TITLE: On Abelian group product of a Free and a Finite group
QUESTION [5 upvotes]: I'd need help with this problem:
Let $H$ be an abelian group.
Let $T \leq H$ and $T' \leq H$ be finite subgroups of H.
Let $F \leq H$ and $F' \leq H$ be free subgroups of H.
Suppose $H = T \times F = T' \times F'$. Show that $T = T'$. Does the same conclusion hold for $F$?
I don't really know how to approach the problem. I feel like I should use the Theorem of Classification of finitely generated abelian groups, but H isn't finitely generated. 
Thanks in advance for any help!

REPLY [3 votes]: Given a direct product of Abelian groups $G=T_0\times F_0$ with $T_0$ being finite and $F_0$ free, the torsion subgroup of $G$ is $T_0\times \{0_{F_0}\}$, as every element $x\neq 0_{F_0}$ in $F_0$ has infinite order. That is how you can identify $T_0$. So indeed, $T=T'$ in your problem. 
As for the free part, it is not true (not even for finitely generated Abelian groups). Say $G= (\mathbb{Z}_2,+)\times (\mathbb{Z},+)$. Then you can have $F= \langle (0,1) \rangle$ and $F'= \langle (1,1) \rangle$.