Definable envelopes in groups having a simple theory

[摘要] 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-G(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. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Model theory;Definable envelope;Group with a simple theory;Nilpotent and soluble subgroups;FC-nilpotent and FC-soluble groups