My teacher told me to prove this using the proof by contradiction. So, I am going to assume that there's an infinite group with finitely many subgroups. Then this will lead to finding at least one element with an infinite order, from which I can derive a contradiction.
But I am lost how I can prove this...
I am thinking if the group has its subgroups, then every element in those subgroups must also have infinite order as they are in their original group, aren't they? So, my conclusion is that every element in infinite group's subgroups must have infinite order except identity element.
But how does this contradict the group is infinite?