Co-finite type abelian groups

Question

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?

Answer 1

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.

Answer 2

For the first question, $$\bigoplus_{p\text{ prime}}\mathbb{Z}/p\mathbb{Z}$$ is a counterexample.