Some finiteness conditions on normalizers or centralizers in groups

TL;DR

This paper investigates finiteness conditions on normalizers and centralizers in groups, showing their equivalence in certain classes and characterizing groups satisfying these conditions as cyclic extensions of Dedekind groups.

Contribution

It establishes the equivalence of two finiteness conditions in locally finite and locally nilpotent groups and characterizes the structure of groups satisfying these conditions.

Findings

Conditions are equivalent in locally finite groups.

Groups satisfying these conditions are cyclic extensions of Dedekind groups.

Extension of analysis to periodic locally graded and non-periodic 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.