Quotient Groups of Groups of Certain Classes

Abstract

For an arbitrary variety \(\mathfrak{X}\) of groups and an arbitrary class \(\mathfrak{Y}\) of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with \(\mathfrak{X}\) - and \(\mathfrak{Y}\) -factors (respectively, is a residually \(\mathfrak{Y}\) -group) if G possesses an invariant system with \(\mathfrak{X}\) - and \(\mathfrak{Y}\) -factors (respectively, is a residually \(\mathfrak{Y}\) -group) and N ∈ \(\mathfrak{X}\) (respectively, N is a maximal invariant \(\mathfrak{X}\) -subgroup of the group G).