Localizing groups with action
Let P be a set of primes. A group G is P-local, if for any prime \(p\not\in P\) the map \(G\to G\), \(x\mapsto x^ P\), is bijective. By a result of G. Baumslag and also \textit{P. Ribenboim} [Lect. Notes Math. 740, 444-456 (1979; Zbl 0425.20032)] the category of P-local groups is reflective in the category of groups. The P-localization of G will be denoted by \(G_ P\). Let R be any group and let \({}_ R{\mathcal G}\) be the category of groups with R-action. Let \(G\rtimes R\) be the semidirect product and \(\pi\) : \(G\rtimes R\to G\) the projection. Then the R-group G is called P-local, if for all \(p\not\in P\) and \(r\in R\) the map \(G\to G\), \(x\mapsto \pi ((x,r)^ p)\), is bijective. Main results: (1) The category of P-local R-groups is reflective in the category of R-groups. (2) If \(\gamma\) : \(R\to S\) is a homomorphism of groups, then the obvious functor \(\gamma^*:_ S{\mathcal G}\to_ R{\mathcal G}\) has a left adjoint. Consequence: There is a functor \(G\mapsto \bar G\) from the category of R- groups to the category of \(R_ p\)-local \(R_ p\)-groups and a natural transformation \(G\to \bar G\) such that \((G\rtimes R)_ P\cong \bar G\rtimes R_ P\). Application: If \(\Omega\) X is the loop space of a based connected CW-complex, then \(\Omega\) \(X\to \Omega X\), \(z\mapsto z^ p\), is a homotopy equivalence for any \(p\not\in P\) iff \(\pi_ 1(X)\) is P- local and \(\pi_ i(X)\) is a P-local \(\pi_ 1(X)\)-group for all \(i\geq 2\). Result (1) is a special case of a theorem of the author, \textit{C. Casacuberta} and \textit{M. Pfenninger} [C. R. Acad. Sci. Paris, Ser. I 310, 207-210 (1990)].
scientific article; zbMATH DE number 4135022