Quotients and Homomorphisms for Beginners: Part 5, The Isomorphism Theorems and Their Significance

Preface. This post has been in my drafts for over a month. Eventually, I will finish this eight part series, but expect other posts to be interspersed. In my drafts are also an introduction to Galois theory, a post on the unreasonable effectiveness of mathematics in the natural sciences (this relates to an expository paper I recently wrote), rationalism, Sylow’s theorems, and the Jordan-Holder program. I also hope to write about my research a bit. We shall see.

In this post we will explore three important theorems regarding isomorphisms and quotient groups. We will introduce these using fairly familiar language and try to gain some intuition for their significance. In a later post we shall view these crucial theorems from a somewhat more abstract and modern point of view, namely that of categories, which one might call the “universal perspective.” For most introductory undergraduate courses, these theorems, sometimes along with the Sylow theorems, which I may write about eventually, mark the grand culmination of all the group theory for the course. Some courses do not even include the second and third theorems; admittedly, these are frequently considered less fundamental than the first.

Before moving on I should note that different texts use different labeling and sometimes do not even use these names (Lang’s Algebra, for instance), or use “Homomorphism” instead and present a more general set of theorems.

Note that \simeq and \cong are equivalent for our purposes. Recently, the author has begun to prefer the latter for isomorphisms in all scenarios, whereas previously that was only the case when \simeq denoted some other equivalence as well.

First Isomorphism Theorem. Let f: G \to G' be a group homomorphism, then \mathrm{Im}(f)  \cong  G / \mathrm{Ker}(f) given by \psi: gH \to f(g).

Proof. 

We will prove the special case where f is surjective. Modifying the proof to the general case is not difficult and is left as an exercise.

Let H = \mathrm{Ker}(f). There is a canonical homomorphism \phi: G/H \to G'. This is well defined, for the image is independent of coset representative. We now prove this map is indeed a homomorphism. In particular, we show \phi(gH \cdot hH) = \phi(gH) \star \phi(hH). By direct computation,

\phi(gH \cdot hH) = \phi( (gh)H ) = f(gh) = f(g) f(h) = \phi(gH) \phi(hH)

where f(gh)=f(g)f(h) is because f is assumed a homomorphism.

Clearly, \phi is onto; showing this explicitly is hardly worth the space at this point. We now need only prove \phi is injective, which will show it is a bijection. We use a well-known fact that a group homomorphism is with trivial kernel is injective. (Note that H is the identity in G/H.) Suppose x \in G/H, then x is of the form $latex gH$, so f(g)=\phi(x). This means g \in H and x = H, meaning the kernel of \phi is trivial. A well-known theorem states that a homomorphism with trivial kernel is injective.

\blacksquare

I do not want this to be unmotivated or uninteresting, so I will explain why we would even think of this theorem or care about it as well what it means in more plain English.

Whenever we are considering mappings, f_i:G_i \to H_i, we want to consider composition maps. It is natural to see if we can’t go from G_1 \mapsto H_1 \mapsto H_2, for example. There are many legitimate applications for this sort of thing which you should be comfortable with from elementary algebra. This makes it more obvious why we might want to examine something like the first isomorphism theorem.

The first isomorphism theorem just says that if you want to go from G to H, you can always go through G/\mathrm{ker} (f) in a very natural way. This is a beautiful theorem. It has analogs for most other algebraic structures we care about (e.g., vector spaces), and it is an example of something important categorically, which we will discuss in the next (and final) post.

Second Isomorphism Theorem. Let H < G and let N be normal in G. Then, H \cap N is normal in H and H/H \cap N \cong HN/N. 

Proof.  Omitted (Exercise)

Third Isomorphism Theorem. Let K \subset H be two normal subgroups of G. Then, G/H \cong (G/K)/(H/K).

Proof. Omitted (Exercise)

At first, I thought I would prove both of these theorems, but I have decided not to for two reasons. First, the proofs are pretty easy from the first isomorphism theorem and make for great practice. Second, these are neither frequently useful not frequently considered fundamental or deep. The third isomorphism theorem is also very easy to remember (“cancel” the K), and the proof is easy enough to recreate at will.

 

In the sixth and final post of this series, we will reset the first isomorphism theorem in more modern language: that of category theory. We will see a useful property of the group-theoretic quotient, see how the same idea applies in other areas, and see where the alternative name “factor group” comes from.

 

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