Ascending chains of finitely generated subgroups

Mark Shusterman

Research output: Contribution to journalArticlepeer-review


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 H≠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).

Original languageEnglish
Pages (from-to)240-250
Number of pages11
JournalJournal of Algebra
StatePublished - 1 Feb 2017


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


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