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!

Cheers