Tame minimal simple groups of finite Morley rank

Cherlin, Gregory; Jaligot, Eric

doi:10.1016/j.jalgebra.2003.12.027

Cited by 37 publications

(119 citation statements)

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“…An excellent, though somewhat aging, reference is [BN94], while [ABC08] may be of help for more recent developments. The fundamental papers [CJ04], [BCJ07], [Del08] may help focus on some of the insights for this article.…”

Section: Preliminariesmentioning

confidence: 99%

“…Their analysis, that widely varies from one configuration to another, nevertheless permits impressive achievements as soon as the definable subgroup structure is sufficiently rich, a phenomenon reminiscent of the Feit-Thompson theorem in finite group theory. The fundamental articles [CJ04], [Del08], [Fré08] are good examples of this. They deal with torsion elements in a similar way to finite group theory, make use of genericity arguments and approach such notions as the Weyl group in this abstract context.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper, we follow the line of [CJ04], [Del08] and [BD09] while using results and techniques from [Fré08] in order to provide a uniform approach to the analysis of the connected minimal simple groups of finite Morley rank. This approach is based on a four-case division whose criteria are the structure of the Weyl group and the intersections of maximal definable connected solvable subgroups (Borel subgroups) and depends on the following central theorem that will be proven in the third section:…”

Section: Introductionmentioning

confidence: 99%

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On Weyl groups in minimal simple groups of finite Morley rank

2013

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Abstract. We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type. IntroductionThe theory of groups of finite Morley rank has become a well-established branch of the model theory of groups. Most research is centered around the analysis of the infinite simple ones and more precisely the following conjecture independently stated by Gregory Cherlin and Boris Zil'ber.Algebraicity Conjecture (Cherlin-Zil'ber): An infinite simple group of finite Morley rank, seen as a pure group structure, is a linear algebraic group over an algebraically closed field.The most important advances in the analysis of the infinite simple groups of finite Morley rank have drawn upon three mathematical resources: the theory of linear algebraic groups over algebraically closed fields, the classification of the finite simple groups, and the analysis of the inherent structure of groups of finite Morley rank using a kind of geometric approach that relying on genericity arguments.As in the classification of the finite simple groups, the most difficult questions, some of which are still open, have arisen in the analysis of "small" simple groups, in particular, the connected minimal simple groups. These are the ones whose proper, definable, connected subgroups are solvable. Their analysis, that widely varies from one configuration to another, nevertheless permits impressive achievements as soon as the definable subgroup structure is sufficiently rich, a phenomenon reminiscent of the Feit-Thompson theorem in finite group theory.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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On Weyl groups in minimal simple groups of finite Morley rank

2013

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“…The next lemma encapsulates a form of genericity argument, based on the irreducibility of connected groups, which has played a significant role since the first analysis of "bad groups." Fact 1.44 [13,Lemma 3.4]. Let G be a connected group of finite Morley rank, and B a proper definable connected subgroup of finite index in its normalizer in G, such that g∈G B g is generic in G. Assume that x ∈ N G (B) \ B is of finite order modulo B, and let x B be the setwise product ( {yB: y ∈ x }).…”

Section: Genericitymentioning

confidence: 99%

Limoncello

Altınel

Cherlin

2005

Journal of Algebra

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“…The first case of Theorem 2 provides an important example of another line of genericity argument which can be interpreted as control of the Weyl group in a degenerate environment. This goes back to the analysis of groups of Morley rank 2 in [13] and was also very effective in the analysis of bad groups as well as in [14].…”

Section: Introductionmentioning

confidence: 97%