Profinite completion and double-dual : isomorphisms and counter-examples

2008/01/18 by Colas Bardavid, Bardavid, Colas
Mathematics · #20B #20E18 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.0801.2955

openalex publication_date 2008/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define, for any group G, finite approximations ; with this tool, we give a new presentation of the profinite completion π : G → G of an abtract group G. We then prove the following theorem : if k is a finite prime field and if V is a k-vector space, then, there is a natural isomorphism between V (for the underlying additive group structure) and the additive group of the double-dual V^**. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case.

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