Suppose that the order of some finite Abelian group is divisible by 10. Prove that the group has a cyclic subgroup of order 10.

My proof so far:

Suppose G is a finite Abelian group with 10 | |G|. By a theorem, I know that G contains a subgroup with order 10. Now, a subgroup of a Abelian group is also Abelian (is that right? I recall that from an exercise I did), and since G can be written as both of these groups are cyclic, so G contains a cyclic subgroup.

Is that right?

Thanks.