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.
Best Answer
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 $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.