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).