Cardinality and Nilpotency of Localizations of Groups and G-modules Assaf Libman August 29, 2001 Abstract We consider the effect of a coaugmented idempotent functor J in the the category of groups or G-modules where G is a fixed group. We are interested in the ‘extent’ to which such functors change the structure of the objects they are applied to. Some positive results are obtained and examples are given concerning the cardinality and structure of J (A) in terms of the cardinality and structure of A, where the latter is a torsion abelian group. For non-abelian groups some partial results and examples are given connecting the nilpotency classes and the varieties of a group G and J (G). Similar but stronger results are obtained in the category of G-modules. This paper deals with the behavior of localization functors in the categories of groups, abelian groups and G-modules for a fixed group G. Nevertheless, most of the results apply to coaugmented idempotent functors. The main interest is in the extent to which such functors preserve the algebraic or even the underlying set structure of the objects to which they are applied. It is shown (see 2.8) that the cardinality of an abelian torsion group cannot increase arbitrarily after localization. For a general group the group structure may vastly change after localization (see 3.4) but for certain nilpotent groups the opposite happens (3.3). Similar results hold for G-modules (4.2 and 4.5). I am indebted to E. Dror Farjoun for his help in encouragement and ideas. I would also like to thank Carles Casacuberta for his remarks and suggestions concerning the statements and the proofs in this paper. 1 Idempotent Coaugmented Functors Let C be a category. A coaugmented functor F is a functor F : C→C together with a natural transformation a: Id → F called coaugmentation. The coaug- mentation is called idempotent if a FX ,F (a X ): FX → FFX are equal and are an isomorphism. Given a coaugmented idempotent functor F ,a local object is an 1