Suppose $B$ is an abelian group such that for every integer $n\ge 1$, the $n$-torsion subgroup $B[n]$ is finite.


Let $B\_{\rm tor} = \varinjlim\_{n\ge 1} B[n]$ be the torsion subgroup of $B$.


Is it true that, necessarily, there exists an integer $d\ge 0$ such that


$$B\_{\rm tor} \simeq (\mathbf{Q}/\mathbf{Z})^d\oplus F,$$
for $F$ a finite group?


What if we replace $B\_{\rm tor}$ by $B[\ell^{\infty}] = \varinjlim\_{n\ge 1} B[\ell^n]$ for a single prime $\ell$, and $(\mathbf{Q}/\mathbf{Z})^d\oplus F$ by $(\mathbf{Q}\_{\ell}/\mathbf{Z}\_{\ell})^d\oplus F\_{\ell}$ for $F\_{\ell}$ a finite $\ell$-group?



Co-finite type abelian groups
The first question has been answered by Jeremy Rickard. Another counter-example is the Prüfer group $\mathbb Z[\ell^{\infty}]$ (this is a subgroup of a group of the form you ask for, but it is not of that form).


For the second: you are taking a torsion $\ell$-group (your $B[\ell^{\infty}]$) with finite socle, so that you have a group which is an essential extension of its finite socle. Such groups are called finitely cogenerated and they are known to be the subgroups of a finite direct sums of Prüfer groups. Hence, as you are considering an $\ell$-group, it is of the form $\mathbb Z(\ell^{\infty})^k\oplus F$, for $F$ a finite $\ell$-group.


Let me remark that, in the general case, if you ask that $|Soc(B)|=\sum\_{\text{$p$ prime}}|B[p]|<\infty$, then the torsion part of $B$ will be of the form
$$B\_{tor}\cong \mathbb Z[\ell\_1^{\infty}]\oplus\ldots\oplus \mathbb Z[\ell\_n^{\infty}]\oplus F$$
with $F$ finite. 


