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- Olivier Frécon
- 2007

The Cherlin-Zil’ber Conjecture states that all simple groups of finite Morley rank are algebraic. We prove that any minimal counterexample to this conjecture has a unique conjugacy class of Carter subgroups, which are defined as being the definable connected nilpotent subgroups of finite index in their normalizers, and which are analogous to Cartan… (More)

- Olivier Frécon
- J. Logic & Analysis
- 2010

We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such… (More)

- Olivier Frécon
- 2010

With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F )×Ks +, where… (More)

- Olivier Frécon
- 2013

We analyze the abstract structure of algebraic groups over an algebraically closed field K, using techniques from the theory of groups of

- Olivier Frécon
- J. Symb. Log.
- 2004

We study the structure of subgroups of minimal connected simple groups of finite Morley rank. We first establish a Jordan decomposition for a large family of minimal connected simple groups including those with a non-trivial Weyl group. We then show that definable, connected, solvable subgroups of such a simple group are the semi-direct product of their… (More)

- Olivier Frécon
- 2015

We consider an o-minimal expansion M0 = (R0, <,+, · · · ) of a real closed field, and a real closed field R, complete in the sense of D. Scott, containing R0 as a dense subfield. We show that M0 has an elementary extension M = (R,<,+, · · · ) with domain R. Moreover, such a structure M with domain R is unique.

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… (More)

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