On Residual Separability of Subgroups in Split Extensions

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) i...

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Bibliographic Details
Main Author: A. A. Krjazheva
Format: Article
Language:English
Published: Yaroslavl State University 2015-08-01
Series:Моделирование и анализ информационных систем
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Online Access:https://www.mais-journal.ru/jour/article/view/268
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Summary: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.
ISSN:1818-1015
2313-5417