On the existence of countable unlimited groups (Q6641639)

The authors denote the group of finitary permutations on \(\mathbb{N}\) by \(\mathrm{Fin}(\mathbb{N})\). Observe that\N\Na permutation \( g \in S(N) \) is called limited if\N\[\N\omega(g) = \max_{\alpha \in N} |\alpha - \alpha g| < \infty.\N\]\NThe authors call a group \textit{limited} if it is isomorphic to a subgroup of\N\[\NG = \mathrm{Lim}(\mathbb{N}).\N\]\NClearly, \(\mathrm{Fin}(\mathbb{N})\) is a normal subgroup of \(\mathrm{Lim} (\mathbb{N})\). Examples of limited groups include countable free groups and countable locally finite groups, although not every countable locally finite group is contained in \(\mathrm{Fin}(\mathbb{N})\), (see [\textit{I. D. Ado}, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 50, 15--17 (1945; Zbl 0061.03002)]).\N\NIn analogy with the famous question due to D. Suprunenko about the subgroups of \(\mathrm{Fin}(\mathbb{N})\) (see [\textit{D. A. Suprunenko} and \textit{K. A. Hirsch} (ed.), Matrix groups. Translated from the Russian by Israel Program for Scientific Translations. Translation edited by K. A. Hirsch. American Mathematical Society (AMS), Providence, RI (1976; Zbl 0317.20028)]), the authors study the group \( G = \mathrm{Lim}(N) \), consisting of \textit{limited permutations} of natural numbers.\N\NIn this interesting paper, the authors prove that the additive group of rationals is unlimited. Moreover, they prove that a \(2\)-complete torsion free abelian group is unlimited.