Normalité

Details

Supervisors

Van der Linden, Tim

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques

Abstract

enThe purpose of this thesis was to generalize the notion of normal subgroups via the theory of categories. First of all, we introduce the definition of normal subobjects. We proved that in the category of groups, this notion of normal subgroups coincided with the concept of normal subobjects. Then, it is well-know, that in the category of groups, we can define the notion of normal subgroups via different equivalents ways. Therefore, we tried to define all these different ways individually. Then another goal of this thesis was to proof under which conditions each notion is equivalent to the normal subobject. Of course, we checked each time this categorical characterization in the category of groups. To do so, we needed to define new concepts such as: regular categories, exact categories, Mal'tsev categories, unital categories, protomodular categories and semi-abelian categories. And at the end, we introduced the notion of commutators to give a last characterization of normal subobjects via the commutator. We only gave a short introduction to this theory via the Higgins' commutator and the Huq's commutator.