query: $PSL_2(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2 \ast \mathbb{Z}_3$
positive_passages.docid: 27472
positive_passages.text: Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$
negative_passages.docid: 142352
negative_passages.text: how to prove $G_1$ and $G_2$ aren't isomorphic?
negative_passages.docid: 107687
negative_passages.text: How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$?
negative_passages.docid: 107163
negative_passages.text: Proof that $S_3$ isomorphic to $D_3$
negative_passages.docid: 33781
negative_passages.text: Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?
negative_passages.docid: 6171
negative_passages.text: Isomorphism between $\Bbb{Z}[\sqrt{d}]$ and a quotient of $\Bbb{Z}[X].$
negative_passages.docid: 158315
negative_passages.text: Are $M_2= \langle(\sqrt{2}^{\sqrt{2}},\infty),\cdot \rangle$ , $M_1 = \langle(2,\infty),\cdot\rangle$ isomorphic structures?
negative_passages.docid: 95764
negative_passages.text: Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 134840
negative_passages.text: Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$
negative_passages.docid: 128300
negative_passages.text: Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.
negative_passages.docid: 78118
negative_passages.text: Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$?
negative_passages.docid: 57434
negative_passages.text: $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong A_4$
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: 37928
negative_passages.text: Isomorphism between $D_{2n}/Z(D_{2n})$ and $D_n$
negative_passages.docid: 55900
negative_passages.text: Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?
negative_passages.docid: 128096
negative_passages.text: Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$