- What does $\cong$ sign represent? - Mathematics Stack Exchange
In geometry, $\cong$ means congruence of figures, which means the figures have the same shape and size (In advanced geometry, it means one is the image of the other under a mapping known as an "isometry", which provides a formal definition of what "same shape and size" means) Two congruent triangles look exactly the same, but they are not the
- Difference between ≈, ≃, and ≅ - Mathematics Stack Exchange
In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators B
- Does there exist a ring R for which non-isomorphic groups always have . . .
Does there exist a ring $R$ such that for any two (finite) groups $G_1$ and $G_2$ $$RG_1\cong RG_2\implies G_1\cong G_2$$ $R\ne\mathbb {C},\mathbb {Q}$ by Maschke's theorem, or any other field by this answer
- Proof of $ (\mathbb {Z} m\mathbb {Z}) \otimes_\mathbb {Z} (\mathbb {Z . . .
Originally you asked for $\mathbb {Z} (m) \otimes \mathbb {Z} (n) \cong \mathbb {Z} \text {gcd} (m,n)$, so any old isomorphism would do, but your proof above actually shows that $\mathbb {Z} \text {gcd} (m,n)$ $\textit {is}$ the tensor product
- Non-associative operations - Mathematics Stack Exchange
While the comments above have merit, it's still important to notice the technical differences between $=$ (equality) and $\cong$ (isomorphism), especially for learners of algebra
- $G \\times H \\cong G \\times K$ , then $ K \\cong H$
If $G\times H\cong G\times K$ is finitely presented and $H\cong\mathbb {Z}$ is infinite cyclic then $K$ need not be infinite cyclic I review the proof of this (it is dead easy!) in an answer to the original question, here
- abstract algebra - Prove that $\mathbb Z_ {m}\times\mathbb Z_ {n} \cong . . .
Prove that $\mathbb Z_ {m}\times\mathbb Z_ {n} \cong \mathbb Z_ {mn}$ implies $\gcd (m,n)=1$ This is the converse of the Chinese remainder theorem in abstract algebra
- Notations involving squiggly lines over horizontal lines
The symbol $\cong$ can in principle be used to designate an isomorphism in any category (e g , isometric, diffeomorphic, homeomorphic, linearly isomorphic, etc )
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