Variations sur un thème de Aldama et Shelah

Cédric Milliet (1, 2, 3)

1 ICJ - Institut Camille Jordan
2 GSU - Université Galatasaray
3 Faculty of Mathematics and Computer Science [Konstanz]

We consider a group G that does not have the independence property and study the definability of certain subgroups of G using parameters from a fixed elementary extention G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X∩G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

Mots clés: definable and type definable envelope; Model theory; independence property; shattering type; VC-dimension; Abelian; nilpotent; and soluble subgroups; nice group; definable and type definable envelope.

Domaines: Logique [math.LO]

Cédric Milliet. Variations sur un thème de Aldama et Shelah. The Journal of Symbolic Logic, 2016, 81 (1), pp.96-126. ⟨10.1017/jsl.2014.67⟩. ⟨hal-00980379v2⟩