query: Prove that $G$ is cyclic group of prime order
positive_passages.docid: 22301
positive_passages.text: If $G$ has no proper subgroup, then $G$ is cyclic of prime order
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: 23885
negative_passages.text: Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is 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: 76317
negative_passages.text: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
negative_passages.docid: 79995
negative_passages.text: G is group of order pq, pq are primes
negative_passages.docid: 39976
negative_passages.text: If a group G has only the trivial group and G itself as its subgroup, can we be sure that G has prime order?
negative_passages.docid: 134820
negative_passages.text: $p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
negative_passages.docid: 37913
negative_passages.text: Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
negative_passages.docid: 117206
negative_passages.text: $|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
negative_passages.docid: 4402
negative_passages.text: Group of order $pqr$, $p < q < r$ primes
negative_passages.docid: 125640
negative_passages.text: Group of order $p^2$ is abelian.
negative_passages.docid: 81864
negative_passages.text: Finite group of order $mn$ with $\gcd(m,n) = 1$ .
negative_passages.docid: 103423
negative_passages.text: Normal subgroup where G has order prime