query: is the group cyclic, if Normal Subgroup and Quotient group are cyclic.
positive_passages.docid: 17077
positive_passages.text: Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.
negative_passages.docid: 32620
negative_passages.text: Is every group of order $21$ cyclic?
negative_passages.docid: 9232
negative_passages.text: Group with proper subgroups infinite cyclic
negative_passages.docid: 144258
negative_passages.text: How to show a group is cyclic?
negative_passages.docid: 13734
negative_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 58014
negative_passages.text: Prove or disprove: The automorphism group of a finite cyclic group must be cyclic.
negative_passages.docid: 4883
negative_passages.text: Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
negative_passages.docid: 73211
negative_passages.text: Cyclic subgroups dealing with prime numbers Question
negative_passages.docid: 63136
negative_passages.text: Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
negative_passages.docid: 23533
negative_passages.text: All subgroups normal $\implies$ abelian group
negative_passages.docid: 121810
negative_passages.text: Prove that the product of two infinite cyclic groups is not infinite cyclic.
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: 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: 122266
negative_passages.text: equivalent definition of normal subgroup
negative_passages.docid: 102146
negative_passages.text: If a group mod its commutator subgroup is cyclic, then the group is abelian?
negative_passages.docid: 116125
negative_passages.text: A group $G$ with $G'$ abelian and every abelian normal subgroup finite
negative_passages.docid: 109236
negative_passages.text: Subgroups containing kernel of group morphism to an abelian group are normal.