On infinite Alperin groups

A group $G$ is called Alperin group if any 2-generated subgroup of $G$ has a cyclic commutator subgroup. We prove the existence of Alperin torsion-free groups and Alperin groups, generated by involutions, with free abelian second commutator subgroups of any finite and countable rank. Also we prove that nilpotent torsion-free Alperin group has nilpotence class $\leq 2$. The last theorem of the article implies that the following condition is insufficient for a group $G$ to be Alperin group:
$$\text{for any } a,b \in G \text{ commutator } [a,b,b] \text{ is a power of } [a,b].$$