# Right exact localizations of groups

## TL;DR

This paper introduces and studies right exact localizations of groups, showing their properties, examples, and implications for nilpotent groups and specific localizations, and discusses a related conjecture.

## Contribution

It defines right exact localizations of groups, explores their properties, and provides examples including known localizations, also addressing a conjecture of Farjoun.

## Key findings

- Right exact localizations preserve nilpotent groups.
- For finite p-groups, the localization map is an epimorphism.
- Examples include Baumslag's P-localization, Bousfield's HR-localization, and Levine's localization.

## Abstract

We introduce several classes of localizations (idempotent monads) on the category of groups and study their properties and relations. The most interesting class for us is the class of localizations which coincide with their 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 nilpotent groups 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 $P$-localization with respect to a set of primes $P,$ Bousfield's $HR$-localization, Levine's localization, Levine-Cha's $\mathbb Z$-localization) are right exact. At the end of the paper we discuss a conjecture of Farjoun about Nikolov-Segal maps and prove a very special case of this conjecture.