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We introduce several classes of localizations (idempotent monads) on thecategory of groups and study their properties and relations. The mostinteresting class for us is the class of localizations which coincide withtheir zero derived functors. We call them right exact (in the sense of Keune).We prove that a right exact localization $L$ preserves the class of nilpotentgroups and that for a finite $p$-group $G$ the map $G\to LG$ is an epimorphism.We also prove that some examples of localizations (Baumslag's $P$-localizationwith respect to a set of primes $P,$ Bousfield's $HR$-localization, Levine'slocalization, Levine-Cha's $\mathbb Z$-localization) are right exact. At theend of the paper we discuss a conjecture of Farjoun about Nikolov-Segal mapsand prove a very special case of this conjecture.

Right exact localizations of groups