query: Prime Order $\Rightarrow$ Cyclic $\Rightarrow$ Abelian
positive_passages.docid: 13734
positive_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 4402
negative_passages.text: Group of order $pqr$, $p < q < r$ primes
negative_passages.docid: 61546
negative_passages.text: Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
negative_passages.docid: 68733
negative_passages.text: If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
negative_passages.docid: 34972
negative_passages.text: order of non abelian group can't be what?
negative_passages.docid: 74187
negative_passages.text: $\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
negative_passages.docid: 63136
negative_passages.text: Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
negative_passages.docid: 79055
negative_passages.text: Orders of a symmetric group
negative_passages.docid: 23885
negative_passages.text: Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
negative_passages.docid: 33025
negative_passages.text: What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
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: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 115449
negative_passages.text: Inclusion of Fields whose order is a prime power
negative_passages.docid: 23394
negative_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 9393
negative_passages.text: Order of elements is lcm-closed in abelian groups
negative_passages.docid: 17178
negative_passages.text: Order of $\phi(g)$ divides the order of $g$
negative_passages.docid: 22301
negative_passages.text: If $G$ has no proper subgroup, then $G$ is cyclic of prime order