For an abelian group $G$, let $G[\operatorname{tors}]$ be its torsion subgroup.


Consider the **torsion sequence**:


$0 \rightarrow G[\operatorname{tors}] \rightarrow G \rightarrow G/G[\operatorname{tors}] \rightarrow 0$.



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> For which torsion abelian groups $T$ is it the case that for all abelian groups $G$ with $G[\operatorname{tors}] \cong T$, the torsion sequence splits?
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I know some sufficient conditions:


1. $T$ is **divisible**. Indeed, this holds iff $T$ is injective as a $\mathbb{Z}$-module iff *any* short exact sequence $0 \rightarrow T \rightarrow G \rightarrow G/T \rightarrow 0$ splits.


Thus divisibility is necessary and sufficient if one considers arbitrary short exact sequences, but in the special case $T = G[\operatorname{tors}]$ divisibility is not necessary. The torsion sequence also splits if:


2. $T$ has **bounded order**: $T = T[n]$ for some $n \in \mathbb{Z}^+$. (For this see e.g. see Corollary 20.14 of [these notes](http://people.brandeis.edu/%7Eigusa/Math131b/torsion.pdf) of K. Igusa.)


I do know *some* examples where the torsion sequence does not split, e.g., when $G = \prod\_{n=1}^{\infty} \mathbb{Z}/p^n \mathbb{Z}$.


But in fact I am interested in the case in which $T$ has "cofinite type", i.e., $T$ can be injected into $(\mathbb{Q}/\mathbb{Z})^n$ for some $n \in \mathbb{Z}^+$. (I am making up the terminology here; if I ever knew what the infinite abelian group people call this, it's not coming to mind at the moment.)


So for instance the simplest case that I don't know at the moment would be something like $T = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Q}\_p/\mathbb{Z}\_p$.


Not that it makes any difference as to what the answer is, but I would be very pleased to hear that the torsion sequence splits whenever $G[\operatorname{tors}]$ has "cofinite type". If you care why, see Theorem 5 [here](http://alpha.math.uga.edu/%7Epete/2011GaTechTalk.pdf).



When is the torsion subgroup of an abelian group a direct summand?
These are the (torsion) [cotorsion groups](http://eom.springer.de/C/c110460.htm). The following follows from a theorem of Baer:



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> A torsion abelian group is cotorsion if and only it is direct sum of a divisible torsion abelian group and an abelian group of bounded exponent.
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The original paper of R. Baer is "The subgroup of the elements of finite order of an abelian group", Ann. of Math. 37 (1936), 766-781. (See in particular Theorem 8.1.)


[I have made Baer's paper available [here](http://alpha.math.uga.edu/%7Epete/Baer36.pdf). --PLC]


