query: Suppose $G$ is a group with order $n$. Assume $(n,\phi(n)) = 1$. Prove that $G$ is Abelian group?
positive_passages.docid: 147517
positive_passages.text: Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.
negative_passages.docid: 86823
negative_passages.text: proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
negative_passages.docid: 134376
negative_passages.text: Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
negative_passages.docid: 1364
negative_passages.text: Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
negative_passages.docid: 125559
negative_passages.text: In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
negative_passages.docid: 27703
negative_passages.text: an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
negative_passages.docid: 15526
negative_passages.text: Question on groups of order $pq$
negative_passages.docid: 3756
negative_passages.text: If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
negative_passages.docid: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 17510
negative_passages.text: Proving a set is an abelian group.
negative_passages.docid: 125582
negative_passages.text: Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
negative_passages.docid: 74586
negative_passages.text: Prove that every group of order $4$ is abelian
negative_passages.docid: 42389
negative_passages.text: If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 141125
negative_passages.text: Group with order $p^2$ must be abelian . How to prove that?
negative_passages.docid: 157841
negative_passages.text: Existence of a non-abelian group of order $p^n$.