I found this question in Arhangel'skii's book Topological Groups and Related Structures coauthored by Tkachenko. The first chapter of the book is devoted to algebraic preliminaries.
The question actually reads:
"Give an example of an infinite abelian group all proper subgroups of which are finite."
What I have done is: Every element of this group has finite order, else we could find an infinite proper subgroup, namely the group generated by $x²$ if $x$ has infinite order.
I think this can be strengthened: every element should have a prime order. Although I haven't proved this.
Intuitively this group cannot be and infinite product of smaller groups, because you could take the product of the even group factors and find an infinite proper subgroup.
Well, this is it, a highly non-trivial problem. Thanks in advance.