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.