query: $(\mathbb R^{\times},\cdot)$ isn't isomorphic to $(\mathbb R,+)$
positive_passages.docid: 18160
positive_passages.text: How to show that the group of nonzero real numbers under multiplication is not isomorphic to the group of real numbers under addition?
negative_passages.docid: 96363
negative_passages.text: How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?
negative_passages.docid: 88645
negative_passages.text: Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.
negative_passages.docid: 79056
negative_passages.text: Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
negative_passages.docid: 85032
negative_passages.text: $Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
negative_passages.docid: 157840
negative_passages.text: Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?
negative_passages.docid: 57269
negative_passages.text: A special bijection between $\mathbb{R}\times \mathbb{R}$ and $\mathbb{R}$
negative_passages.docid: 5996
negative_passages.text: Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
negative_passages.docid: 144501
negative_passages.text: $\operatorname{Hom}(X \times Z, Y) \cong \operatorname{Hom}(X, \operatorname{Map}(Z, Y))$ is not true in $\textbf{Top}$
negative_passages.docid: 68014
negative_passages.text: Prove that $\mathfrak{su}(2)$ is not isomorphic to $\mathfrak{sl}(2,\mathbb R)$
negative_passages.docid: 108044
negative_passages.text: Isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent
negative_passages.docid: 81229
negative_passages.text: Show that $(C([a,b]),\|{\cdot}\|_2)$ is not a complete normed vector space
negative_passages.docid: 133820
negative_passages.text: The product topology on $\mathbb R_d\times \mathbb R$ vs. the dictionary order topology on $\mathbb R\times \mathbb R$
negative_passages.docid: 26764
negative_passages.text: Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)
negative_passages.docid: 149973
negative_passages.text: Showing the following coordinate rings are not isomorphic
negative_passages.docid: 98587
negative_passages.text: $\mathbb Z [X] / (X)$ isomorphic to $\mathbb Z[X] / (X+1)$ isomorphic to $\mathbb Z [X] / (X+2015)$
negative_passages.docid: 4156
negative_passages.text: What is $\Bbb R^{\times}$? [unit group, ring to "times" power]