query: $G$ is abelian group of order $pq$, where $p$ and $q$ prime. Prove that $G$ cyclic
positive_passages.docid: 25835
positive_passages.text: Structure of groups of order $pq$, where $p,q$ are distinct primes.
negative_passages.docid: 131706
negative_passages.text: Is my proof that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes correct?
negative_passages.docid: 108830
negative_passages.text: Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
negative_passages.docid: 76446
negative_passages.text: Why are cyclic groups of order $p$ (prime) unique up to isomorphism?
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: 89382
negative_passages.text: In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
negative_passages.docid: 98145
negative_passages.text: If $G$ is cyclic then $G/H$ is cyclic?
negative_passages.docid: 117468
negative_passages.text: $o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
negative_passages.docid: 67973
negative_passages.text: Prove that a finite abelian group is simple if and only if its order is prime.
negative_passages.docid: 152481
negative_passages.text: If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 116080
negative_passages.text: Let G be a finite group with more than one element. Show that G has an element of prime order
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: 13734
negative_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 38853
negative_passages.text: Cyclic group of prime order
negative_passages.docid: 41731
negative_passages.text: Question about the order of the product of two elements in an abelian group