Examples of an abelian group whose torsion subgroup is not a direct summand

Question

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.

Answer 1

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.