Weird group isomorphism problem

1. Mar 3, 2009

y2kevin

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

Thanks.

2. Mar 3, 2009

e(ho0n3

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?

3. Mar 3, 2009

y2kevin

It came up during class following the problem ZxZ/<(7,37)>. Apparently these two problems share some similar characteristics.

4. Mar 5, 2009

anyone?

5. Mar 5, 2009

e(ho0n3

Have you tried working out the problem with ZxZ instead of Z? I'm pretty sure it should be ZxZ.