A note on superstable groups

2005 ◽  
Vol 70 (2) ◽  
pp. 661-663 ◽  
Author(s):  
Jerry Gagelman
Keyword(s):  
Chain Condition ◽  
Morley Rank ◽  
Stable Group ◽  
Minimal Structure ◽  
Finite Morley Rank ◽  
Descending Chain Condition

AbstractIt is proved that all groups of finite U-rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o-minimal structure is totally transcendental of finite Morley rank.

An infinite superstable group has infinitely many conjugacy classes

1991 ◽  
Vol 56 (2) ◽  
pp. 618-623 ◽  
Author(s):  
I. Aguzarov ◽  
R. E. Farey ◽  
J. B. Goode
Keyword(s):  
Simple Group ◽  
Finite Rank ◽  
Conjugacy Classes ◽  
Morley Rank ◽  
Stable Group ◽  
The Third ◽  
Rank One ◽  
Preliminary Version ◽  
Finite Morley Rank ◽  
Basic Reference

We begin with some notes concerning the genesis of this paper. A preliminary version of it was written by the third author, who was moved by the desire to correct a mistake in Poizat [1987, p. 97], and to refresh some other minor results of the same book, concerning equations which are satisfied generically in a stable group. The book in question will be considered here as our basic reference on stable groups, and these other results will be discussed elsewhere.This preliminary version contained §§1, 2 and 3 of the present paper, restricted to the context of groups of finite Morley rank. It was observed that a counterexample of finite rank to the above theorem would be an extreme refutation of a conjecture by Zil'ber and Cherlin (cyrillic alphabet order), that may be not so solid as was believed some time ago, which states that a simple group of finite rank should be an algebraic group. Additional motivation for the problem was seen in Reineke's theorem (Reineke [1975]), stating that a connected group of rank one is abelian—the cornerstone for the study of superstable groups—whose proof rests on the fact that a group with two conjugacy classes either has only two elements, or has infinite chains of centralizers (a property that violates stability).

Schur-Zassenhaus theorem revisited

1994 ◽  
Vol 59 (1) ◽  
pp. 283-291 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Ali Nesin
Keyword(s):  
Solvable Groups ◽  
Hall Subgroup ◽  
Unipotent Radical ◽  
Morley Rank ◽  
Stable Group ◽  
Similar Question ◽  
Closed Fields ◽  
Finite Morley Rank ◽  
Divisible Subgroup ◽  
Algebraically Closed Fields

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.Theorem 2. Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.This result has been proved in [1] with the additional assumption that G is connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.The second author has shown in [11] that a nilpotent ω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.

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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