Groups Of Finite Morley Rank Research Articles

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.

On Weyl groups in minimal simple groups of finite Morley rank

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type.

Pseudofinite groups as fixed points in simple groups of finite Morley rank

We prove that if the group of fixed points of a generic automorphism of a simple group of finite Morley rank is pseudofinite, then this group is an extension of a (twisted) Chevalley group over a pseudofinite field. On the way to obtain this result, we classify non-abelian definably simple pseudofinite groups of finite centralizer dimension (using the ideas of Wilson (1993) [29]).

Groups of finite Morley rank with a pseudoreflection action

In this work, we give two characterisations of the general linear group over an algebraically closed field as a group G of finite Morley rank acting on an abelian connected group V of finite Morley rank definably (in the sense that G⋉V is a group of finite Morley rank in which G and V are definable), faithfully and irreducibly. We prove that if the pseudoreflection rank of G is equal to the Morley rank of V, then V has a definable vector space structure over an algebraically closed field, G≅GL(V) and the action is the natural action. The same result holds also under the assumption of Prüfer 2-rank of G being equal to the Morley rank of V.

Groups of Finite Morley Rank with Solvable Local Subgroups

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.

On groups of finite Morley rank with a split BN-pair of rank 1

We study groups of finite Morley rank with a split BN-pair of Tits rank 1 in the case where the normal complement to B ∩ N in B is infinite and abelian. For such groups, we give conditions ensuring that the standard action of G on the cosets of B is isomorphic to the natural action of PSL 2 ( F ) on P 1 ( F ) for F an algebraically closed field. In particular, we show that SL 2 ( F ) and PSL 2 ( F ) are the only infinite quasisimple L ⁎ -groups of finite Morley rank possessing a split BN-pair of Tits rank 1 where the normal complement to B ∩ N in B is infinite abelian. Our approach is through the theory of Moufang sets and is tied to work attempting to classify the abelian Moufang sets of finite Morley rank.

Linear representations of soluble groups of finite Morley rank

Sufficient conditions are given for groups of finite Morley rank having nontrivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. Along the way, using a notion of definable weight space, we prove that certain connected soluble groups of finite Morley rank with torsion-free derived subgroup can be embedded in groups of finite Morley rank whose Fitting subgroups have definable abelian supplements.

Weyl groups of small groups of finite Morley rank

We examine Weyl groups of minimal connected simple groups of finite Morley rank of degenerate type. We show that they are cyclic, and lift isomorphically to subgroups of the ambient group.

Automorphism groups of small simple groups of finite Morley rank

IfGGis a minimal connected simple group of finite Morley rank with a nontrivial Weyl group, then its connected definable automorphism groups are inner.

Small groups of finite Morley rank with involutions

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 several aspects reminiscent of “small” groups of finite Morley rank of odd type.

Splitting in solvable groups of finite Morley rank

We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such that G = U o T , if and only if, for any field K of finite Morley rank, the connected definable subgroups of K∗ are pseudo-tori. Also we build a centerless connected solvable group G of finite Morley rank with no definable representation over a direct sum of interpretable fields. 2000 Mathematics Subject Classification 03C45, 20A15 (primary); 03C60 (secondary)

Book Review: Simple groups of finite Morley rank

Book Review: Simple groups of finite Morley rank

SEMISIMPLE TORSION IN GROUPS OF FINITE MORLEY RANK

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

Cosets, genericity, and the Weyl group

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 for the abstract theory of Weyl groups in groups of finite Morley rank.

Actions of Groups of Finite Morley Rank on Small Abelian Groups

AbstractWe classify actions of groups of finite Morley rank on abelian groups of Morley rank 2: there are essentially two, namely the natural actions ofSL(V)andGL(V)withVa vector space of dimension 2. We also prove an identification theorem for the natural module of SL2in the finite Morley rank category.

Steinberg's torsion theorem in the context of groups of finite Morley rank

Groups of finite Morley rank generalize algebraic groups; the simple ones have even been conjectured to be algebraic. Parallel to an ambitious classification program towards this conjecture, one can try to show direct equivalents of known results on algebraic groups in the context of groups of finite Morley rank. This is done here with Steinberg's theorem on centralizers of semi-simple elements.

Signalizers and balance in 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. The most successful approach to this conjecture has been Borovik's program analyzing a minimal counterexample, or simple K ∗ -group. We show that a simple K ∗ -group of finite Morley rank and odd type is either algebraic of else has Prüfer rank at most two. This result signifies a switch from the general methods used to handle large groups, to the specilized methods which must be used to identify PSL 2 , PSL 3 , PSp 4 , and G 2 .

On the linearity of torsion-free nilpotent groups of finite Morley rank

It is proved that every torsion-free nilpotent group of finite Morley rank is isomorphic to a matrix group over a field of characteristic zero.

A GENERATION THEOREM FOR GROUPS OF FINITE MORLEY RANK

We deal with two forms of the "uniqueness cases" in the classification of large simple K*-groups of finite Morley rank of odd type, where large means the 2-rank m2(G) is at least three. This substantially extends results known for even larger groups having Prüfer 2-rank at least three, so as to cover the two groups PSp4and G2. With an eye towards more distant developments, we carry out this analysis for L*-groups, a context which is substantially broader than the K* setting.

Groupes géométriques de rang de Morley fini

AbstractWe consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.