Given any class of groups \({\mathcal X}\), we can define another class \(L({\mathcal X})\) consisting of those groups in which the normal closure of each element is in \({\mathcal X}\). The property of being in the class \(L ({\mathcal X})\) is called the Levi property generated by \({\mathcal X}\). It is not difficult to see that, if \({\mathcal X}\) is a variety, so also is \(L({\mathcal X})\). However, as an example shows, not all varieties of groups are Levi varieties.
One interesting question considered here is the relationship between the laws of \({\mathcal X}\) and those of \(L({\mathcal X})\). If the law \(w(x_ 1, \dots, x_ n)\) holds in \({\mathcal X}\), then certainly the law \(w(x^{y_ 1}, x^{y_ 2}, \dots, x^{y^ n})\) holds in \(L({\mathcal X})\), and, indeed, by suitable conjugation, this can be transformed to the \(n\)- variable law \(w(x^{y_ 1}, x^{y_ 2}, \dots, x^{y_{n-1}},x)\). But are these the only sort of laws that hold in \(L({\mathcal X})\)? An example is given where such laws do not determine the Levi variety, but it is also shown that, when the laws of \({\mathcal X}\) are outer-commutator laws (in the sense of Philip Hall) then the corresponding laws do indeed give a basis for the laws of \(L({\mathcal X})\).
For the entire collection see [Zbl 0801.00023].