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

We define, for any group G, finite approximations ; with this tool, we give a new presentation of the profinite completion b π : G → b 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 b 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.