We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove a main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases and study torsion.
We study existentially closed CSA-groups. We prove that existentially closed CSA-groups without involutions are simple and divisible, and that their maximal abelian subgroups are conjugate. We also prove that every countable CSA-group without involutions embeds into a finitely generated one having the same maximal abelian subgroups, except maybe the… (More)
We prove conjugacy and generic disjointness of generous Carter subgroups in groups of finite Morley rank. We elaborate on groups with a generous Carter subgroup and on a minimal counterexample to the Gener-icity Conjecture.
We consider groups G interpretable in a supersimple finite rank theory T such that T eq eliminates ∃ ∞. It is shown that G has a definable soluble radical. If G has rank 2, then if G is pseudofinite it is soluble-by-finite, and partial results are obtained under weaker hypotheses, such as unimodularity of the theory. A classification is obtained when T is… (More)
In continuation of [JOH04, OH07], we prove that existentially closed CSA-groups have the independence property. This is done by showing that there exist words having the independence property relative to the class of torsion-free hyperbolic groups.
In a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite… (More)
By analogy with Thompson's classification of nonsolvable finite N-groups, we classify groups of finite Morley rank with solvable local subgroups of even and of mixed type. We also consider miscellaneous aspects concerning " small " groups of finite Morley rank of odd type.
We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The first-one concerns generic types and the non-superstability of such groups in many cases. The second-one concerns the connectedness of most free products of groups without amalgamation. First-order theories of free products of… (More)