Submission date: 9 January 2012.

Abstract:

Let G be a group with simple theory. For any nilpotent subgroup N of class n, there is a definable nilpotent group E of class at most 2n finitely many translates of which cover N. The group E is definable with parameters in N. If S is a soluble subgroup of G of derived length l, there is a definable soluble group F of derived length at most 2l finitely many translates of which cover S. The group F is definable with parameters in S.

Mathematics Subject Classification: 03C45, 03C60, 20F16, 20F18, 20F19, 20F24.

Keywords and phrases: group with simple theory - nilpotent group - soluble group - definable envelope - FC-nilpotent group - FC-soluble group.

Full text HAL.