Cellular covers of groups

Abstract

In this paper we discuss the concept of cellular cover for groups, especially nilpotent and finite groups. A cellular cover is a group homomorphism c: G → M such that composition with c induces an isomorphism of sets between Hom(G, G) and Hom(G, M). An interesting example is when G is the universal central extension of the perfect group M. This concept originates in algebraic topology and homological algebra, where it is related to the study of localizations of spaces and other objects. As explained below, it is closely related to the concept of cellular approximation of any group by a given fixed group. We are particularly interested in properties of M that are inherited by G, and in some cases by properties of the kernel of the map c.

On kernels of cellular covers

“…M be a cellular cover. In previous papers we have noticed that G inherits several important properties from M : First the kernel K D ker c is central in G, that is, G is a central extension of M; and further, if M is nilpotent, then G is nilpotent of the same class; if M is finite then so is G. In addition, we have classified all possible covers of divisible abelian groups ( [CFGS]) and showed that when M is abelian the kernel K is reduced and torsion-free ( [FGS,Theorem 4.7]). The case when M is abelian was independently investigated in [BD] and [D].…”

“…In particular, if F ¤ 1, then, since F Ä K, Hom.G; K/ ¤ 0; contradicting [FGS,Lemma 3.5 (c)]. Thus F D 1, so G D G 1 K 2 and hence jGj ġ.…”

“….g/ ng/ is in Hom.G; K/, so by (iii) it is the zero map and it follows that is multiplication by n. This shows that End.G/ Š Z. [FGS,Lemma 3.6] shows that c is a cellular cover.…”

Cellular covers of totally ordered abelian groups

“…We continue the investigations on cellular covers of groups initiated by Farjoun, Göbel and Segev [3], and continued by Chachólski, Farjoun, Göbel and Segev [2] for divisible abelian groups. In this note we consider the case where all the groups are totally ordered abelian groups.…”

Localizations, colocalizations and non additive ∗-objects

“…For example the concept, borrowed from topology, of cellular approximation in arbitrary category is a particular case of a colocalization, fact remarked for example in [4]. Some results concerning the cellular approximation may be deduced in a formal, categorical way by stressing this duality.…”