Simple Groups of Finite Morley Rank

Abstract

We clarify quasi-Frobenius configurations of finite Morley rank. 1. We remove one assumption in an identification theorem by Zamour while simplifying the proof. 2. We show that a strongly embedded quasi-Frobenius configuration of odd type, is actually Frobenius. 3. For dihedral configurations, one has dim G = 3 dim C. These results rely on an interesting phenomenon of closure of non-generic matter under taking centralisers.

Recognizing PGL$_3$ via generic 4-transitivity

“…As mentioned in the introduction, background on groups of finite Morley rank can be found in [Poi87], [BN94], and [ABC08]. In this section, we collect some specialized results about groups of small Morley rank.…”

“…We will work in the finite Morley rank category and defer to [Poi87], [BN94], and [ABC08] for the necessary background. The question we address is interesting even when restricted to the algebraic context, and the reader without knowledge of groups of finite Morley rank is encouraged to, if necessary, translate "rank" to "dimension" and "definable" to "constructible."…”

Fusion systems

“…Fusion systems over p-unipotent groups of finite Morley rank. In [ABC,§ X.4.1], the authors speculate about working out a theory of saturated fusion systems over p-unipotent groups of finite Morley rank, especially when p = 2. Just as for fusion systems over pro-p-groups, this would require restrictions on the subgroups and morphisms in the category: not all subgroups would be objects, and not all injective homomorphisms between objects are allowed.…”

Special abelian Moufang sets of finite Morley rank in characteristic 2

“…We instead appeal to the main theorem of [1] which states that a simple group of finite Morley rank either is isomorphic to an (a‰ne) algebraic group over an algebraically closed field of characteristic 2 or has no infinite elementary abelian 2-subgroup.…”

“…However, it should be noted that a much shorter proof which does not require the hypothesis that End H ðUÞ be infinite can be obtained by appealing to the deep main theorem of [1]. The result is that a simple group of finite Morley rank either is isomorphic to an algebraic group over an algebraically closed field of characteristic 2 or has no infinite elementary abelian 2-subgroup.…”