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).