Is there an infinite group with a finite number of subgroups?

Suppose that G is an infinite group with a finite number of subgroups. We have two possibilities, G contains an infinite cyclic subgroup or it does not. If we are in the first case then this subgroup, being isomorphic to ℤ, has in turn an infinite number of subgroups, absurd. Then we must be in the second case, but then taking an element of the group, say g, its order must be finite (if it were not the cyclic generated by it would be isomorphic to Z) then we can construct infinite subgroups in this way, we take y in G and x ∈ G \ <y> then <x> is different from <y> , so we create an infinite number of subgroups, absurd. Since we always prevent an absurdity G cannot exist.