Maximal quasivarieties of groups

We consider a lattice Lq(qG) of quasivarieties contained in the quasivariety qG, generated by a polycyclic-by-finite group G. It is proved that the lattice contains a finite set of coatoms (i.e., proper maximal elements) and that each of its elements distinct from qG is contained in some coatom. We construct an example of a finitely generated solvable group B of derived length 3, whose quasivariety lattice Lq(qB) is freed of coatoms. © 1998 Plenum Publishing Corporation.