you tried working out the problem with ZxZ instead of Z? I'm pretty sure it should be ZxZ.
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