If a group $G$ has only finitely many subgroups, then show that $G$ is a finite group. If a group $G$ has only finitely many subgroups, then show that $G$ is a finite group.  I have no idea on how to start this question. Can anyone guide me?

Suppose $G$ were infinite.  If $G$ contains an element of infinite order, then _.
Otherwise, every element of $G$ has finite order, and if $x_1, x_2, \ldots, x_n$ are any finite set of elements of $G$, then the subgroups $\langle x_i \rangle$ cover only finitely many elements of $G$.  Therefore __.
