Ascending chains of finitely generated subgroups

JOURNAL OF ALGEBRA

卷:471

Article

Shusterman, Mark¹ 

[1] Tel Aviv Univ, Open Space Room 2,Schreiber Bldg Math,Levanon St, Tel Aviv, Israel

关键词: Profinite groups;    Pro-p groups;    Limit groups;    Rank gradient;    Chain conditions;    Commensurators;    Greenberg-Stallings property;   

DOI  :  10.1016/j.jalgebra.2016.09.023

来源: Elsevier

【 摘 要 】

We show that a nonempty family of n-generated subgroups of a pro-p group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-p groups. To demonstrate this, we show that in various pro-p groups Gamma(e.g. free pro-p groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup H not equal 1 is the greatest subgroup of Gamma containing H as an open subgroup. We also show that an ascending chain of n-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takabssi, Higman, and Kapovich Myasnikov). (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free