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 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.

On the Schur-Zassenhaus theorem for groups of finite Morley rank

1992 ◽  
Vol 57 (4) ◽  
pp. 1469-1477 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Hall Subgroup ◽  
Morley Rank ◽  
Finite Group Theory ◽  
Hall Subgroups ◽  
Finite Morley Rank ◽  
A Finite Group

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact 1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group, where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p-Sylow for any prime p, or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H. Since ℂ* is abelian, these complements cannot be conjugated to each other.

On 2-Step Solvable Groups of Finite Morley Rank

1990 ◽  
Vol 110 (2) ◽  
pp. 479
Author(s):  
Kathryn Enochs ◽  
Ali Nesin
Keyword(s):  
Solvable Groups ◽  
Morley Rank ◽  
Finite Morley Rank

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.

scholarly journals SEMIABELIAN VARIETIES OVER SEPARABLY CLOSED FIELDS, MAXIMAL DIVISIBLE SUBGROUPS, AND EXACT SEQUENCES

2014 ◽  
Vol 15 (1) ◽  
pp. 29-69 ◽  
Author(s):  
Franck Benoist ◽  
Elisabeth Bouscaren ◽  
Anand Pillay
Keyword(s):  
Group Scheme ◽  
Algebraic Methods ◽  
Closed Field ◽  
Morley Rank ◽  
Finite Degree ◽  
Closed Fields ◽  
Divisible Subgroup ◽  
Uniform Treatment ◽  
Exact Sequences

Given a separably closed field $K$ of characteristic $p>0$ and finite degree of imperfection, we study the $\sharp$ functor which takes a semiabelian variety $G$ over $K$ to the maximal divisible subgroup of $G(K)$. Our main result is an example where $G^{\sharp }$, as a ‘type-definable group’ in $K$, does not have ‘relative Morley rank’, yielding a counterexample to a claim in Hrushovski [J. Amer. Math. Soc. 9 (1996), 667–690]. Our methods involve studying the question of the preservation of exact sequences by the $\sharp$ functor, and relating this to issues of descent as well as model-theoretic properties of $G^{\sharp }$. We mention some characteristic 0 analogues of these ‘exactness-descent’ results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right, as well as providing a uniform treatment of the characteristic 0 and characteristic $p$ cases of ‘exactness descent’.

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).

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