On structures preserved by idempotent transformations of groups and homotopy types

On the homotopy type of p -completions of infra-nilmanifolds

“…Are the resulting spaces again finitely covered by nilpotent spaces? To these questions, already raised in [5], we find an answer in the present paper.…”

Idempotent functors and localizations in categories of modules and Abelian groups

“…Papers [8,11] contain various versions of the following theorem in the case, where S is the ring of integers. (1) We can define a multiplication on the module LS such that LS becomes a ring, and this is the unique multiplication for which f is a ring homomorphism.…”

“…In module theory, localizations with respect to torsions are studied. In recent years, new applications of localization functors were presented in homotopy theory (see [11]). In recent studies in homotopy theory and group theory, the main attention is given to properties that are preserved by idempotent functors.…”

“…In recent studies in homotopy theory and group theory, the main attention is given to properties that are preserved by idempotent functors. In the category of groups, idempotent functors preserve Abelian groups and nilpotent groups of class ≤ 3; in the category of Abelian groups idempotent functors preserve rings, commutative rings, and modules [11].…”

Finite simple groups and localization

“…A group homomorphism ϕ : H → G is called in turn a localization if there exists a localization functor (L, η) such that G = LH and ϕ = η H : H → LH (but we note that the functor L is not uniquely determined by ϕ). In this situation, we often say that G is a localization of H. A very simple characterization of localizations can be given without mentioning localization functors: A group homomorphism ϕ : H → G is a localization if and only if ϕ induces a bijection ϕ * : Hom(G, G) ∼ = Hom(H, G) (0.1) as mentionned in [Cas,Lemma 2.1]. In the last decade several authors (Casacuberta, Farjoun, Libman, Rodríguez) have directed their efforts towards deciding which algebraic properties are preserved under localization.…”