The precise center of a crossed module

Abstract

Abstract. We generalize the definition of the precise center of a group to the crossed modules context. We construct the Ganea map for the homology of crossed modules, and we study the connections between the precise center of a crossed module and the Ganea map. We extend some other known notions from group theory such as capable and relatively capable groups, capable pairs and unicentral groups with the definitions of capable and unicentral crossed modules. Finally we show how to apply these constructions to solve some open questions in the theory of crossed modules.

CHARACTERIZING n-ISOCLINIC CLASSES OF CROSSED MODULES

“…Proof. In view of [1,Proposition 3], μ is monic and the groups Y , F and F/μ(Y ) are free. So, we can assume that μ is the inclusion map.…”

“…Proof. It is well known that the category CM of crossed modules is equivalent to the category Cat 1 of cat 1 -groups (see [20]). (Recall that a cat 1 -group is a triple (C, ν, ω), where C is a group and ν, ω : C −→ C are group homomorphisms satisfying the conditions νω = ω, ων = ν, and [ker ν, ker ω] = 1.)…”

The non-abelian tensor product of normal crossed submodules of groups

“…The algebraic study of the category of crossed modules was initiated by Norrie [18] and has led to a substantial algebraic theory contained essentially in the following papers: [1,7,11,12,13,15,20,21]. In particular, Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules.…”