TITLE: On subgroups of a group
QUESTION [5 upvotes]: We know that if $G$ is a finite group, then all subgroups of $G$ are finite  and the number subgroups of $G$ is finite. Now 
1) If all proper subgroups of $G$ are finite, then is $G$ finite?
2) If the number subgroups of $G$ is finite, then is $G$ finite?
Thank you

REPLY [2 votes]: I think the Prüfer group serves as a counterexample for 1, let me attempt 2:
Assume $G$ is infinite, it therefore contains at least a countable number of elements. Let $g_i$ be these elements. Look then at $(g_i)$, the subgroups generated by the $g_i$. If all of these subgroups are finite, then there are an infinite number of subgroups, regardless of whether or not there are  $(g_i) = (g_j)$ for $i \neq j$. If a single one of these subgroups are infinite, then it is isomporphic to $\mathbb{Z}$, which has an infinite number of subgroups. Therefore 2 is true.