Weird group isomorphism problem

  • Thread starter y2kevin
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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

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?
It came up during class following the problem ZxZ/<(7,37)>. Apparently these two problems share some similar characteristics.
Have you tried working out the problem with ZxZ instead of Z? I'm pretty sure it should be ZxZ.

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