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Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be thesmallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugatescover $G$. In this paper we determine all residually finite discrete groupswith finite NCC and give an almost complete characterization of profinitegroups with finite NCC. As a consequence we can show, for example, that for any$k\in\mathbb N$ and any prime $p>3$ the number of finite $p$-groups $P$ withmore than $3$ generators and $NCC(P)=k$ is finite.

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