Almost group theory

TL;DR

This paper introduces the concepts of almost centralizer and almost commutator, and applies them to study a class of groups called Mc-groups, establishing properties like nilpotency of the Fitting subgroup and generalizing Hall's theorem.

Contribution

It defines almost centralizer and almost commutator, and extends classical group theory results to Mc-groups, broadening understanding of their structure.

Findings

Fitting subgroup of Mc-groups is nilpotent.

Generalization of Hall's theorem to ind-definable almost nilpotent subgroups.

Basic properties of almost centralizer and almost commutator established.

Abstract

The notion of almost centralizer and almost commutator are introduced and basic properties are established. They are used to study $\widetilde{\mathfrak M}\_c$-groups, i. e.groups for which every descending chain of centralizers each having infinite index in its predecessor stabilizes after finitely many steps. The Fitting subgroup of such groups is shown to be nilpotent and a theorem of Hall for nilpotent groups is generalized to ind-definable almost nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups.