Poly-separated and <i>ω</i>-stable nilpotent groups

Abstract

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π -torsion group of bounded exponent and a π -radicable group. It is easy to see that an abelian π -separated group can in fact be decomposed as a direct product of bounded and π -radicable factors (Lemma 1.1 below). A poly- π -separated group is one which can be obtained from π -separated groups by forming a finite series of group extensions. We will show here: Theorem 1. Every poly-π-separated nilpotent group is π-separated . The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω -stable simple groups are algebraic, it is useful to have a structure theory for solvable ω -stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω -stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω -stable group of finite Morley rank is nilpotent. As a model-theoretic application of Theorem 1 we obtain: Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0- definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk( B ∩ D ) ≤ rk D . Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite . In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω -stable abelian groups as exactly the separated ones.