Morley Rank Research Articles

Sous-groupes de Carter dans les groupes de rang de Morley fini

RésuméA Carter subgroup is a self-normalizing locally nilpotent subgroup. For studying these subgroups in groups of finite Morley rank, we introduce the new notion of a locally closed subgroup. We show that every solvable group of finite Morley rank has a unique conjugacy class of Carter subgroups.

BN-pairs of finite Morley rank where $B$ is solvable

Let G be a simple group of finite Morley rank with a definable irreducible BN-pair of (Tits) rank 2 where B is solvable and let P be the associated generalized n-gon. If n is odd and B connected, then n = 3 and G is definably isomorphic to PSL3(K) for some algebraically closed field K. Furthermore, n 14 if T = B N 6 1. We also give sucient conditions for G to be a simple algebraic group.

A GENERIC IDENTIFICATION THEOREM FOR GROUPS OF FINITE MORLEY RANK

The paper contains a final identification theorem for the ‘generic’ K*-groups of finite Morley rank.

A signalizer functor theorem for groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Towards this end, the development of the theory of groups of finite Morley rank has achieved a good theory of Sylow 2-subgroups. It is now common practice to divide the Cherlin–Zilber conjecture into different cases depending on the nature of the connected component of the Sylow 2-subgroup, known as the Sylow◦ 2-subgroup. We shall be working with groups whose Sylow◦ 2-subgroup is divisible, or odd type groups. To date, the main theorem in the area of odd type groups is Borovik’s trichotomy theorem. The “trichotomy” here is a case division of the minimal counterexamples within odd type. More technically, Borovik’s result represents a major success at transferring signalizer functors and their applications from finite group theory to the finite Morley rank setting. The major difference between the two settings is the absence of a solvable signalizer functor theorem. This forced Borovik to work only with nilpotent signalizer functors, and the trichotomy theorem ends up depending on the assumption of tameness to assure that the necessary signalizer functors are nilpotent. The present paper shows that one may obtain a connected nilpotent signalizer functor from any sufficiently non-trivial solvable signalizer functor. This result plugs seamlessly into Borovik’s work to eliminate the assumption of tameness from his trichotomy theorem. In the meantime, a new approach to the trichotomy theorem has been developed by Borovik [7], based on the “generic identification theorem” of Berkman and Borovik [5]. Borovik uses his original signalizer functor arguments, and incorporates the result of the present paper. The paper is organized as follows. The first section will develop a limited characteristic zero notion of unipotence to complement the usual p-unipotence theory. The section on centralizers and generation which follows will establish some background needed in the rest of the paper. In Section 4 we prove our main result on signalizer functors, and in Section 5 we discuss some applications. With Borovik’s kind permission, we include a proof of the nilpotent signalizer functor theorem as an appendix. The results of Section 3 are based in part on a section of an unpublished version of [3].

Parabolic 2-local subgroups in groups of finite Morley rank of even type

Parabolic 2-local subgroups in groups of finite Morley rank of even type

An identification theorem for groups of finite Morley rank and even type

The paper contains a construction of a definable BN-pair in a simple group of finite Morley rank and even type with a sufficiently good system of 2-local parabolic subgroups. This provides ‘the final identification theorem’ for simple groups of finite Morley rank and even type.

On groups of finite Morley rank of even type

On groups of finite Morley rank of even type

A notion of rank in set theory without choice

Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory.

On Suzuki 2-groups of finite Morley rank

On Suzuki 2-groups of finite Morley rank

Borovik-Poizat rank and stability

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat rank function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… un groupe de Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1. A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5.

On Lascar rank and Morley rank of definable groups in differentially closed fields

AbstractMorley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.

Split BN-pairs of finite Morley rank

Let G be a simple group of finite Morley rank with a definable BN-pair of (Tits) rank 2 where B=UT for T=B∩N and U a normal subgroup of B with Z(U)≠1. By [9] (Forum Math. 13 (2001) 853) the Weyl group W=N/T has cardinality 2n with n=3,4,6,8 or 12. We prove here:Theorem 1.Ifn=3, thenGis interpretably isomorphic toPSL3(K) for some algebraically closed fieldK.Theorem 2.SupposeZ(U) contains someB-minimal subgroupA⩽Z(U) withRM(A)⩾RM(Pi/B) for both parabolic subgroupsP1andP2. Thenn=3,4 or 6 andGis interpretably isomorphic toPSL3(K), PSp4(K) orG2(K) for some algebraically closed fieldK.Theorem 3.IfUis nilpotent andn≠8, thenGis interpretably isomorphic to eitherPSL3(K), PSp4(K) orG2(K) for some algebraically closed fieldK.

Compact complex manifolds with the DOP and other properties

AbstractWe point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has U-rank different from Morley rank. We also give a sufficient condition for a Kähler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which satisfy these conditions.

Pushing Up and C(G, T) in Groups of Finite Morley Rank of Even Type

Pushing Up and C(G, T) in Groups of Finite Morley Rank of Even Type

Some model-theoretic results in the algebraic theory of quadratic forms

This paper studies some model-theoretic properties of special groups of finite type. Special groups are a first-order axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia, which is essentially equivalent to abstract Witt rings. More precisely, we consider elementary equivalence, saturation, elementary embeddings, quantifier elimination, stability and Morley rank.

On the definability of verbal subgroups

We show that if G is a group of finite Morley rank, then the verbal subgroup is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then Gn= is definable.

Full Frobenius Groups of Finite Morley Rank and the Feit-Thompson Theorem

AbstractWe show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis ofFT-groups, if one tries to extend the Feit-Thompson theorem to groups of finite Morley rank.

The Classical Involution Theorem for Groups of Finite Morley Rank

This paper gives a partial answer to the Cherlin–Zil'ber Conjecture, which states that every infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field. The classification of the generic case of tame groups of odd type follows from the main result of this work, which is an analogue of Aschbacher's Classical Involution Theorem for finite simple groups.

Fields of finite Morley rank

AbstractIfKis a field of finite Morley rank, then for any parameter setA⊆Keqthe prime model overAis equal to the model-theoretic algebraic closure ofA. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).

Fusion of 2-elements in groups of finite Morley rank

The Alperin-Goldschmidt Fusion Theorem [1, 5], when combined with pushing up [7], was a useful tool in the classification of the finite simple groups. Similar theorems are needed in the study of simple groups of finite Morley rank, in the even type case (that is, when the Sylow 2-subgroups are of bounded exponent, as in algebraic groups over fields of characteristic 2). In that context a body of results relating to fusion of 2-elements and the structure of 2-local subgroups is needed: pushing up, and the classification of groups with strongly or weakly embedded subgroups, or have strongly closed abelian subgroups (c.f, [2]). Two theorems of Alperin-Goldschmidt type are proved here. Both are needed in applications.The following is an exact analog of the Alperin-Goldschmidt Fusion Theorem for groups of finite Morley rank, in the case of 2-elements:Theorem 1.1. Let G be a group of finite Morley rank, and P a Sylow 2-subgroup of G. If A, B ⊆ P are conjugate in G, then there are subgroups Hi ≤ Pand elementsxi ∈ N(Hi) for 1 ≤ i ≤ n, and an elementy ∈ N(P), such that for all i:1. Hi is a tame intersection of two Sylow 2-subgroups;2. CP(Hi) ≤ Hi;3. N(Hi)/Hiis 2-isolatedand(a) (b) .