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Mathematics LibreTexts

5.2: Isomorphism Theorem

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!

Theorem 5.2.0: Isomorphism Theorem

Let \(\rho: G\rightarrow H\) be a homomorphism with kernel \(K\) and image \(I\). Then \(I\sim G/\mathord K\).

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

\[\frac{ \frac{n}{m} }{ \frac{p}{m} }=\frac{n}{p}.\]

A Prelude to Categories

Some of the important objects in this chapter are similar to things you've probably encountered in linear algebra: kernel, image, and product. This isn't an accident: vector spaces are also algebraic structures. For almost any algebraic structure, we begin with the actual structure, and then consider special functions between the objects with that kind of structure. In groups, these are homomorphisms, and in linear algebra the special maps are linear functions.

In most of these contexts, the idea of kernel and image still make sense, and one can form products and quotients, and even derive a version of the isomorphism theorems.

Contributors

  • Tom Denton (Fields Institute/York University in Toronto)