A signalizer functor theorem for groups of finite Morley rank

Abstract

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 The results of Section 3 are based in part on a section of an unpublished version of [3].