Bad groups of finite Morley rank

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. Abstract. We prove the following theorem. Let G be a connected simple bad group (i.e. of finite Morley rank, nonsolvable and with all the Borel subgroups nilpotent) of minimal Morley rank. Then the Borel subgroups of G are conjugate to each other, and if B is a Borel subgroup of G, then G = UgeGB9, NG(B) = B, and G has no involutions.

Tame minimal simple groups of finite Morley rank

“…The structure of Sylow 2-subgroups in a bad group is dramatically trivial: Fact 1.3 [10,14,22]. A simple bad group has no involutions.…”

Bad Fields in Positive Characteristic

“…In the context of simple groups, the analogy seems to be very close, which led Gregory Cherlin [2] and Boris Zilber to conjecture that a simple (non-abelian) group of finite Morley rank should be an algebraic group over an algebraically closed field. Early work identified two main obstacles to proving this conjecture, namely the possible existence of a definable section which is either a bad group (a simple non-abelian group whose proper definable connected subgroups are nilpotent; see [2,3,7,9]), or which forms the additive group of a bad field (see below for a definition), and whose normaliser definably surjects onto a proper infinite multiplicative subgroup of that field. Groups that do not thus interpret either structure are called tame; Borovik and Nesin [1] have initiated a programme to classify tame simple groups of finite Morley rank in analogy with the classification of finite simple groups of Lie type.…”