Ascending chains of finitely generated subgroups

作者：Shusterman Mark^*

来源：Journal of Algebra, 2017, 471: 240-250.

DOI：10.1016/j.jalgebra.2016.09.023

• Profinite groups
• Pro-p groups
• Limit groups
• Rank gradient
• Chain conditions
• Commensurators
• Greenberg-Stallings property

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).

• 出版日期2017-2-1