Weyl groups of small groups of finite Morley rank

Burdges, Jeffrey; Deloro, Adrien

doi:10.1007/s11856-010-0087-9

Cited by 8 publications

(19 citation statements)

References 20 publications

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“…Now, we move on to the problem of self-normalization of Borel subgroups. We will need several results from [Del08] and [BD09], that we will present in a form more suitable for our needs.…”

Section: T ) It Then Follows From Proposition 32 That W (G) ≃ N G (mentioning

confidence: 99%

“…As for [BD09], the second part of the following fact, rather than the cyclicity of the Weyl group, will be needed in the sequel. The results of [BD09, §3] do not need that the group G be degenerate, but just that |W (G)| be odd.…”

Section: T ) It Then Follows From Proposition 32 That W (G) ≃ N G (mentioning

confidence: 99%

“…Nevertheless, we will give a slightly longer but direct argument for two reasons. The first is that the quick ending is in fact longer in that it uses the full force of [BD09], which we do not need here. The second and more important reason is that in Section 6, as explained after Fact 6.4, it will be crucial to have a self-normalization argument that deals with the special case when the Weyl group is of odd order in order to avoid referring to [Del08].…”

Section: Proof -By Fact 24 B Is Non-trivial So N G (B)mentioning

confidence: 99%

“…The second and more important reason is that in Section 6, as explained after Fact 6.4, it will be crucial to have a self-normalization argument that deals with the special case when the Weyl group is of odd order in order to avoid referring to [Del08]. We achieve this goal by reducing the proof of Theorem 3.12, at the end of its first paragraph, to the case where the Weyl group of the ambient group is of odd order, and then avoid using any argument that necessitates the use of [Del08], in particular the full force of [BD09]. The care about the hypothesis in Fact 3.8, that assumes the Weyl group to be of odd order, is also part of these efforts.…”

Section: Proof -By Fact 24 B Is Non-trivial So N G (B)mentioning

confidence: 99%

“…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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“…Now, we move on to the problem of self-normalization of Borel subgroups. We will need several results from [Del08] and [BD09], that we will present in a form more suitable for our needs.…”

Section: T ) It Then Follows From Proposition 32 That W (G) ≃ N G (mentioning

confidence: 99%

Section: T ) It Then Follows From Proposition 32 That W (G) ≃ N G (mentioning

confidence: 99%

Section: Proof -By Fact 24 B Is Non-trivial So N G (B)mentioning

confidence: 99%

Section: Proof -By Fact 24 B Is Non-trivial So N G (B)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Weyl groups in minimal simple groups of finite Morley rank

2013

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show abstract