query: The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$
positive_passages.docid: 28913
positive_passages.text: How to find a generator of a cyclic group?
negative_passages.docid: 133019
negative_passages.text: Nilpotent elements in $\mathbb Z_n$
negative_passages.docid: 102721
negative_passages.text: Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$
negative_passages.docid: 151621
negative_passages.text: $G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
negative_passages.docid: 47179
negative_passages.text: Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
negative_passages.docid: 96680
negative_passages.text: Generators of symmetric group $S_n$
negative_passages.docid: 3317
negative_passages.text: Subgroup generated by $2$ and $7$ in $(\mathbb Z,+)$
negative_passages.docid: 128710
negative_passages.text: Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$
negative_passages.docid: 29269
negative_passages.text: fundamental group of $GL^{+}_n(\mathbb{R})$
negative_passages.docid: 98352
negative_passages.text: If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$
negative_passages.docid: 37414
negative_passages.text: Question on Groups $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
negative_passages.docid: 5069
negative_passages.text: $GL_n(\mathbb F_q)$ has an element of order $q^n-1$
negative_passages.docid: 33387
negative_passages.text: Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$
negative_passages.docid: 146028
negative_passages.text: $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.