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Problem 92 (The First Isomorphism Theorem Proof)

Prove the First Isomorphism Theorem.

Theorem (The First Isomorphism Theorem)

Let $f:G\to H$ be a surjective homomorphism with kernel $K$. Because we know that $f(x)=f(y)$ for any $y\in Kx$ (elements in the same coset of the kernel have the same image under $f$), then we can define a map $\phi:G/K\to H$ by defining $\phi(Kg)=f(g)$. This map $\phi$ is always an isomorphism.

The following pages link to this page.

Problem.TheFirstIsomorphismTheoremProof
Schedule.20161128
Schedule.20161130
Schedule.20161205
Schedule.20171129
Schedule.AllProblems

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Page last modified on November 27, 2017, at 07:40 AM

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HomePage Current Problems All Problems Pictures ProblemsByWeek List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Problem The First Isomorphism Theorem Proof Please Login to access more options. Problem 92 (The First Isomorphism Theorem Proof) Prove the First Isomorphism Theorem. Theorem (The First Isomorphism Theorem) Let $f:G\to H$ be a surjective homomorphism with kernel $K$. Because we know that $f(x)=f(y)$ for any $y\in Kx$ (elements in the same coset of the kernel have the same image under $f$), then we can define a map $\phi:G/K\to H$ by defining $\phi(Kg)=f(g)$. This map $\phi$ is always an isomorphism. The following pages link to this page. Problem.TheFirstIsomorphismTheoremProof Schedule.20161128 Schedule.20161130 Schedule.20161205 Schedule.20171129 Schedule.AllProblems Edit Page History Source Attach File Backlinks List Group Page last modified on November 27, 2017, at 07:40 AM
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