Monthly Archives: April 2019

The Fundamental Theorem of Finite Abelian Groups

It’s been a while since I’ve posted, mainly because I’ve been trying to work through a single theorem: the fundamental theorem of finite abelian groups. I read through the proof as presented in Gallian, and then, because it was so concise and dense, I decided to try to work through the presentation of the proof in Carter’s Visual Group Theory. I had hoped that this second exposition of the proof would be clearer, but in reality, Carter presents the proof as a series of exercises which guide the reader through creating the proof. Unfortunately, while I made it through most of these exercises, the last one was rather opaque, and, although I think I’d have been able to make it through it, I decided that I wanted to move on to the next topic (Sylow theorems) so that I’d have time to spend on Sylow before I have to wrap up the independent study by the end of spring exams.

The statement of the fundamental theorem of finite abelian groups is as follows: given an abelian group GG can be expressed uniquely as the direct product of cyclic groups of prime-power order. In a sense, one can view this theorem as a version of the fundamental theorem of arithmetic (which states that every natural number has a unique decomposition as the product of powers of primes) for abelian groups.

I won’t include every exercise from Visual Group Theory that I worked on since many of them are rather verbose and fairly straightforward. I will include one, however, that is more interesting and that I found more difficult.

This exercise ultimately concludes that, even if the image is not cyclic, we can apply the same homomorphism to the image of the original homomorphism, and continue applying this homomorphism to the image of the previous application of the homomorphism until we finally get the trivial group. The group in this chain of images that comes right before the trivial group must be a cyclic group of order p, and so then part (a) of this exercise then guarantees that all of the groups in the chain are cyclic (by applying the result once for each homomorphism in the chain). This eventually shows that G itself must be cyclic. Thus, part (a) is the heart of the exercise—the rest of the parts just deal with defining the chain of homomorphisms and groups that eventually allows us to apply the result of part (a).

The overall chain of reasoning of the proof is as follows: First, we show that any abelian group can be decomposed into the direct product of (not necessarily cyclic) abelian groups whose orders are powers of primes. We then show that any abelian group whose order is a power of a prime into the direct product of cyclic groups of prime-power order. The exercise I described above establishes a crucial fact that is instrumental in the second part of the proof.

 

Homomorphisms and the Isomorphism Theorems

In our most recent meeting, we discussed a pretty hefty chapter that covers homomorphisms and the first isomorphism theorem (along with some other results), which is one of the more important theorems from group theory. Then we worked on two exercises from the chapter that asked us to use the first isomorphism theorem to prove the second and third isomorphism theorems.

To begin, I’ll just include a statement of the first isomorphism theorem:

Now, we were tasked with the following two exercises:

Our line of reasoning for the second isomorphism theorem was as follows:

And for the third:

 

Cosets and Lagrange’s Theorem

I’m just now getting around to writing this post, although I worked through these exercises last week. These exercises come from the chapter on cosets and Lagrange’s theorem.

My solution is as follows (included as an image so that I could more easily include well-formatted math):

The exercise that we couldn’t solve is this one:

Although it looks fairly straightforward, it really isn’t, at least using any methods that I know so far. We struggled to figure out what facts or theorems to take advantage of in order to prove the desired result. (I even tried looking up solutions online, and all of them rely on results that I haven’t proven yet.) This highlights how often times the most difficult part of solving an exercise isn’t executing a proof once you have a plan of attack, but coming up with the plan of attack in the first place.