query: Prove or disprove this statement. If G is a group in which every proper subgroup is cyclic, then G is cyclic
positive_passages.docid: 75986
positive_passages.text: If all Subgroups are Cyclic, is group Cylic?
negative_passages.docid: 59533
negative_passages.text: If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
negative_passages.docid: 92430
negative_passages.text: Show that the group is cyclic.
negative_passages.docid: 127306
negative_passages.text: Is the group of units of a finite ring cyclic?
negative_passages.docid: 151727
negative_passages.text: If $|G/H|=4$ then $G$ is union of three proper subgroups
negative_passages.docid: 14890
negative_passages.text: Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
negative_passages.docid: 38853
negative_passages.text: Cyclic group of prime order
negative_passages.docid: 40993
negative_passages.text: Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.
negative_passages.docid: 23533
negative_passages.text: All subgroups normal $\implies$ abelian group
negative_passages.docid: 84501
negative_passages.text: Proof involving Cyclic group, generator and GCD
negative_passages.docid: 118894
negative_passages.text: Cyclic subgroup of dihedral group
negative_passages.docid: 36887
negative_passages.text: Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
negative_passages.docid: 124150
negative_passages.text: Faithful representation implies group is cyclic
negative_passages.docid: 107935
negative_passages.text: If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is cyclic
negative_passages.docid: 34552
negative_passages.text: Proper subgroups of non-cyclic p-group cannot be all cyclic?
negative_passages.docid: 9232
negative_passages.text: Group with proper subgroups infinite cyclic