On Residual Separability of Subgroups in Split Extensions
https://doi.org/10.18255/1818-1015-2015-4-500-506
Abstract
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.
About the Author
A. A. Krjazheva
Voronezh State Technology University
Russian Federation
Russian Federation
graduate student
Universitetskaya str., 7, Voronezh, 394016, Russia
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Review
For citations:
Krjazheva A.A. On Residual Separability of Subgroups in Split Extensions. Modeling and Analysis of Information Systems. 2015;22(4):500-506. (In Russ.) https://doi.org/10.18255/1818-1015-2015-4-500-506
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