1. The problem statement, all variables and given/known data Let G be an abelian group and let H and K be finite cyclic subgroups with |H|=r and |K|=s. Show that if r and s are relatively prime, then G contains a cyclic subgroup of order rs 2. Relevant equations Fundemental theorem of cyclic group which states that the order of any cyclic subgroup of some cyclic group G must be the divisor of the number of elements, n, in G. 3. The attempt at a solution Well the big trouble I'm having here (i think) is that I can't apply the fundamental theorem of cyclic group since I don't know if G is cyclic. I just know that it's abelian. The problem doesn't even state if G is finite (problem 6-56 of fraleigh). I know that every cyclic group is abelian. But I also know that the converse is not true in general. Is there any way for this problem to deduce if G is cyclic? If I knew G was cyclic then this problem is (almost) trivial. Or am I barking up the wrong tree?