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

TL;DR

This work introduces the cyclic covering number ${ m NCC}(G)$ as a measure of how a group can be covered by conjugates of cyclic subgroups, and develops a robust framework ${ m CC}(G,\Phi)$ to handle automorphism-group actions. It proves a complete classification for infinite discrete residually finite groups: such a group with finite ${ m NCC}$ is necessarily cyclic or dihedral, and it gives an almost complete description for profinite groups, showing finite ${ m NCC}$ implies a virtually nilpotent/pro-nilpotent structure with open subgroups falling into a small number of rigid types (finite, procyclic, prodihedral for $p=2$, or open subgroups of ${ m PGL}_1(D)$). The proofs proceed by reducing to virtually solvable and then virtually nilpotent cases, using $p$-adic analytic methods, dimension-series arguments, and Lie-algebra techniques for pro-$p$ groups, then assembling the profinite picture from finite quotients. The results connect to topology via classifying spaces for families of subgroups, addressing questions of JPL and LRRV, and indicate strong structural constraints on groups with finite NCC, with implications for both algebraic and topological contexts. Overall, the paper sharpens our understanding of how finiteness of cyclic coverings dictates group structure in both discrete and profinite settings, and it links group-theoretic invariants to geometric-topological classification problems.

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.

Theorems & Definitions (106)

• Definition
• Remark
• Theorem 1.1
• Corollary 1.2
• Remark
• Theorem 1.3
• Remark
• Proposition 1.4
• Theorem 1.5
• Definition