Let be an infinite group. Then the subgroups generated by its elements (this is why it is on cyclic groups) is . Notice there are infinite many different because is an infinite group. If we can show that for then there are infinitely many subgroups (namely, one for each generating element).Originally Posted bytopsquark

Now we know the following:

If an infinite group then infinite subgroups

Take contrapositive:

If finite subgroups then finite group.

Q.E.D.

I did not prove that I assumed it is true (it is definitely not true for finite groups).