Musings, and groups lacunæ
A warning before I begin: this post will be rather more stream-of-consciousness than most.
Firstly, I want to welcome everyone who’s reading my scribblings. I didn’t really expect more than a handful of people to look at me regularly, but in the last few days this thing has really taken off. Credit must be given where due, though: if I have received so many hits it is by being linked to by giants.
After toying with some of the site statistics tools I’ve seen a number of the links in, and what (if anything) is being said there. I must say it’s been generally positive. One commentary, though, sticks in my mind. In a comment here Anatoly Vorobey expresses some doubts about my project of describing group theory, among other things, to a generally interested lay audience. I’ll go into his misgivings in general and his specific points after the jump.
I want to reiterate that I am not trying to give a course here. I don’t expect someone to just read my notes and be able to pass a final for abstract algebra 1. What I’m trying to do is give a flavor of the subject while telling as few outright lies as possible. Think of it as living at a similar level of discourse to Discovery Science Channel programs, and aimed at the same sort of generally interested audience.
One point Mr. Vorobey makes is the difficulty of imagining what other people might have difficulty with in the material. Of course, this is true. It’s why I have always hated when professors teach out of their own texts, since that leaves no alternate viewpoint from which a student can triangulate on the material itself. I think that this is alleviated here by the fact that a weblog, while not a face-to-face dialogue, is still a conversation of sorts. Things I may have overlooked can be brought up, as Mr. Vorobey did, and I can return to them to clarify points my original descriptions left muddled.
Another point is that abstract algebra is usually taught only after certain other courses are passed, not so much for the groundwork laid therein, but for the “mathematical sophistication” (he uses the term “maturity”) gained. I understand this concept as well. We use it as the reason for knocking incoming freshmen whose calculus BC courses were weak on series back down into calculus 2 rather than placing them into multivariable calculus straight away. We don’t use series as such, but the maturity that comes with exposure is really needed to master the next course.
However, as I’ve said, I’m not trying to teach a course here. I’m trying to give a thumbnail sketch of a vast and complicated subject and to introduce some of its glossary. This is also why I started with groups rather than go all the way down to set theoretic foundations and build up like Bourbaki would have done. The current short goal is to get to quandles so I can explain the terms in my first rant. Groups are a nice place to get on, and are general enough that I’m sure they’ll come up over and over and over again, so having these notes down early will come in handy.
I also find it nettlesome — though this may be just my reading — that the doubts seem to assume that I don’t know the difficulties this project entails. I may not have made it clear, but I am an academic, not an industrial mathematician (knock wood!). Fully half of the academic mission is education, and I take that half as seriously as the research itself. Not to blow my own horn too much, but I’m a damn fine teacher. Not only do I communicate material, but judging from the handful of students each semester who decide to add the mathematics major (or add a supporting sequence in another major) I manage to communicate something of the beauty and intrinsic interest of the subject.
Along those lines, I’ve used this conversational style before among my friends with good results. No, they don’t pick up enough to do serious work, but they get a rough idea of a subject’s outlines. It’s that sort of conversation I’m trying to replicate here. The back-and-forth doesn’t come so quickly, but the notes are there, written out to read over again, and the slower pace will hopefully give the freshly leavened ideas time to rise. Have I been blessed with extraordinarily perceptive friends? I doubt it. No slight to them, but I think anyone who’s taking the time to have a conversation at a café or to read someone’s weblog is already somewhat interested, and their interest engenders perception enough for what I’m trying to communicate.
Speaking of friends of mine, one in particular I want to mention, since I usually have him in mind when I’m writing this sort of thing. I don’t want to invade his privacy, so I’ll just call him “Naked Dan”. I met Naked Dan back in my first year of college, living in the same dorm. He was very sharp, taking multivariable calculus his first semester despite having no need for the extra math credit in his film studies major. I remember waking up one morning to an instant message asking if I had any classes that day I couldn’t blow off. I told him I had none, and fifteen minutes later we were in my car bound for a university an hour or so away because Naked Dan had heard Roger Penrose was going to be there. I lost contact with Naked Dan for a long time, but coincidentally ran into him just over this winter break. That very night I was tossing out rough sketches of higher categories and he was throwing back metaphors drawn from film editing.
Naked Dan had no need to learn multivariable calculus or to hear the author of The Emperor’s New Mind. He just wanted to know because he found it interesting to know. It’s the same sort of person who watches Nova or the Discovery Channel, and the same sort of person I’m aiming these notes at. When I write a sketch of group theory I think of how I’d explain it to Naked Dan.
It’s an experiment that I know may or may not work at all. On the other hand, I didn’t expect anyone much at all to come here, and I had almost 1,500 hits yesterday. Sometimes things work out better than you expect.
Now, to the specific points raised.
- That a set is a collection of entities treated as having no structure, and what that means
- That an algebraic structure is generally a set + some operations defined axiomatically, and how that connects to familiar objects not previously so considered
- Why sticking a and b together is called “composition”, what does it mean that it’s another element of the group (it’s just a and b written together!).
- What’s ‘associative’ and why it’s important.
- What does it mean to have an operation that’s not given explicitly, but only constrained axiomatically, and how does one do things with it if you don’t know what it means.
- ugh, rational numbers… I remembered once what those are exactly… oh you mean fractions!
- How the hell did you get from permutations of letters to some mysterious “objects”?
First off, I think that in a thumbnail sketch my description of a set as a collection of things is sufficient. Plenty of math majors get through without thinking much more about sets than that.
To give a group, the first thing I have to do is say what the members of the group are. Then I have to say how to stick two of them together — “compose” them — to get another. If I start with a and b, we write their composition as ab. Then this way of sticking things together has to obey certain rules.
It’s important to note that the way we write group operations is just a description. If the group is the integers with addition I can take 2 and 3, compose them to get 2+3, and see that it equals 5. I’m not just writing out abstract symbols and setting them next to each other. The numerals 2 and 3 denote relatively well-understood concepts, and addition takes those two numbers and puts them together into a new number: 5. Before I do that calculation, though, I can write “2+3″ to mean “whatever the result of composing 2 and 3 by addition will be”.
We can add pretty much right away, so it might be confusing why focus on the “unevaluated” addition. Well what if we don’t know one of the values? From high school algebra we remember using variables. We can denote “whatever the unknown value is, three more than that” by “x+3″. Now we can’t evaluate the addition because we don’t know what x is. We can still talk about the result by using this notation, though. By using more variables we can leave more and more unspecified. We can say, “take two unknowns and compose them by addition” by writing “x+y“.
Abstract algebra is just pushing this sort of thing one step further. What if I don’t just want to leave the values unspecified, but I want to even leave the operation unspecified? We usually just write group elements next to each other, but for now I’ll use a new symbol with (hopefully) no preexisting meaning to denote an “unknown operation”. We say “take unknown elements a and b from some group and compose them with the group’s composition” by writing “a◊b“. This is great because now we can write down expressions about elements of groups without specifying the group!
Think about it like this: when we left a number unspecified in high school algebra we got a template for arithmetic expressions. We write “x+3″, which calls to mind a list of expressions, one for each value of x. We have “1+3″, “2+3″, “3+3″, and so on, and we solve algebra equations by asking what values of x will lead to true statements in arithmetic?
Now I can leave the operation unspecified and get a template for algebraic expressions. When I write “a◊b“, I can get one algebraic expression for each choice of group. If I pick the integers with addition I get “a+b“, where a and b are variables taking integer values. If I pick positive fractions with multiplication I get “a*b“, where a and b are variables taking positive fractional values.
High school algebra lets us make statements abstracted over all choices of a number to substitute for x. Group theory lets us make statements abstracted over all choices of a set with a notion of composition obeying the group laws.
As to those laws, associativity is the most important. What it means is that if I have a long list of members of the group to compose together, all that matters is the ordered list. No matter how I parenthesize the list to say, “compose those two first, then those, then those…” I’ll get the same answer.
- All these products are the same if the composition is associative
and we can take any two and show that they’re equal by using the associative law for the group composition.
This post is already getting huge, so I’ll leave alone the points about permutations and rational numbers (though I’ll try to remind that “rational numbers = fractions” in the future) until such time as I make posts more specifically about them.
Again, thanks for all the comments, and feel free to ask for more details if you want them.