Quasi groupes de Frobenius dimensionnels

TL;DR

This paper classifies quasi-Frobenius groups with involutions using model-theoretic dimension concepts, linking their internal structure to classical groups like GA1(C), PGL2(C), and SO3(R).

Contribution

It provides a classification of quasi-Frobenius groups with involutions within a dimensional framework, connecting their structure to classical groups under certain conditions.

Findings

Classification of quasi-Frobenius groups with involutions.

Identification conditions for classical groups in a dimensional setting.

Analysis of incidence geometry induced by involutions.

Abstract

We are interested in a class of groups, quasi-Frobenius groups (with involutions), whose internal structure generalizes that of the classical groups GA1(C), PGL 2(C) and SO3(R) : a subgroup and its conjugates, of finite index in their normalizer and trivial mutual intersection, cover "generically" the ambient group. From the perspective of model theory, we work with the hypothesis of the existence of a good notion of dimension on definable sets (we must distinguish between the o-minimal case and the ranked case). We pay special attention to the ranked case. By studying the geometry of incidence induced by involutions, we sketch a classification of quasi-Frobenius groups and thus determine under which conditionsclassical groups can be identified in a dimensional framework -- -- Nous nous int\'eressons \`a une classe de groupes, les quasi-groupes de Frobenius (avec involutions), dont la structure interne g\'en\'eralise celle des groupes classiques GA1(C), PGL2(C) et SO3(R) : un sous-groupe et ses conjugu\'es, d'indice fini dans leur normalisateur et d'intersection mutuelle triviale, recouvrent "g\'en\'eriquement" le groupe ambiant. Dans la perspective de la th\'eorie des mod\`eles, nous travaillons avec l'hypoth\`ese de l'existence d'une bonne notion dimension sur les ensembles d\'efinissables (il faut distinguer le cas o-minimal et le cas rang\'e). Nous accordons une attention particuli\`ere au cas rang\'e. En \'etudiant la g\'eom\'etrie d'incidence induite par les involutions, nous esquissons une classification des quasi-groupes de Frobenius et nous d\'eterminons ainsi sous quelles conditions des groupes classiques peuvent \^etre identifi\'es dans un cadre dimensionnel.