A covering of a group $G$ a family $\{S\_i\}\_{i \in I}$ of subsets of $G$ such that $G = \displaystyle \bigcup \_{i \in I} S\_i$.

Why is true that: A group covered by finitely many cyclic subgroups is either cyclic or finite?

Remark:
Is true that by Baer (see D. Robinson, Finiteness Conditions and Generalized Soluble Groups, p.105) that:
Theorem 4.16: A group is central-by-finite if and only if it has a finite covering consisting of abelian subgroups?