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

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

2008-01-18

arxiv.org/abs/0801.2955v1

Mathematics

Group Theory

Scientific paper

We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hat{\pi} : G \to \hat{G}$ 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 $\hat{V}$ (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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