query: $G$ is a finite group, prove that $G$ is abelian when $G/C(G)$ is cyclic.
positive_passages.docid: 23394
positive_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 17510
negative_passages.text: Proving a set is an abelian group.
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: 151025
negative_passages.text: If G is a group of order n=35, then it is cyclic
negative_passages.docid: 150585
negative_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
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: 142894
negative_passages.text: Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
negative_passages.docid: 135039
negative_passages.text: ${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
negative_passages.docid: 65900
negative_passages.text: Show that a finite group with certain automorphism 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: 57421
negative_passages.text: If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian?
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: 147574
negative_passages.text: Prove of a finite cyclic group of order $n$ is isomorphic tto $(\mathbb Z_n,+)$.
negative_passages.docid: 60208
negative_passages.text: Proof of property of abelian group
negative_passages.docid: 102146
negative_passages.text: If a group mod its commutator subgroup is cyclic, then the group is abelian?
negative_passages.docid: 8623
negative_passages.text: Finite subgroups of the multiplicative group of a field are cyclic