Variations sur un thème de Aldama et Shelah
Résumé
We study the definability of certain subgroups of a group G that does not have the independence property. If a (type) definable subset $\mathbf X$ of an elementary extension $\mathbf G$ of G has property P, we call its trace $\mathbf X\cap G$ 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 $\mathbf X\cap G$ of G such that $\mathbf X$ generates a soluble subgroup of~$\mathbf G$ 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 $\ell$.
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