A note on superstable groups

It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o-minimal structure is totally transcendental of finite Morley rank. Motivation for this paper is a problem concerning stable groups definable in ominimal structures. One theorem is that any definably simple pure groupG = 〈G, ·〉 that is stable and definable in an o-minimal structure must be ù-stable of finite Morley rank. (Recall that an infinite group G = 〈G, ·, . . . 〉 is definably simple provided there are no infinite, proper, normal subgroups of G that are definable in G.) This result is a corollary of a theorem in [5], which states that, for any definably simple pure group G that is definable in an o-minimal structure, there is a pure field K = 〈K,+, ·〉 that is bi-interpretable with G, and K is either real closed or algebraically closed. Stability of G prohibits the case that K is real closed, and the theorem in [5] furthermore ensures thatG is a linear algebraic group overK , hence finite Morley rank. The author has been able to generalize this result as follows. Theorem 1. Any stable group that is definable in an o-minimal structure is totally transcendental of finite Morley rank. Proof. Theorem 1 from [2] implies that any stable structure interpretable in an o-minimal structure is superstable of finite U -rank. A theorem of Pillay [6] is that every group definable in an o-minimal structure has the d.c.c. on definable subgroups. Theorem 2 below yields the desired conclusion. a The author would like would like to thankAnand Pillay, first for his guidance as a research adviser, and second for the conversations that led to a strategy for proving Theorem 2. Thanks also go to the referee, who suggested illuminating simplications to parts of the argument. Our conventions are the usual ones for model theory of groups. To say that G is a stable group we mean that G together with all its structure is definable in the monster model of some stable theory. (So all definable and type-definable objects in G are over parameter sets that are “small” with respect to the saturation degree.) By aminimal type we mean a non-algebraic type that has a unique non-algebraic completion over every set containing its parameters; this makes sense even when the type is incomplete. A minimal set is just the set of realizations of a minimal type. Received August 10, 2004; accepted January 24, 2005. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$1.00