Submission date: 14 April 2014.

Abstract:

We study the definability of certain subgroups of a group G that does not have the independence property. If a (type) definable subset X of an elementary extension G of G has property P, we call its trace X\cap G over G an externally (type) definable P set. We show the following. Centralisers of subsets of G are externally definable subgroups. Cores of externally definable subgroups and iterated centres of externally definable subgroups are externally definable subgroups. Normalisers of externally definable subgroups are externally type definable subgroups and externally definable (as sets). A soluble subgroup S of derived length \ell is contained in an S-invariant externally type definable soluble subgroup of G of derived length \ell. The subgroup S is also contained in an externally definable subset X\cap G of G such that X generates a soluble subgroup of G of derived length \ell. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property. A soluble subgroup S of G of derived length \ell is contained in an externally type definable soluble subgroup of derived length \ell.

Mathematics Subject Classification: 03C45, 03C60

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

Full text HAL 00980379: pdf.