Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.

Question

Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.

This is a question from page $181$, Chapter $3.4$ Elements of Modern Algebra ($8^{th}$ Edition) by Linda Gilbert.

My intuition tells me that the given statement is false, however, I cannot seem to find a rigorous argument (or counterexample) to disprove the statement. Can anyone please point me in the right direction?

I know that if a group $G$ is cyclic, then every subgroup is cyclic. This is not an if and only if statement - hence my intuition telling me the above mentioned statement is false.

Answer 1

The question is easily settled even without an explicit counterexample. There exist non cyclic finite abelian groups, so a non cyclic finite abelian group of minimal cardinality provides the counterexample. Indeed its proper subgroups must be cyclic, by minimality.

Of course, the counterexample can be shown: the Klein group $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.

Answer 2

A more exotic infinite counterexample is any Prufer p-group.