On stabilization of E_k chains

Abstract

We study special subgroups of infinite groups that generalize double centralizers. We analyze sufficient conditions for descending chains of such subgroups to stop after finitely many steps. We discuss whether this phenomenon can happen in the class of groups satisfying chain condition on centralizers.

Keywords:

• Definable envelopes
• ${\mathfrak{M}}_{\mathit{c}}$-groups
• verbal topology

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F22
• 20B35
• 20A15

Acknowledgements

The author is indebted to her PhD supervisors Erdal Karaduman and Tuna Altınel for their continuous support and infinite patience. The author thanks to Mathematics Department of Univérsite Claude Bernard Lyon-1 for their hospitality during the years 2015–2016. The author also would like to thank to the referee for valuable comments and suggestions that improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by TÜBİTAK, the Scientific and Technological Research Council of Turkey, through its programs 2214A and 2211E.