query: Does $A \times B \cong A \times C$ imply that $B \cong C$?
positive_passages.docid: 105951
positive_passages.text: Does $G\times K\cong H\times K$ imply $G\cong H$?
negative_passages.docid: 133915
negative_passages.text: In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true?
negative_passages.docid: 149553
negative_passages.text: Is it always true that $(A_1 \cup A_2) \times (B_1 \cup B_2)=(A_1\times B_1) \cup (A_2 \times B_2)$
negative_passages.docid: 5223
negative_passages.text: $\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 154698
negative_passages.text: Union of Cartesian products: $X \times (Y \cup Z)= (X \times Y) \cup(X\times Z)$
negative_passages.docid: 6161
negative_passages.text: The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable
negative_passages.docid: 151621
negative_passages.text: $G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
negative_passages.docid: 139934
negative_passages.text: In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
negative_passages.docid: 15184
negative_passages.text: $C \otimes A \cong C \otimes B$ does not imply $A \cong B$
negative_passages.docid: 115876
negative_passages.text: $D_6 \cong D_3\times \mathbb Z_2$
negative_passages.docid: 90239
negative_passages.text: $\mathbb{Z} \times \mathbb{Z}$ is cyclic.
negative_passages.docid: 63267
negative_passages.text: Under what conditions on the sets $A$ and $B$ will it be true that $A\times B= B\times A$
negative_passages.docid: 80684
negative_passages.text: Counter example for $(A \times B) \cap (C \times D) = (A \cap C ) \times (B \cap D)$
negative_passages.docid: 41099
negative_passages.text: If $A \subseteq C$ and $B \subseteq D$ then $A \times B \subseteq C \times D$
negative_passages.docid: 158355
negative_passages.text: If $A$ commutes with $B$ and $B$ commutes with $C$, then $A$ commutes with $C$
negative_passages.docid: 135701
negative_passages.text: An example of a group such that $G \cong G \times G$