query: $\mathbb{Z}$ is not isomorphic to $\mathbb{Q}$
positive_passages.docid: 95764
positive_passages.text: Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
negative_passages.docid: 92211
negative_passages.text: When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?
negative_passages.docid: 67374
negative_passages.text: Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?
negative_passages.docid: 76501
negative_passages.text: How can we show that $\mathbb Q$ is not a free $\mathbb Z$-module?
negative_passages.docid: 126927
negative_passages.text: $\mathbb Q$ is not the direct product of two non-trivial groups.
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 14003
negative_passages.text: $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
negative_passages.docid: 26661
negative_passages.text: Ring homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$
negative_passages.docid: 11710
negative_passages.text: How to show that $\mathbb{Q}$ is not $G_\delta$
negative_passages.docid: 41035
negative_passages.text: Mapping that proves $\mathbb{Z} \subset \mathbb{Q}$
negative_passages.docid: 6171
negative_passages.text: Isomorphism between $\Bbb{Z}[\sqrt{d}]$ and a quotient of $\Bbb{Z}[X].$
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: 25050
negative_passages.text: To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.
negative_passages.docid: 33781
negative_passages.text: Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?
negative_passages.docid: 57310
negative_passages.text: Prove $\mathbb{Z}$ is not a vector space over a field
negative_passages.docid: 88645
negative_passages.text: Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.
negative_passages.docid: 135241
negative_passages.text: Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group