Conjugacy of Carter Subgroups in Groups of Finite Morley Rank

Frécon, Olivier

doi:10.1142/s0219061308000713

Cited by 14 publications

(27 citation statements)

References 22 publications

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“…In [Jal06], Jaligot proved the conjugacy of generous Carter subgroups of groups of finite Morley rank. The most general conjugacy result for Carter subgroups that does not depend on the generosity hypothesis was proven in [Fré08] for K * -groups, that is the groups of finite Morley rank all of whose simple definable proper sections are algebraic. Fact 2.18 (4) is the important particular case that deals with the minimal connected simple groups and that we will need in this paper.…”

Section: An Abelian Connected Group a Of Finite Morley Rank Is Indecomentioning

confidence: 99%

“…When the Weyl group is non-trivial, it determines many structural aspects of the ambient group as was exemplified in the classification of simple groups of even type or in [Del08]. On the other hand, when it is trivial, the ambient group has very high chances of being torsion-free, and the arguments tend to use the geometry of G as in [Fré08].…”

Section: Tetrachotomy Theoremmentioning

confidence: 99%

“…• The minimal connected simple groups with a nongenerous Carter subgroup are of type (2) and are analyzed in [Fré08].…”

Section: Tetrachotomy Theoremmentioning

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: An Abelian Connected Group a Of Finite Morley Rank Is Indecomentioning

confidence: 99%

Section: Tetrachotomy Theoremmentioning

confidence: 99%

“…• The minimal connected simple groups with a nongenerous Carter subgroup are of type (2) and are analyzed in [Fré08].…”

Section: Tetrachotomy Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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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Linearity of groups definable in o-minimal structures

Frécon

2016

Sel. Math. New Ser.

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We consider an arbitrary o-minimal structure M and a definably connected definable group G. The main theorem provides definable real closed fields R 1 , . . . , R k such that G/Z(G) is definably isomorphic to a direct product of definable subgroups of GLn 1 (R 1 ), . . . , GLn k (R k ), where Z(G) denotes the center of G. From this we derive a Levi decomposition for G, and show that [G, G]Z(G)/Z(G) is definable and definably isomorphic to a direct product of semialgebraic linear groups over R 1 , . . . , R k .

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