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).