query: Proving $|G|=pq$ and $p>q$ , $q$ does not divide $p-1$ $\implies$ $G$ is cyclic , without using Cauchy's and Sylow's theorems
positive_passages.docid: 8746
positive_passages.text: Groups of order $pq$ without using Sylow theorems
negative_passages.docid: 72144
negative_passages.text: Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
negative_passages.docid: 117206
negative_passages.text: $|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
negative_passages.docid: 121810
negative_passages.text: Prove that the product of two infinite cyclic groups is not infinite cyclic.
negative_passages.docid: 95240
negative_passages.text: Show that $({\mathbb{Q}},+)$ is not finitely generated using the Fundamental Theorem of Finitely Generated Abelian Groups.
negative_passages.docid: 13807
negative_passages.text: Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S
negative_passages.docid: 158106
negative_passages.text: Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
negative_passages.docid: 26449
negative_passages.text: $|G|=24$ prove that $G$ is not simple
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: 115726
negative_passages.text: For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
negative_passages.docid: 64130
negative_passages.text: Prove that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$
negative_passages.docid: 6119
negative_passages.text: If both $P$ and $Q$ are true , how can I tell that $P$ implies $Q$?
negative_passages.docid: 140412
negative_passages.text: If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
negative_passages.docid: 135039
negative_passages.text: ${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
negative_passages.docid: 84501
negative_passages.text: Proof involving Cyclic group, generator and GCD
negative_passages.docid: 145112
negative_passages.text: Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative