query: If G/H is a cyclic group, then G is a cyclic group
positive_passages.docid: 75254
positive_passages.text: Properties possessed by $H , G/H$ but not G
negative_passages.docid: 92430
negative_passages.text: Show that the group is cyclic.
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 152071
negative_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
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: 34902
negative_passages.text: If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 147517
negative_passages.text: Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
negative_passages.docid: 27436
negative_passages.text: Does $G\cong G/H$ imply that $H$ is trivial?
negative_passages.docid: 155219
negative_passages.text: Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
negative_passages.docid: 38853
negative_passages.text: Cyclic group of prime order
negative_passages.docid: 144258
negative_passages.text: How to show a group is cyclic?
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 84501
negative_passages.text: Proof involving Cyclic group, generator and GCD
negative_passages.docid: 127306
negative_passages.text: Is the group of units of a finite ring cyclic?