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Étude des quasi-groupes de Frobenius et des K-boucles de rang de Morley fini

Abstract : This thesis is devoted to the study of certain groups of finite Morley rank and odd type. In the first chapter, we consider the problem of the classification of ranked sharply 2-transitive groups from the algebraic point of view of near-domains and near-fields. We study ranked near-fields of characteristic equal to two and under a rather strong technical hypothesis, we prove that they are algebraically closed fields. A characterization of the near-domains in terms of the properties of the half-embedding of the additive K-loop is also sketched. In a second chapter, we establish a number of results on the model theory of uniquely 2-divisible omrga-stable K-loops. The link with the notion of "symétron" introduced by B. Poizat is exploited. We give in this context an analogue of Lascar’s analysis of groups of finite Morley rank. In a third chapter, the class of quasi-Frobenius groups - which contains in particular SO(3, R), PGL(2, C) and GA(1, C) - will be in focus. The question of the identification of these classical groups in a context that extends ranked universes and o-minimal structures is discussed in detail. In a fourth and last chapter, we examine the link between the classification project of ranked quasi-Frobenius groups and definably linear groups over a ranked field. A characterization of definably linear bad pairs in zero characteristic is proposed. We also consider the problem of the solvability of split quasi-Frobenius groups of odd type.
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Contributor : Samuel Zamour Connect in order to contact the contributor
Submitted on : Tuesday, August 30, 2022 - 10:59:16 PM
Last modification on : Saturday, September 24, 2022 - 3:36:05 PM

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  • HAL Id : tel-03765160, version 1

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Samuel Zamour. Étude des quasi-groupes de Frobenius et des K-boucles de rang de Morley fini. Théorie des groupes [math.GR]. Université Claude Bernard Lyon 1, 2022. Français. ⟨NNT : ⟩. ⟨tel-03765160⟩

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