query: A cyclic group with finite order $k$ is isomorphic to group $\mathbb{Z}/k\mathbb{Z}$
positive_passages.docid: 147574
positive_passages.text: Prove of a finite cyclic group of order $n$ is isomorphic tto $(\mathbb Z_n,+)$.
negative_passages.docid: 137949
negative_passages.text: If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
negative_passages.docid: 61550
negative_passages.text: Existence of elements of order $p^k$ in a finite group
negative_passages.docid: 129485
negative_passages.text: A subgroup of a cyclic group is cyclic - Understanding Proof
negative_passages.docid: 93169
negative_passages.text: Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups
negative_passages.docid: 151047
negative_passages.text: a non trivial example: $G/H$ is isomorphic to $G$
negative_passages.docid: 2304
negative_passages.text: What is the quotient of a cyclic group of order $n$ by a cyclic subgroup of order $m$?
negative_passages.docid: 152954
negative_passages.text: Extensions of finite cyclic groups
negative_passages.docid: 129239
negative_passages.text: Additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity
negative_passages.docid: 114916
negative_passages.text: $\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
negative_passages.docid: 150204
negative_passages.text: Isomorphisms between the groups $U(10), U(5)$ and $\mathbb{Z}/4\mathbb{Z}$
negative_passages.docid: 36882
negative_passages.text: Two finite abelian groups with the same number of elements of any order are isomorphic
negative_passages.docid: 13734
negative_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 112993
negative_passages.text: Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
negative_passages.docid: 83710
negative_passages.text: Order of a homomorphism of groups
negative_passages.docid: 55393
negative_passages.text: Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$
negative_passages.docid: 95764
negative_passages.text: Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.