up to isomorphism to $\{Q^i_m\}_{i\in\mathbb{N}}$ i.e, every maximal subgroup of $G^{\mathscr{F}}$ is isomorphic to $Q^i_m$ for some $i\in\mathbb{N}$ and for every $i$ there exists a maximal subgroup of $G^{\mathscr{F}}$ that is isomorphic to $Q^i_m$.

Taking $\mathscr{F}=\{\mathbb{Q}\}$ in previous theorem we recover the following theorem by V.S. Guba,

[@Gu86 Theorem 1] There exists a non trivial finitely generated torsion free divisible group.

By taking $\mathscr{F}=\{\mathbb{Z}\}$ we get following,

[@Ol93 Corollary 1] Every non cyclic torsion free hyperbolic group $G$ has a non cyclic quotient $\bar{G}$ such that every proper non trivial subgroups of $\bar{G}$ is infinite cyclic.

A group $G$ has unique product property whenever for all pairs of non empty finite subsets $A$ and $B$ of $G$ the set of products $AB$ has an element $g\in G$ with a unique representation of the form $g=ab$ with $a\in A$ and $b\in B$. In [@RS87], Rips and Segev gave the first examples of torsion-free groups without the unique product property. Other examples of torsion free groups without unique product can be found in [@Pas77; @Pro88; @Car14; @Al91]. As a corollary of Theorem \[rankoneintro\] we get following,

For every rank one abelian group $Q_m$, there exists a non elementary, torsion free, property $(T)$, 