query: Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
positive_passages.docid: 152662
positive_passages.text: Proving that any finite cyclic group of order $n$ is isomorphic to $$
negative_passages.docid: 36137
negative_passages.text: Prove $GL_2(\mathbb{Z}/2\mathbb{Z})$ is isomorphic to $S_3$
negative_passages.docid: 154878
negative_passages.text: Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 83654
negative_passages.text: Order of an element in a finite cyclic group
negative_passages.docid: 46116
negative_passages.text: Proof of the First Isomorphism Theorem for Groups
negative_passages.docid: 29445
negative_passages.text: Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.
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: 23394
negative_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 62635
negative_passages.text: Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
negative_passages.docid: 23279
negative_passages.text: Group Theory Isomorphism
negative_passages.docid: 132280
negative_passages.text: Let $G$ be a group of order 35. Show that $G \cong Z_{35}$
negative_passages.docid: 4648
negative_passages.text: Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$
negative_passages.docid: 97833
negative_passages.text: Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$
negative_passages.docid: 142894
negative_passages.text: Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
negative_passages.docid: 87946
negative_passages.text: Proving that the unities of a ring form a group under multiplication
negative_passages.docid: 67772
negative_passages.text: How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$