On quasivarieties of groups and equations over groups

A proof is given that the quasivariety of groups generated by finite and torsion-free groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit-Thompson theorem on the solvability of finite groups of odd order as well as Kostrikin-Zelmanov results on the restricted Burnside problem, and applies technical details of a recent construction of weakly finitely presented periodic groups.