4

According to Wikipedia there are abelian groups $G$ such that the short exact sequence $TG\to G\to G/TG$ is not split, where $TG$ is the torsion subgroup of $G$. However Wikipedia does not give any examples of such and I also can't think of any. What are some examples of such?

Edit: KCd gave an uncountable example. Would be interesting to know the answer for if there exists a countable such abelian group too.

Carla_
  • 361
  • 2
    I'm surprised I couldn't find a more on-point duplicate, but the two linked questions address both the basic standard example that KCd mentioned and a countable example with an stronger property (that some endomorphism of the torsion subgroup does not extend to the whole group). – Eric Wofsey Apr 09 '23 at 00:56

1 Answers1

3

Use $G = \prod_p \mathbf Z/p\mathbf Z$ where $p$ runs over the primes. See Example 4.7 here. See also Remark 4.8, Example 4.9, and Remark 4.10.

KCd
  • 55,662
  • Nice example. I dont see how 4.9 and 4.10 are relevant here. Also do you happen to know if there exists a countable group answer to the question? – Carla_ Apr 09 '23 at 00:24
  • 1
    You can get a countable counterexample by taking a countable subgroup for which the same argument still works (for instance, the preimage in $G$ of a countable divisible subgroup of $G/TG$). I give an explicit such subgroup at https://math.stackexchange.com/a/2411921/86856 (where I show it has a stronger property than just its torsion subgroup not being a direct summand). – Eric Wofsey Apr 09 '23 at 00:47
  • @Carla_ the relevance of 4.9 and 4.10 is that it shows without assuming the modules are over a PID (like $\mathbf Z$) you can find finitely generated examples where the torsion submodule is not a direct summand. – KCd Apr 09 '23 at 03:41