1. If every proper subgroup of a group G is cyclic, then G itself must be cyclic.
I know that if a group G is cyclic, then every subgroup of G is also cyclic. However does the converse hold? And how would you prove it?
2. A group with only a finite number of subgroups must be finite.
I think this statement is true, but I can't seem to gather a concrete proof for it.
Any help would be greatly appreciated!