Ascending chains of finitely generated subgroups

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• 作者：Mark Shusterman ^{markshus@mail.tau.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Profinite groups ; Pro-p groups ; Limit groups ; Rank gradient ; Chain conditions ; Commensurators ; Greenberg&ndash ; Stallings property
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：471
• 期：Complete
• 页码：240-250
• 全文大小：319 K

文摘

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 Γ (e.g. free pro-p   groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup cccfcc01bb618b47" title="Click to view the MathML source">H≠1$H\ne 1$ is the greatest subgroup of Γ 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 Takahasi, Higman, and Kapovich–Myasnikov).