query: When will two groups be isomorphic???
positive_passages.docid: 29051
positive_passages.text: If I know the order of every element in a group, do I know the group?
negative_passages.docid: 84869
negative_passages.text: Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic?
negative_passages.docid: 83710
negative_passages.text: Order of a homomorphism of groups
negative_passages.docid: 112993
negative_passages.text: Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
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: 34606
negative_passages.text: Which groups have precisely two automorphisms
negative_passages.docid: 152481
negative_passages.text: If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
negative_passages.docid: 87766
negative_passages.text: Using Cayley's Theorem to find an isomorphic subgroup.
negative_passages.docid: 157915
negative_passages.text: Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$
negative_passages.docid: 99538
negative_passages.text: Non-trivial groups pairwise non-isomorphic with isomorphic direct products
negative_passages.docid: 150204
negative_passages.text: Isomorphisms between the groups $U(10), U(5)$ and $\mathbb{Z}/4\mathbb{Z}$
negative_passages.docid: 100149
negative_passages.text: If there are injective homomorphisms between two groups in both directions, are they isomorphic?
negative_passages.docid: 151862
negative_passages.text: Is any subgroup of a direct product isomorphic to a direct product of subgroups?
negative_passages.docid: 30236
negative_passages.text: isomorphic groups related in direct product.
negative_passages.docid: 137949
negative_passages.text: If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
negative_passages.docid: 120155
negative_passages.text: How to show that this group presentation is isomorphic to dihedral group?
negative_passages.docid: 146356
negative_passages.text: To prove two sets are isomorphic is it sufficient to find 2 injections or 2 surjections?