A Property of Small Groups

A group is small if it has countably many pure n-types for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisible group with one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-by-abelian subgroup. As a corollary, a group with small and simple theory has an infinite definable finite-by-abelian subgoup. A connected group of Morley rank one is abelian [20, Reineke]. Better, a connected omega-stable group of minimal Morley rank is abelian, from which it follows that every infinite omega-stable group has a definable infinite abelian subgroup [6, Cherlin]. Berline and Lascar generalised this to superstable groups in [4]. More recently, Poizat introduced d-minimal structures which englobe minimal ones, and proved a d-minimal group to be abelian-by-finite [17]. He went further showing that an infinite group of finite Cantor rank has a definable abelian infinite subgroup [18]. More generally, we show that an infinite weakly small group has an infinite abelian subgroup, which may however not be definable. We then turn to groups definable in a small and simple theory. Recall that an omega-categorical superstable group is abelian-by-finite [3, Baur, Cherlin and Macintyre]. In [23], Wagner showed any small stable infinite group to have an infinite abelian subgroup. Later on, Evans and Wagner proved that an omega-categorical supersimple group is finite-by-abelian-by-finite and has finite SU-rank [7]. We show that an infinite group the theory of which is small and simple has an infinite definable finite-by-abelian subgroup. However it is still unknown whether a stable group must have an infinite abelian subgroup. Definition 1. A theory is small if it has countably many n-types without parameters for every integer n. A structure is small if its theory is so. Note that smallness is preserved by interpretation, and by adding finitely many parameters to the language. Small theories arise when one wishes to count the number of pairwise non-isomorphic countable models of a complete first order theory in a countable language. Such a theory having fewer than the maximal number of pairwise non-isomorphic models is small indeed. 2000 Mathematics Subject Classification. 03C45, 03C60, 20E45, 20F18, 20F24.