query: Every subgroup $H$ of $G$ with $H\ne G$ is cyclic. Is $G$ cyclic?
positive_passages.docid: 155219
positive_passages.text: Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
negative_passages.docid: 151025
negative_passages.text: If G is a group of order n=35, then it is cyclic
negative_passages.docid: 14003
negative_passages.text: $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
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: 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: 22301
negative_passages.text: If $G$ has no proper subgroup, then $G$ is cyclic of prime order
negative_passages.docid: 124455
negative_passages.text: Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
negative_passages.docid: 155054
negative_passages.text: $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
negative_passages.docid: 144065
negative_passages.text: Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
negative_passages.docid: 129485
negative_passages.text: A subgroup of a cyclic group is cyclic - Understanding Proof
negative_passages.docid: 2104
negative_passages.text: Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
negative_passages.docid: 23394
negative_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 117206
negative_passages.text: $|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
negative_passages.docid: 39469
negative_passages.text: If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
negative_passages.docid: 135039
negative_passages.text: ${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
negative_passages.docid: 89382
negative_passages.text: In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.