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Subgroup properties of Demushkin groups†

Published online by Cambridge University Press:  05 October 2015

ILIR SNOPCE  and
PAVEL A. ZALESSKII
ILIR SNOPCE
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, 21941-909 Rio de Janeiro, RJ, Brasil. e-mail: ilir@im.ufrj.br
PAVEL A. ZALESSKII
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasília, DF, Brasil. e-mail: pz@mat.unb.br
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Abstract

We prove that a non-solvable Demushkin group satisfies the Greenberg–Stallings property, i.e., if H and K are finitely generated subgroups of a non-solvable Demushkin group G with the property that H ∩ K has finite index in both H and K, then H ∩ K has finite index in 〈H, K〉. Moreover, we prove that every finitely generated subgroup H of G has a ‘root’, that is a subgroup K of G that contains H with |K : H| finite and which contains every subgroup U of G that contains H with |U : H| finite. This allows us to show that every non-trivial finitely generated subgroup of a non-solvable Demushkin group has finite index in its commensurator.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 1 , January 2016 , pp. 1 - 9
Copyright
Copyright © Cambridge Philosophical Society 2015 

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Footnotes

†

Research partially supported by CNPq

References

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