On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

TL;DR

This paper characterizes discrete and profinite groups that can be covered by finitely many conjugates of cyclic or procyclic subgroups, identifying all residually finite groups with this property and nearly classifying profinite groups.

Contribution

It provides a complete classification of residually finite discrete groups with finite NCC and an almost complete characterization for profinite groups.

Findings

All residually finite discrete groups with finite NCC are classified.

An almost complete characterization of profinite groups with finite NCC is provided.

The concept of NCC helps understand group coverings by conjugates of cyclic subgroups.

Abstract

Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$. In this paper we determine all residually finite discrete groups with finite NCC and give an almost complete characterization of profinite groups with finite NCC.