ω-STABILITY AND MORLEY RANK OF BILINEAR MAPS, RINGS AND NILPOTENT GROUPS

2017 ◽  
Vol 82 (2) ◽  
pp. 754-777 ◽  
Author(s):  
ALEXEI G. MYASNIKOV ◽  
MAHMOOD SOHRABI
Keyword(s):  
Algebraic Structure ◽  
Nilpotent Groups ◽  
Morley Rank ◽  
Bilinear Maps ◽  
Complete Structure ◽  
Structure Theorems ◽  
Finite Morley Rank

AbstractIn this paper we study the algebraic structure of ω-stable bilinear maps, arbitrary rings, and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

On solvable centerless groups of Morley rank 3

1993 ◽  
Vol 58 (2) ◽  
pp. 546-556
Author(s):  
Mark Kelly Davis ◽  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
General Structure ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Closed Field ◽  
Morley Rank ◽  
Nonnilpotent Group ◽  
Finite Morley Rank ◽  
Torsion Free Subgroup ◽  
Torsion Free

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

Poly-separated and ω-stable nilpotent groups

1991 ◽  
Vol 56 (2) ◽  
pp. 694-699 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Nilpotent Group ◽  
Torsion Group ◽  
Nilpotent Groups ◽  
Simple Groups ◽  
Structure Theory ◽  
Morley Rank ◽  
Central Product ◽  
Derived Subgroup ◽  
Finite Morley Rank ◽  
Algebraically Closed Fields

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π-torsion group of bounded exponent and a π-radicable group. It is easy to see that an abelian π-separated group can in fact be decomposed as a direct product of bounded and π-radicable factors (Lemma 1.1 below). A poly-π-separated group is one which can be obtained from π-separated groups by forming a finite series of group extensions. We will show here:Theorem 1. Every poly-π-separated nilpotent group is π-separated.The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω-stable simple groups are algebraic, it is useful to have a structure theory for solvable ω-stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω-stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω-stable group of finite Morley rank is nilpotent.As a model-theoretic application of Theorem 1 we obtain:Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0-definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk(B ∩ D) ≤ rk D.Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite.In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω-stable abelian groups as exactly the separated ones.

scholarly journals On Groups of Finite Morley Rank with Weakly Embedded Subgroups

1999 ◽  
Vol 211 (2) ◽  
pp. 409-456 ◽  
Author(s):  
Tuna Altınel ◽  
Alexandre Borovik ◽  
Gregory Cherlin
Keyword(s):  
Morley Rank ◽  
Finite Morley Rank

scholarly journals Lascar and Morley ranks differ in differentially closed fields

1999 ◽  
Vol 64 (3) ◽  
pp. 1280-1284 ◽  
Author(s):  
Ehud Hrushovski ◽  
Thomas Scanlon
Keyword(s):  
Fundamental Issue ◽  
Morley Rank ◽  
Closed Fields ◽  
The Family ◽  
Finite Morley Rank

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. Section 1 consists of some general lemmas relating the above issues. Section 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium.

scholarly journals Signalizers and balance in groups of finite Morley rank

2009 ◽  
Vol 321 (5) ◽  
pp. 1383-1406 ◽  
Author(s):  
Jeffrey Burdges
Keyword(s):  
Morley Rank ◽  
Finite Morley Rank

scholarly journals Tame minimal simple groups of finite Morley rank

2004 ◽  
Vol 276 (1) ◽  
pp. 13-79 ◽  
Author(s):  
Gregory Cherlin ◽  
Eric Jaligot
Keyword(s):  
Simple Groups ◽  
Morley Rank ◽  
Finite Morley Rank
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