Definable envelopes in groups having a simple theory

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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 NG(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 NG(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.