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VARIATIONS SUR UN THÈME DE ALDAMA ET SHELAH

Published online by Cambridge University Press:  22 January 2016

CÉDRIC MILLIET
CÉDRIC MILLIET*
Affiliation:
UNIVERSITÉ GALATASARAY FACULTÉ DE SCIENCES ET DE LETTRES DÉPARTEMENT DE MATHÉMATIQUES ÇIRAĞAN CADDESI N°36 34357 ORTAKÖY, İSTANBUL,TÜRKIYE UNIVERSITÄT KONSTANZ FACHBEREICH MATEMATIK UND STATISTIK 78457 KONSTANZ, GERMANY PÔLE DE MATHÉMATIQUES DE L’INSA DE LYON BÂTIMENT LÉONARD DE VINCI 21, AVENUE JEAN CAPELLE 69621 VILLEURBANNE , FRANCEE-mail: [email protected]
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Abstract

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 extension 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.

Keywords

03C4503C60Model theoryindependence propertyshattering typeVC-dimensionabeliannilpotentand soluble subgroupsnice subgroupdefinable and type definable envelope
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 96 - 126
Copyright
Copyright © The Association for Symbolic Logic 2016 

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