A finiteness condition on centralizers in locally nilpotent groups

TL;DR

This paper characterizes infinite locally nilpotent groups where each non-normal cyclic subgroup has a finite index in its centralizer, and extends the analysis to non-periodic groups with bounded index conditions.

Contribution

It provides a detailed classification of such groups and generalizes the conditions to non-periodic groups with bounded indices.

Findings

Characterization of infinite locally nilpotent groups with finite centralizer indices.

Extension of results to non-periodic groups with bounded indices.

Insight into the structure of groups satisfying these finiteness conditions.

Abstract

We give a detailed description of infinite locally nilpotent groups G such that the index |C_G (x) : <x>| is finite, for every non-normal cyclic subgroup <x> of G. We are also able to extend our analysis to all non-periodic groups satisfying a variation of our condition, where the requirement of finiteness is replaced with a bound.