Cyclic subgroups of an Abelian group

dancavallaro

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?

e(ho0n3

Yes you are. Since 4 and 6 divide n, so must 12. (Think of the prime factorization of n).