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# 5.3: Isomorphism Theorem

• • Tom Denton
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We've observed a few cases now where we: 1. Define a homomorphism $$\rho: G\rightarrow H$$, and then 2. Notice that $$G/\mathord K \sim H$$, where $$K$$ is the kernel of $$\rho$$. This isn't an accident!

The proof is just to build a correspondence between the cosets of the kernel $$gK$$ and elements of the image $$I$$. Indeed, in any coset $$gK$$ all elements map to the same element of the image. $$\rho(gk)=\rho(g)\rho(k)=\rho(g)1=\rho(g)$$ for any $$k\in K$$.

This suggests a homomorphism from the set of cosets to the image: set $$\phi(gK)=\rho(g)$$. This is a homomorphism, since $$\phi(ghK)=\rho(gh)=\rho(g)\rho(h)=\phi(gK)\phi(hK)$$.

The map $$\phi$$ is also one-to-one: if $$\phi(gK)=\phi(hK)$$, we have $$\rho(g)=\rho(h)$$, so that $$1=\rho(g^{-1}h)$$, meaning $$g^{-1}h\in K$$. Then $$h=g(g^{-1}h)\in gK$$, which tells us that $$gK=hK$$, since cosets are either equal or disjoint.

The map $$\phi$$ is onto, since any element in the image may be written as $$\rho(g)$$ for some $$g$$, which is also the image of $$gK$$ under $$\phi$$. Therefore, the map $$\phi$$ is an isomorphism.

TODO: Pictures!

This theorem is often called the "First Isomorphism Theorem." There are three isomorphism theorems, all of which are about relationships between quotient groups. The third isomorphism theorem has a particularly nice statement: $$(G/\mathord N)/\mathord (H/\mathord N) \sim G/\mathord H$$, which one can relate to the the numerical identity

$\frac{ \frac{n}{m} }{ \frac{p}{m} }=\frac{n}{p}.$