query: If there monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$ then there is an isomorphism $h : A \rightarrow B$
positive_passages.docid: 115502
positive_passages.text: Analogue of the Cantor-Bernstein-Schroeder theorem for general algebraic structures
negative_passages.docid: 73652
negative_passages.text: Exhibiting an isomorphism between two finite fields
negative_passages.docid: 117131
negative_passages.text: Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$
negative_passages.docid: 100149
negative_passages.text: If there are injective homomorphisms between two groups in both directions, are they isomorphic?
negative_passages.docid: 67199
negative_passages.text: Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$
negative_passages.docid: 89190
negative_passages.text: Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)
negative_passages.docid: 30236
negative_passages.text: isomorphic groups related in direct product.
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 11202
negative_passages.text: If $g \circ f$ is injective, so is $g$
negative_passages.docid: 126644
negative_passages.text: Simple module and homomorphisms
negative_passages.docid: 90649
negative_passages.text: Consider $f(x)=mx$ and $g(x) =bx$ in which $m>0$ and $b>0$. Does there exist any homeomorphism $h$ such that $h \circ f$ = $g \circ h$
negative_passages.docid: 22995
negative_passages.text: Let f:G to H be 1-1 homomorphism show that order of f(a) divides the order of G.
negative_passages.docid: 157597
negative_passages.text: Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
negative_passages.docid: 25439
negative_passages.text: f(a) = inverse of a is an isomorphism iff a group G is Abelian
negative_passages.docid: 72694
negative_passages.text: Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
negative_passages.docid: 101064
negative_passages.text: Showing the triviality of homomorphisms