Ascending chains of finitely generated subgroups

2016/01/09 by Mark Shusterman, Shusterman, Mark
Mathematics · #20E18 #20E26 #20E36 #20F05 #20F65 #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #math.AC #math.GR #msc:20E18 #msc:20E26 #msc:20E36 #msc:20F05 #msc:20F65

paper · pdf · doi:10.48550/arxiv.1601.02135

arxiv created 2016/01/09 · arxiv updated 2016/01/12

Abstract

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 sequence 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).

Related

• Pro-p groups with constant generating number on open subgroups
• Counting irreducible modules for profinite groups
• A p-group with positive Rank Gradient
• Virtual retraction and Howson's theorem in pro-p groups
• Finite groups can be generated by a pi-subgroup and a pi'-subgroup