Some finiteness conditions on normalizers or centralizers in groups
Abstract
We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) |C_G(x):<x>| is finite for every non-normal cyclic subgroup <x> of G. We show that (i) and (ii) are equivalent in the classes of locally finite groups and locally nilpotent groups. In both cases, the groups satisfying these conditions are a special kind of cyclic extensions of Dedekind groups. We also study a variation of (i) and (ii), where the requirement of finiteness is replaced with a bound. In this setting, we extend our analysis to the classes of periodic locally graded groups and non-periodic groups. While the two conditions are still equivalent in the former case, in the latter the condition about normalizers is stronger than that about centralizers.
Keywords
Cite
@article{arxiv.1412.7330,
title = {Some finiteness conditions on normalizers or centralizers in groups},
author = {Gustavo A. Fernandez-Alcober and Leire Legarreta and Antonio Tortora and Maria Tota},
journal= {arXiv preprint arXiv:1412.7330},
year = {2016}
}
Comments
This paper has been revised and split into three parts. This version is the third part of the original paper. The first two parts are arXiv:1510.03733 and arXiv:1510.08251, respectively