On properties of (weakly) small groups

A group is small if it has countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite weakly small group has an infinite abelian subgroup, which may not be definable. A small nilpotent group is the central product of a definable divisible group with a definable one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finite-byabelian subgroup. First corollary : a weakly small group with simple theory has an infinite definable finite-by-abelian subgoup. Secondly, in a group with simple theory, a normal solvable group A of derived length n is contained in an A-definable almost solvable group of class n. A connected group of Morley rank 1 is abelian [20, Reineke]. Better, in an omegastable group, a definable connected group of minimal Morley rank is abelian. This implies that every infinite omega-stable group has a definable infinite abelian subgroup [6, Cherlin]. Berline and Lascar generalised this result to superstable groups in [4]. More recently, Poizat introduced d-minimal structures (englobing minimal ones) and structures with finite Cantor rank (including both d-minimal and finite Morley ranked structures). Poizat 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 in this paper that an infinite weakly small group has an infinite abelian subgroup, which may not be definable however. We then turn to weakly small groups with a simple theory. Recall that an א0categorical superstable group is abelian-by-finite [3, Baur, Cherlin and Macintyre]. In [23], Wagner showed any small stable infinite group to have a definable infinite abelian subgroup of the same cardinality. Later on, Evans and Wagner proved that an א0-categorical supersimple group is finite-by-abelian-by-finite and has finite SU rank [7]. We shall show that an infinite group the theory of which is small and simple has an infinite definable finite-by-abelian subgroup. However we still do not known whether a stable group must have an infinite abelian subgroup or not. 2000 Mathematics Subject Classification. 03C45, 03C60, 20E45, 20E99, 20F18, 20F24.