query: If $G$ s.t. $|G|=15$ has only one subgroup of order $3$ and only one of order $5$, then $G$ is cyclic. Generalize to $|G|=pq$, for $p,q$ primes.
positive_passages.docid: 142894
positive_passages.text: Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
negative_passages.docid: 73211
negative_passages.text: Cyclic subgroups dealing with prime numbers Question
negative_passages.docid: 57887
negative_passages.text: When are the product of cyclic groups also cyclic?
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: 44152
negative_passages.text: If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
negative_passages.docid: 65926
negative_passages.text: Group of order $1575$ having a normal sylow $3$ subgroup is abelian.
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: 155219
negative_passages.text: Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
negative_passages.docid: 14528
negative_passages.text: Proof that all abelian simple groups are cyclic groups of prime order
negative_passages.docid: 21673
negative_passages.text: If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
negative_passages.docid: 6030
negative_passages.text: Group of order 15 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: 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: 134455
negative_passages.text: Any group of order $15$ is abelian(without sylow theorem)
negative_passages.docid: 50706
negative_passages.text: $|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
negative_passages.docid: 90453
negative_passages.text: How can I find cyclic subgroups of $S_3$?
negative_passages.docid: 132475
negative_passages.text: Simple method the show that a group of order 15 is cyclic