1. The problem statement, all variables and given/known data If G is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must G contain? 2. Relevant equations A cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors. 3. The attempt at a solution I know that a cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors, so I'm inclined to say that G must have cyclic subgroups with orders 1, 2, and 3. But I also have a hunch that the subgroups of orders 4 and 6 combine in some way, so maybe there's also a cyclic subgroup of order lcm(4,6) = 12? Am I on the right track here?