query: Are $\langle \mathbb R,+\rangle $ and $\langle \mathbb R\setminus \{0\},*\rangle $ isomorphic?
positive_passages.docid: 145561
positive_passages.text: About the additive group and the multiplicative group of a field
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: 132725
negative_passages.text: Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 44843
negative_passages.text: What is the difference between $|0\rangle $ and $0$?
negative_passages.docid: 151721
negative_passages.text: Uhlmann's Theorem: proof of $\text{tr}(A^{\dagger} B) = \langle m | A \otimes B |m\rangle $
negative_passages.docid: 98352
negative_passages.text: If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$
negative_passages.docid: 121449
negative_passages.text: An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$
negative_passages.docid: 92211
negative_passages.text: When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?
negative_passages.docid: 156712
negative_passages.text: Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$
negative_passages.docid: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
negative_passages.docid: 9140
negative_passages.text: Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$
negative_passages.docid: 84114
negative_passages.text: Does there exist a surjective homomorphism from $(\mathbb R,+)$ to $(\mathbb Q,+)$ ?
negative_passages.docid: 37904
negative_passages.text: Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.
negative_passages.docid: 127047
negative_passages.text: How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
negative_passages.docid: 140628
negative_passages.text: Are $(\Bbb R,+)$ and $(\Bbb Q, +)$ elementary equivalent?
negative_passages.docid: 115270
negative_passages.text: What are the generators of $(\mathbb{R},+)$?