Levi classes generated by nilpotent groups (Q2729364)

Let \(L({\mathcal M})\) be the class of all groups \(G\) for which the normal closure \((x)^G\) of every element \(x\) belongs to the class \(\mathcal M\); \(L({\mathcal M})\) is called the Levi class generated by \(\mathcal M\). It is known that if \(\mathcal M\) is a variety then \(L({\mathcal M})\) is a variety; if \(\mathcal M\) is a quasivariety then \(L({\mathcal M})\) is a quasivariety of groups. Let \(\mathcal N\) and \({\mathcal N}_0\) be the classes of finitely generated nilpotent groups and of torsion-free, finitely generated nilpotent groups, respectively. Let \(q{\mathcal K}\) (or \(qG\) if \({\mathcal K}=\{G\}\)) be the quasivariety generated by a class \(\mathcal K\) of groups and let \(\overline{\mathcal K}\) be the set of all finitely generated groups in \(\mathcal K\). The author proves that \(q{\mathcal N}_0\subset L(q{\mathcal N}_0)\) and \(q{\mathcal N}\subset L(q{\mathcal N})\), and that \(L(q{\mathcal N}_0)\not=qL({\mathcal N}_0)\) and \(L(q{\mathcal N})\not=qL({\mathcal N})\). It is shown that the quasivarieties \(L(q{\mathcal N})\) and \(L(q{\mathcal N}_0)\) are closed under free products, and that each contains at most one maximal proper subquasivariety. It is also proved that \(L({\mathcal M})\) is closed under free products if so is \(\mathcal M\).