On properties of (weakly) small groups

Cédric Milliet (1, 2)

1 ICJ - Institut Camille Jordan
2 Département de Mathématiques

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-by-abelian 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.

Mots clés: nilpotent group; Small group; weakly small group; Cantor-Bendixson rank; local chain condition; infinite abelian subgroup; group in a simple theory; infinite finite-by-abelian subgroup; nilpotent group.

Domaines: Logique [math.LO]

Cédric Milliet. On properties of (weakly) small groups. The Journal of Symbolic Logic, 2012, 77 (1), pp.94-110. ⟨10.2178/jsl/1327068693⟩. ⟨hal-00582391⟩