query: If G/N is cyclic, then G is abelian.
positive_passages.docid: 23394
positive_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 92430
negative_passages.text: Show that the group is cyclic.
negative_passages.docid: 14528
negative_passages.text: Proof that all abelian simple groups are cyclic groups of prime order
negative_passages.docid: 11113
negative_passages.text: Group With an Endomorphism That is "Almost" Abelian is Abelian.
negative_passages.docid: 102146
negative_passages.text: If a group mod its commutator subgroup is cyclic, then the group is abelian?
negative_passages.docid: 76317
negative_passages.text: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 151619
negative_passages.text: A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic
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: 84
negative_passages.text: For what $n$ is $U_n$ cyclic?
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: 132725
negative_passages.text: Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
negative_passages.docid: 40287
negative_passages.text: Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian
negative_passages.docid: 125640
negative_passages.text: Group of order $p^2$ is abelian.
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 59007
negative_passages.text: Suppose $|G| = 105$. Show that it is abelian if it has a normal $3$-Sylow subgroup.