Sunday, June 05, 2005

Math Post the Third: Groups and Axioms

At last, I'm able to tackle my actual topic of interest: group theory. Groups are one of the simplest (not easiest) algebraic structures around, but in spite of the ease of giving an axiomatic definition, I'm going to start with examples before we start; it'll make things much easier.

The first example of practically everything in algebra is the same: the integers, those guys I've been talking about for a week, positive and negative whole numbers. I'll ignore multiplication for a moment and focus on the fact that you can add two integers together. And that, my friends, is algebra.

The canonical second example of a group is still just adding numbers, but adding them on a clock face. This is called "integers modulo twelve," or mod twelve for short, and the idea is just that twelve is the same as zero and otherwise we add normally (so 3+11=2*, and so on). Obviously twelve can be replaced by whatever number I want here; the concept will remain the same.

Less additively, imagine drawing a square on a table and putting down a square piece of paper with labelled corners (the same front and back) that fits directly on top of it. If you wanted to, you could pick up the paper and put it down with the corners in a different position. In fact, there's something like a "multiplication" that you can do on these motions: make a new motion by taking two old motions, doing one, then doing the other. This just physically realizes the symmetries of the square (rotate, reflect, and so on). In fact, symmetries of things (broadly construed) provide a wonderful source of groups (experts may here recall the old saw about the catagorical definition of group**, which I'll probably explain at some point). Another way to see this is to think about a Rubik's cube -- beyond just being a cube and having those symmetries, there are others you can get from the operation of twisting a face, since twisting a face keeps the cube a cube and just moves the colors around.

More abstruse examples abound. For anyone who recalls matrices, the set of n by n matrices with nonzero determinant is a group with matrix multiplication; so is the set of matrices with determinant equal to one (both of these are actually symmetry groups in disguise). The permutations of a set also constitute a group, called of course a permutation group. If you write down words composed of the letters a, b, a^-1, and b^-1, stipulating that a letter and its inverse cancel if they appear next to each other, you get a group by concatenating (and then reducing) words; just write one, then the other. And this is only the beginning; stranger examples will appear soon enough.

As you might expect, a definition that actually captures all of these examples is likely to be a little out-there. It'll involve many symbols when I give it, but first I'd like to say a word or two about axioms. Obviously there is a lot of theory you can give about all of these examples individually, but they do all involve some common features, and capturing them in and then working with an axiomatic defintion permits you to apply old theory in new situations (for the computer scientists in the audience, think of a well-documented code library). While it'll be a while before examples of this come up (though they will, fear not), it'll be handy to start collecting axiomatic structures now.

So, a group is a set, say G, with a multiplication, written here by juxtaposition of elements, which has:

  • Associativity: given any three elements g,h, and k in G, g(hk)=(gh)k.
  • A unit element: there is an element e in G such that for any g in G, eg=ge=g.
  • Inverses: for any g in G, there is an element g^-1 in G such that g(g^-1)=(g^-1)g=e.

Associativity just means that you don't have to worry about where you put the parentheses in a long list of elements being multiplied together; this is usually either pretty hard or pretty easy to prove (there are two or three examples above which require individual lengthy proofs of this; the others can be handled easily together if you're slightly clever). The other two are easier to understand; in fact, I'm going to leave understanding what they mean in all the above examples as an exercise. You can answer in comments for the harder ones (though please don't post a first response if you already know this stuff) the following question: for each example of a group listed above, what is the unit element, what is an inverse element, and how would you get one? It shouldn't take more than a few minutes of thought, but it's a good place to start getting a feel for how to think about this stuff.

Next time: subgroups, cosets, quotients, and one of my favorite cute little theorems ever.

*Technically, that equals sign should have three bars in it, signifying that it means equality modulo something, not normal integer equality. It's not strictly required, since the addition is really taking place within the mod twelve group structure anyhow, but it's sometimes nice to have.

**"A group is a category with only one object in which every morphism is an isomorphism."


Blogger Vito said...

You left out closure. That's really important if you look at things like addition on 3Z or multiplication on GL2*.

2:09 PM  
Blogger Dennis said...

Closure is actually something I'll talk about next time that's only implicitly part of the actual definition of a group -- I mentioned for each of the examples how multiplying two somethings gives you back another something, and in fact I was wrestling with defining a multiplication as a function GxG->G, which appears in any completely formal definition, but I didn't want to introduce new notation before I absolutely needed it. You'll note that your examples of closure are for subgroups, not groups per se.

3:49 PM  

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