I came up with a new group.

Let where .

Then under complex multiplication will form a group.

I hope this is group is not isomorphic to any of the known ones. And hopefully has some interesting properties.

We know the following:

1) is an abelian group.

2) is a countable group.

3*) has a cyclic subgroup for every finite order.

What are the answers to the following questions:

1)Is is free abelian group?

2)Is a finitely generated abelian group?