Permutation groups of finite Morley rank

Borovik, Alexandre; Cherlin, Gregory

doi:10.1017/cbo9780511735219.003

Cited by 31 publications

(60 citation statements)

References 65 publications

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“…Our main general reference for groups of finite Morley rank is [4]. All group and ring actions in this paper are on the right.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Various field interpretation results in nonabelian soluble groups of finite Morley rank by Zil ber [4], later work by Nesin and Enochs [7], [12] on centreless metabelian groups of finite Morley rank, and ongoing work of Frécon form a consistent line of development. Indeed, no counterexample to the following conjecture is known to the authors:…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

Linear representations of soluble groups of finite Morley rank

Altınel

Wilson²

2010

Proc. Amer. Math. Soc.

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Abstract. Sufficient conditions are given for groups of finite Morley rank having nontrivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. Along the way, using a notion of definable weight space, we prove that certain connected soluble groups of finite Morley rank with torsion-free derived subgroup can be embedded in groups of finite Morley rank whose Fitting subgroups have definable abelian supplements.

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“…Our main general reference for groups of finite Morley rank is [4]. All group and ring actions in this paper are on the right.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Linear representations of soluble groups of finite Morley rank

Altınel

Wilson²

2010

Proc. Amer. Math. Soc.

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“…Nevertheless, we assume some familiarity with the basics of groups of finite Morley rank. 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%

“…In order to appreciate its importance, it suffices to consult Section 10.5 of [BN94], [ABC08] or [BCJ07]. We will be content with saying that a simple group of finite Morley rank with a strongly embedded subgroup is conjectured to be isomorphic to PSL 2 (K) where K is algebraically closed of characteristic 2, and the strongly embedded subgroups are the Borel subgroups.…”

Section: A Classification Theoremmentioning

confidence: 99%

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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“…The Euclidean rotation group keeps appearing as a phantom "minimal" configuration in my work in model-theoretic algebra. Some details can be found in [257,Chapter 8].…”

Section: Fancy Being Euclid?mentioning

confidence: 99%