Self-derived localizations of groups.

We introduce several classes of localizations (idempotent monads) on the category of groups and study their properties and relations. The most interesting class is the class of localizations which coincide with their zero derived functors. We call them self-derived localizations. We prove that self-derived localizations preserve the class of nilpotent groups and that for a finite $p$-group $G$ the map $G\to LG$ is an epimorphism. We also prove that Bousfield's $HR$-localization and Baumslag's $P$-localization with respect to a set of primes $P$ are self-derived.