Algebraic topologists have invented a very pretty technique of localizing nilpotent groups. (Garth Warner covers the topic in his book manuscript Topics in Topology and Homotopy Theory). For topologists this construction is a means to an end, but I'm curious if it has any pure group-theoretic applications. Are there any theorems about nilpotent groups that have been proven using the technique? Are there any pre-existing theorems which are best understood from the point of view of localization?
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2$\begingroup$ I think, the ideas of localization go back to Malcev's work (in 1949) on Malcev completion, ncatlab.org/nlab/show/Mal%27cev+completion, which is a basic tool in studying finitely generated nilpotent groups. I suspect that whatever can be learned about nilpotent groups via localization, can be seen by analysing the Malcev completion. $\endgroup$– Igor BelegradekJan 5, 2014 at 18:05
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$\begingroup$ As a mathscinet search (on nilpotent group and localization) reveals, there are lots of group theoretic application. I was unable to find any recent survey though. $\endgroup$– Igor BelegradekJan 6, 2014 at 3:10
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