On subgroups of a group 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

I think, the first one is wrong. Enough to consider $G=\mathbb Z(p^{\infty})$. This group is infinite, each of whose proper subgroups is finite and also cyclic.
