I know that the fundamental Fundamental Theorem of Cyclic Groups stats that if $G$ is cyclic then so is every subgroup. So if I am given a group $G$, then by demonstrating that every non-trivial proper subgroup is cyclic can I conclude that $G$ is cyclic?
I am inclined to say no. This is because the fundamental theorem is not an if and only if statement, but I cannot seem to come up with a counter example.
The Klein four-group is a counterexample, as is the symmetric group on three elements.