Just today, I realised that I had been mis-interpreting the [FTFGAG](https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Classification).  One can speak of the $2^\infty$-torsion subgroup or the $2'$-torsion subgroup, the $3^\infty$-torsion or the $3'$-torsion subgroup, or even just the torsion subgroup of a FGAG or the maximal torsion-free quotient … so surely one can speak of *the* torsion-free subgroup, right?  In fact, when I was corrected on this, my first thought was to reply:  "just take the subgroup consisting of all infinite-order elements", and it only occured to me as I was saying it to wonder how the identity element would squeeze its way into this so-called 'subgroup'.