query: How to prove a group is cyclic
positive_passages.docid: 144258
positive_passages.text: How to show a group is cyclic?
negative_passages.docid: 32620
negative_passages.text: Is every group of order $21$ cyclic?
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 132127
negative_passages.text: Proving a direct product of groups is a group
negative_passages.docid: 38687
negative_passages.text: Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$
negative_passages.docid: 7967
negative_passages.text: Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
negative_passages.docid: 14890
negative_passages.text: Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
negative_passages.docid: 61394
negative_passages.text: The product of finitely many cyclic groups is cyclic
negative_passages.docid: 59533
negative_passages.text: If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
negative_passages.docid: 129485
negative_passages.text: A subgroup of a cyclic group is cyclic - Understanding Proof
negative_passages.docid: 102146
negative_passages.text: If a group mod its commutator subgroup is cyclic, then the group is abelian?
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
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: 23394
negative_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 36887
negative_passages.text: Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
negative_passages.docid: 84823
negative_passages.text: When is the group of units in $\mathbb{Z}_n$ cyclic?
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is cyclic