query: For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?
positive_passages.docid: 157202
positive_passages.text: $A\oplus C \cong B \oplus C$. Is $A \cong B$ when $C$ is finite, A and B infinite.
negative_passages.docid: 61513
negative_passages.text: Prove that all cyclic groups of the same order are isomorphic
negative_passages.docid: 128652
negative_passages.text: Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$
negative_passages.docid: 35188
negative_passages.text: Counterexample: $G \times K \cong H \times K \implies G \cong H$
negative_passages.docid: 35353
negative_passages.text: Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
negative_passages.docid: 152603
negative_passages.text: Group isomorphic to its automorphism group
negative_passages.docid: 139835
negative_passages.text: Every Finite Group is Isomorphic to a Subgroup of $A_n$
negative_passages.docid: 154318
negative_passages.text: Can a field be isomorphic to its subfield?
negative_passages.docid: 45179
negative_passages.text: When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
negative_passages.docid: 81197
negative_passages.text: Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
negative_passages.docid: 53327
negative_passages.text: $C_2 \times C_2$ and $C_4$
negative_passages.docid: 22604
negative_passages.text: Can $R \times R$ be isomorphic to $R$ as rings?
negative_passages.docid: 26749
negative_passages.text: $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
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: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
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: 71232
negative_passages.text: if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$?