I want to solve the following problem (Fuchs, "Infinite Abelian Groups", Vol.$2$, pp. $153$ Ex. $4$):
"Let $A$ be a torsion-free group of finite rank $n$ and $F$, $F'$ free subgroups of $A$ of rank $n$. Then $A/F$ and $A/F'$ are isomorphic to subgroups of $\oplus_{i=1}^n\mathbb{Q}/\mathbb{Z}$ and $A/F\oplus G\cong A/F' \oplus G' $ for suitable finite groups $G$ and $G'$."
How can I prove the existence of such groups $G$ and $G'$?
Thank you in advance for your help.
Best Answer
The subgroup $F\cap F'$ is free abelian. Working in the quotient by this subgroup reduces to proving the following:
In an abelian group $G$, and prime $p$, let $G_p$ be the $p$-primary part (the set of element of order dividing some power of $p$). We have $B/U\simeq\bigoplus_p(B/U)_p\simeq\bigoplus_p B_p/U_p$, where the sum is over primes. For all but finitely many primes $p$, $U_p=V_p=0$. For such a prime, we have $(B/U)_p\simeq B_p\simeq (B/V)_p$. So it is enough to prove result for the finitely many individual remaining primes.
In other words, we can suppose that $B=B_p$, which is an artinian $p$-primary group. Structure of artinian abelian groups says that, for some integer $k$, $B$ is isomorphic to $C_{p^\infty}^k\oplus F_B$, where $F_B$ is a finite abelian $p$-group (namely $F_B$ is isomorphic to the quotient of $B$ by its divisible part, i.e. is the largest finite quotient of $B$). We easily deduce that $B/U$ is isomorphic to $C_{p^\infty}^k\oplus F_{B/U}$ and $B/V$ is isomorphic to $C_{p^\infty}^k\oplus F_{B/V}$. Hence $(B/U)\oplus F_{B/V}$ is isomorphic to $(B/V)\oplus F_{B/U}$.