Definable envelopes in groups having a simple theory

Cédric Milliet (1, 2)

1 ICJ - Institut Camille Jordan
2 Département de Mathématiques

Let G be a group having a simple theory. For any nilpotent subgroup N of class n, there is a definable nilpotent subgroup E of G which is virtually `nilpotent of class at most 2n' and finitely many translates of which cover N. The group E is definable using parameters in N, and normalised by N<sub>G</sub>(N). If S is a soluble subgroup of G of derived length ℓ, there is a definable soluble subgroup F which is virtually `soluble of derived length at most 2ℓ' and contains S. The group F is definable using parameters in S and normalised by N_G(S). Analogous results are shown in the more general setting where the ambient group G is defined by the conjunction of infinitely many formulas in a structure having a simple theory. In that case, the envelopes E and F are defined by the conjunction of infinitely many formulas.

Mots clés: group with simple theory; nilpotent group; soluble group; definable envelope; FC-nilpotent group; FC-soluble group

Domaines: Logique [math.LO]

Cédric Milliet. Definable envelopes in groups having a simple theory. Journal of Algebra, 2017, 492, pp.298-323. ⟨10.1016/j.jalgebra.2017.09.007⟩. ⟨hal-00657716v2⟩