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Jul 1, 2016 · A homework problem asked to find a short exact sequence of abelian groups $$0 \rightarrow A \longrightarrow B \longrightarrow C \rightarrow 0$$ such that $B \cong A \oplus C$ …
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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 …
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 …
Nov 13, 2015 · A symbol I have in my math homework looks like a ~ above a =. (That is, $\\cong$.) What does this mean? I'm studying Congruency at the moment if that helps.
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 …
Sep 28, 2024 · Claim: $\operatorname {Hom}_ {G} (V,W) \cong \operatorname {Hom}_ {G} (\mathbf {1},V^ {*} \otimes W)$ I'm looking for hints as to how to approach the proof of this claim.
Nov 26, 2014 · How can I see that the connected sum $\mathbb {P}^2 \# \mathbb {P}^2$ of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit …
Jan 28, 2021 · Suppose $R$ is a PID, $A,B$ and $C$ are finitely generated $R$ modules and that $A \oplus B \cong A \oplus C$, then $B \cong C$. It seems like we should be able to apply the …
This approach uses the chinese remainder lemma and it illustrates the 'unique factorization of ideals' into products of powers of maximal ideals in Dedekind domains: It follows $-1 \cong 10-1 …
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