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

Abstract Given a discrete (resp. profinite) group G , we define $$\textrm{NCC}(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>NCC</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.