PlanetMath: proof of first isomorphism theorem

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proof of first isomorphism theorem

(Proof)

Let denote $\ker f$ . is a normal subgroup of because, by the following calculation, $gkg^{-1} \in K$ for all $g\in G$ and $k\in K$ (rules of homomorphism imply the first equality, definition of for the second):

$\displaystyle f(gkg^{-1}) = f(g) f(k) f(g)^{-1} = f(g) 1_H f(g)^{-1} = 1_H$

Therefore,

is well defined.

Define a group homomorphism $\theta: G/K \to {\rm im} f$ given by:

$\displaystyle \theta (gK) = f(g)$

We argue that $\theta$ is an isomorphism.

First, $\theta$ is well defined. Take two representative, and , of the same modulo class. By definition, $g_1g_2^{-1}$ is in . Hence, sends $g_1g_2^{-1}$ to (all elements of are sent by to ). Consequently, the next calculation is valid: $f(g_1)f(g_2)^{-1}=f(g_1g_2^{-1})=1$ but this is the same as saying that . And we are done because the last equality indicate that $\theta (g_1K)$ is equal to $\theta (g_2K)$ .

Going backward the last argument, we get that $\theta$ is also an injection: If $\theta (g_1K)$ is equal to $\theta (g_2K)$ then and hence $g_1g_2^{-1}\in K$ (exactly as in previous part) which implies an equality between and .

Now, $\theta$ is a homomorphism. We need to show that $\theta (g_1K\cdot g_2K) = \theta (g_1K)\theta (g_2K)$ and that $\theta ((gK)^{-1}) = (\theta (gK))^{-1}$ . And indeed:

$\displaystyle \theta (g_1K\cdot g_2K)=\theta (g_1g_2K)=f(g_1g_2)=f(g_1)f(g_2)=\theta (g_1K)\theta (g_2K)$

$\displaystyle \theta ((gK)^{-1})=\theta (g^{-1}K)=f(g^{-1})=(f(g))^{-1}=(\theta (gK))^{-1}$

To conclude, $\theta$ is surjective. Take to be an element of ${\rm im} f$ and its pre-image. Since we have that is also the image of of $\theta(gK)$ .

"proof of first isomorphism theorem" is owned by uriw.

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Cross-references: image, surjective, injection, isomorphism, group homomorphism, homomorphism, normal subgroup

This is version 3 of proof of first isomorphism theorem, born on 2002-05-21, modified 2002-05-22.
Object id is 2922, canonical name is ProofOfFirstIsomorphismTheorem.
Accessed 461 times total.

Classification:

AMS MSC:

20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

Pending Errata and Addenda

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