1)If a group is abelian then all its subgroups are normal.

But turn the question around,

If all the subgroups of a group are normal then it is abelian?

I was not able to show that.

Since this is my own question I believe that is is wrong.

I think we should consider cases when the group is finite and infinite. Maybe that will help because if it is finite then it has a composition series (but I am not sure what that leads to).