A Hall Theorem for ω-Stable Groups

Abstract

A theorem of Hall extends the sylow theory to π-subgroups of a finite solvable group for any set of primes π. This theorem has been generalized to solvable periodic linear groups [ 21 ] and to other similar settings [ 19 ]. We consider the same problem in a model-theoretic context. In model theory, the study of ω-stable groups generalizes the classical theory of algebraic groups in the case of a algebraically closed base field, and is closely connected with certain aspects of the general structure theory developed by Shelah, particularly in connection with the more geometric theory of Zilber, Buechler, Hrushovski, Pillay, and others. In this connection, the primary concern is with the structure of simple ω-stable groups of finite rank. One way to approach the study of ω-stable groups is by generalizing some of the methods of finite group theory to this setting. One anticipates that character theory and counting arguments will not be available, though one would hope to recover much of the ‘local analysis’. Unfortunately there is as yet no very satisfactory sylow theory for ω-stable groups, except for the prime 2 [ 4 ]. One of us (Nesin) noticed initially that there is a sylow theorem for connected solvable ω-stable groups of finite Morley rank; this prompted us to look for a generalization along the following lines.