query: Group isomorphism of the nonzero real numbers
positive_passages.docid: 33233
positive_passages.text: Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.
negative_passages.docid: 38153
negative_passages.text: Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
negative_passages.docid: 115663
negative_passages.text: Proving an isomorphism using the isomorphism theorems
negative_passages.docid: 111168
negative_passages.text: Automorphisms of a finite non-abelian group
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: 5906
negative_passages.text: All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $
negative_passages.docid: 121449
negative_passages.text: An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$
negative_passages.docid: 149030
negative_passages.text: Nonisomorphic groups of order 12.
negative_passages.docid: 37351
negative_passages.text: Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$
negative_passages.docid: 126790
negative_passages.text: Basic Isomorphism Question
negative_passages.docid: 6171
negative_passages.text: Isomorphism between $\Bbb{Z}[\sqrt{d}]$ and a quotient of $\Bbb{Z}[X].$
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
negative_passages.docid: 126098
negative_passages.text: Isomorphism of factor groups
negative_passages.docid: 17022
negative_passages.text: Definable real numbers
negative_passages.docid: 126930
negative_passages.text: Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 109801
negative_passages.text: multiplicative group of infinite fields