On Residual Separability of Subgroups in Split Extensions - A. A. Krjazheva (Моделирование и анализ информационных систем)
In 1973, Allenby and Gregoras proved the following statement. Let G be a split extension of a ﬁnitely generated group A by the group B. 1) If in groups A and B all subgroups (all cyclic subgroups) are ﬁnitely separable, then in group G all subgroups (all cyclic subgroups) are ﬁnitely separable; 2) if in group A all subgroups are ﬁnitely separable, and in group B all ﬁnitely generated subgroups are ﬁnitely separable, then in group G all ﬁnitely generated subgroups are ﬁnitely separable. Recall that a group G is said to be a split extension of a group A by a group B, if the group A is a normal subgroup of G, B is a subgroup of G, G = AB and A ∩ B = 1. Recall also that the subgroup H of a group G is called ﬁnitely separable if for every element g of G, which does not belong to the subgroup H, there exists a homomorphism of G on a ﬁnite group in which the image of an element g does not belong to the image of the subgroup H. In this paper we obtained a generalization of the Allenby and Gregoras theorem by replacing the condition of the ﬁnitely generated group A by a more general one: for any natural number n the number of all subgroups of the group A of index n is ﬁnite. In fact, under this condition we managed to obtain a necessary and suﬃcient condition for ﬁnite separability of all subgroups (of all cyclic subgroups, of all ﬁnitely generated subgroups) in the group G.