2023/01/12 by Chaitanya Ambi, Ambi, Chaitanya
Mathematics · #20E18 #20E34 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2301.04837
openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give three necessary and sufficient conditions for a pro-p group to be p-adic analytic. We show that a noetherian pro-p group having finite chain length has a finite rank and conversely. We further deduce that a noetherian pro-p group has a finite rank precisely when it satisfies the weak descending chain condition. Using these results, we resolve a conjecture posed by Lubotzky and Mann in the affirmative within the class of noetherian groups which are countably based. Using these results, we answer a related conjecture about pro-p groups for the case of countably based pro-p groups. Namely, we prove that if every closed but non-open subgroup of a countably based pro-p group has finite rank, then the group is p-adic analytic and conversely.