Show that a group that has only a finite number of subgroups must be a finite group

Question

Show that a group that has only a finite number of subgroups must be a finite group.

I started by assuming that group is infinite.

But, don't understand how I should go from this assumption

Answer 1

Note that each $a\in G$ generates a subgroup $\langle a\rangle$. Noting that

$$G=\bigcup_{a\in G}\langle a\rangle$$

and any elements of infinite order produce a subgroup isomorphic to $\Bbb Z$, so that this subgroup has infinitely many subgroups--i.e. if $\langle a\rangle$ is infinite, then $\langle a^n\rangle, n\in\Bbb N$ is an infinite family of subgroups (a contradiction). But then $G$ is a finite union of finite sets, making it a finite set itself.