1. Sep 9, 2006

mathwonk

I have finally understood algebra. It is all about understanding groups.

first of all we tackle abelian groups, and finitely generated ones at that., we completely classify them as sums of cyclic groups using their structure as Z modules where Z is the integers.

then we ask about non finitely generated abelian groups. Of course we want to use the same methods that have worked so well before, so we ask if we can view them as modules over some other rings, so that maybe they become finitely generated or cyclic over those rings.

From another point of view, it is natural to ask about the maps of an abelian group as well as the group itself, and the maps from G to G form a ring End(G) = Hom(G,G), such that G is a module over End(G). So we seek to identify a suitable subring R of End(G), such that the structure of G as an R module is simpler, but R is itself not too complicated.

For example, if Q is the rational numbers, then Q^n is not a finitely generated abelian group, but it a finitely generated Q module.

After studying abelian groups as modules over various subrings of End(G), we ask about non abelian groups G. Then we look for rings R such that G is the group of units of R and R acts on some natural modules.

presumably this leads to the analysis of group representatioins of G, and modules over the group ring of G.:!!)

Last edited: Sep 9, 2006
2. Sep 10, 2006

fourier jr

i thought it was all modules. start with groups, rings, fields, etc & then find out later on that theyre all modules? thats how i thought of it anyway.

3. Sep 10, 2006

matt grime

Actually, they're all just functors.

Given a group G, let Cat(G) be the category with one object and such that Hom(*,*) = G, then a representation is a functor from Cat(G) to Vect(k,n), the category of n-dimensional vector spaces over k.

4. Sep 10, 2006

HallsofIvy

Shouldn't this be title "Abstract Algebra made Trivial"?

5. Sep 10, 2006

mathwonk

actually my definition of "trivial", is "something i can understand". so "simple" could be harder or easier than "trivial".

Thus I guess you are right Halls, i am only claiming to have understood it, so that is by definition trivial.

6. Sep 10, 2006

mathwonk

lengthy blab of questionable interest

I do not like to follow a book in its logical but unmotivated development of topics. i.e. groups, although they have only one operation, are to me much more difficult than the ring of integers, because they may not be abelian. So I disagree with the usual ordering of topics in an algebra book (groups, rings, fields, modules), which is there more to make life easier for the author than for the learner.

so to me the beginning of the subject is with abelian objects, which one understands as Z modules, and classifies them as such in the finitely generated case.

Then I am always asking myself, why is the next topic naturally what it is?

I.e. I do not put myself in the role of a student asking what the topics were in the book, but in the mindset of a researcher who is asking, as if the topics had never been discovered before, what should come next and why?

E.g. suppose I had just written an NSF grant to classify finitely generated abelian groups, and succeeded by diagonalizing matrices over Z. what would be my next grant proposal, well to remove some of those hypotheses, and yet to use the same techniques that had worked so well.

so i begin to ask how to understand possibly infinitely generated abelian groups via their relation to other rings, and how can I generalize the techniques that worked before.

I am led to ask what tools and properties of Z were used in the proof, such as lack of zero divisors, possibility of division, euclidean algorithm, existence of gcd's and the ability to express a gcd as a linear combination of the two given elements, rational root theorem, the fact all proper primes in Z are maximal,...

I am led in this way to define euclidean rings, principal ideal domains, unique factorization domains, dedekind domains,.....

Then I naturally ask whether the classification of finitely generated abelian groups can be generalized to these cases.

I see immediately that the proof generalizes at once to abelian groups which admit a module structure which is finitely generated over a euclidean domain, and with a few stretches also any p.i.d.

then I might ask if they extend to u.f.d.'s or dedekind domains. once i see that a dedekind domain is integrally closed and everywhere locally a pid i have some hope there as well. it turns out there is a fairly simple generalization of the theorem to that case, which i predicted in this way last week, but had not known of before. (Now for the first time I notice it occurs in the last chapter of Rotman).

Asking how broadly one can use these module techniques, one notices that every abelian group G is a module over its endomorphism ring End(G), hence also over every subring of that ring, and every ring mapping to a subring, and that is all. i.e. an R module structure on an abelian group G is nothing but a ring map R--->End(G).

now I understand why the best books, such as those by jacobson and sah have sections on the endomorphisms rings of abelian groups and of modules, but lesser ones leave unclear how fundamental this is. (Compare Dummit Foote page 347.) Or compare the unenlightening but usual definition of R - module, p.337 of DF, with the one in Sah, p.124 (i.e. as a ring map R--->End(G)).

Since for a field k, k[X] is a euclidean domain, to apply our techniques, we look for abelian groups admitting a ring map from k[X] to End(G). We see that k must act on G, so G must be a k-vector space, and that after that we only need one more endomorphism which we can choose freely.

This fact is clearly laid out in DF but not to me in a way that makes it seem inevitable.

Knowing something is different from understanding it. Ask someone e.g., who thinks they know what an R module is, if they have noticed it is nothing but a ring map R--->End(G).

Thus studying modules over k[X] is the same as studying subalgebras of form k[T] inside End(G), where G is a k vector space. this motivates the usual analysis of canonical forms of matrices. I.e. G may not be cyclic over k, but it may well be cyclic over some ring k[T].

So this approach helps me see how the theory arose. I.e. the desire is to find a subring of End(G) that is small and simple enough to be understood as a ring, but large enough to render the module structure of G simpler than it was as an abelian group, or vector space.

I knew all these applications before, but did not see so clearly how they should have been obvious extensions of the more elementary theory.

This all plays into the categorical idea that we should never be content to look merely at an object G, but should always be aware of the maps on that object, such as End(G).

This helps me remove my bias against non commutative rings and non abelian groups, and raise my awareness of the importance of modules as opposed to just rings in algebraic geometry. It also helps me see how the ideas for studying non abelian groups could arise from a natural extension of those that succeeded for abelian groups.

For example I have never studied or known anything about group representations. Now I wonder how one could ever NOT be led to look at them.

Of course many others know all of these things much better than I, but for me to see how things could have been predicted and discovered helps me to see how to simplify them in my teaching, and perhaps understand them, and maybe even find something new.

I want to show my classes how each discovery leads to the next, and to practice guessing what might come after, and to make conjectures and propose ways to solve them. So I never stop pondering any subject until it seems to me natural and almost inevitable.

And being naturally slow, this goes on for years in my case. I have just understood how integration theory should have been immediately extended from Riemanns case to lebesgues. i.e. what is often presenetd as a flaw in riemanns theory, the fact that limits of some functions are not integrable in his sense, is in fact the key to its solution. i.e. as long as the integrals converge, just define those limit functions to be integrable with integral equal to the limit of those integrals.

from this point of view one would think lebesgue integration would have been discovered 5 minutes after riemanns version. but it takes time to see things clearly. It has taken me in that instance, 45 years.

Of course everyone "undersatands thigns in his/her own way, and this may not help others. It is also true thjat I am constantly forgetting things I knew before as a youngster, and rediscovering them as I age. But I posted this epiphany, since it unifies and organizes for me much of algebra, as well as motivates me to learn more of topics I myself have ignored heretofore.

7. Sep 10, 2006

mathwonk

so I propose the following as a more natural ordering of algebra topics than the usual one:

Abelian theory
integers,
cyclic groups,
finitely generated abelian groups
rings, eucldiean, pid, ufd, integrally closed, krull dimension, noetherian, dedekind rings
modules, especially fin gen over euclidean and pids,
review of polynomials, gauss lemma for ufds, hilbert basis thm for noetherian rings, irreducibility criteria (Eisenstein, Dumas [indecomposable newton polygon implies irreducible polynomial]).
application to rational and jordan canonical forms of matrices, characteristic and minimal polynomials, cayley hamilton.

Non abelian theory
general groups, actions, sylow theorems, direct, semi direct, free products, solvable groups jordan - holder, symmetric and dihedral groups (example of semi direct product)
fields+ constructibility, galois groups, algebraic extensions, algebraic closures, simple extensions
fund thm of galois theory
applications to calculation of galois groups, cyclotomic fields, dirchlets thm for r = 1, existence of extensions of Q with Galois group isomorphic to any finite abelian group.
effect of specialization on galois groups, monodromy groups in topology and algebraic geometry

group representations

Last edited: Sep 10, 2006
8. Sep 10, 2006