After an insanely long hiatus during which I went to college and learned some actual math, I’ve decided to start posting here again about some higher level concepts, like algebra and number theory, and maybe some other fun stuff I decide to mess around with. Since I just finished a course on group theory, and algebra is a lot of fun, I figure I’ll start with some basic ideas about that. I won’t be assuming terribly much of the reader, a modest mathematical background should be enough to follow the posts.
What is a group?
Before I get into the formal definition, let’s talk about symmetry. There are all sorts of symmetries, and pretty much all of them are really nice when it comes to math. Symmetries make your integrals collapse, and your shapes easier to work with. So group theory tries to capture the notion of “symmetry” in some abstract sense. For an example, let’s take a look at a square. Picture your favorite square. What symmetries does it have? Well, first of all, we should figure out what we mean by a symmetry of a square (or polygon). We’ll say a motion or movement is a symmetry of the square if after the motion is completed, the square is in the position it was originally (but perhaps with the corners in different locations). Also, we should require that the motions be rigid: so no cutting the square and moving only the cut piece. So, what are these symmetries then? First off, we can keep the square exactly where it is. This is a bit boring, but it qualifies. We can also rotate the square by 90 degrees counterclockwise about its center, or by 180 degrees, or by 270 degrees. These also qualify, as you can see if you move your favorite square about in your head. We can also draw two imaginary lines connecting opposite corners on the square and two more imaginary lines connecting the midpoints of opposite edges, and flip the square about any one of these lines. After some thought, it seems like we’ve found them all, and we have. These symmetries satisfy some nice properties. First, if we perform any number of them on the square, we get another symmetry! This isn’t hard to see, especially from our definition of a symmetry. Second, we can always undo a symmetry with another symmetry to put the square back into its original position. Keeping these in mind, we’ll now define a group (and see that the symmetries of the square are indeed such an object).
Definition: A group is a set equipped with a binary operation satisfying the following properties:
- * is associative: For all ,
- There exists an identity element: There exists such that for all .
- There exist inverses: For any , there exists such that .
And that, folks, is the formal definition of a group! Let’s look at some examples:
EXAMPLE: The Trivial Group
The simplest group we can imagine is the group that has only one element, the identity. So here, , and the operation is given by (as if we had a choice). This is a group, but it’s pretty much as boring as a group can be, so let’s move on to another example.
EXAMPLE: The Integers,
Henceforth, I’ll be writing to mean the integers; that is, . This set forms a group under addition in the usual sense: we already know addition is associative, so point one is satisfied. We know that for any integer , so we have an identity, . We also have inverses: , and thus is indeed a group! Notice that we have for any , so the operation is not only associative, but commutative. Speaking of commutativity, one might wonder “is a group under multiplication? That’s also associative and commutative.” The answer would be no: while we do have associativity and an identity, we don’t have inverses for most elements: in fact, there are only two elements in that have a multiplicative inverse in .
EXAMPLE: Symmetries of a Square,
Time to come full circle. Or full square. But enough silliness: we’ll call the set of symmetries of a square “the dihedral group of order 8″ and denote it . Now, we want to see that this set is indeed a group! First of all, let’s give judicious names to a few chosen motions: the identity, doing nothing, will of course be , as usual. Besides that, we’ll let be the rotation of the square by 90 degrees counterclockwise, and we’ll let be the flip about the vertical line passing through the center of the square. You might ask “what about the rest of the motions that we talked about above?” to which I would respond, “they can actually all be represented by repeated applications of and .” To get the other rotations, we simply rotate by 90 degrees until we get the amount of rotating we want: for example, the rotation by 180 degrees counterclockwise would be given by , and we will ignore the and simply write it as if we’re multiplying: . Great! So now we have all the rotations: (the trivial rotation), , and . These are half of the motions we talked about above: and since we have another element that we haven’t really looked at yet, it would be a pretty good guess to say that if we apply any of these motions and then apply , we’ll get the other four. So, let’s try it out. (ACHTUNG! Read the multiplication of elements in this group from right to left, that is, if we have , where , , and are all symmetries of a square, we move the square according to first, then , then ). It wouldn’t be unwise to actually get a paper square with the corners labeled A, B, C, and D or whatever your favorite four letters are to see all the motions happening here. gives us the flip about the vertical line, is first a rotation by 90 degrees counterclockwise, and then a flip about the vertical line through the center (not what used to be the vertical line, what is now the “new” vertical line). With a little thought, we see that this is the same as flipping the square about the diagonal going from lower left to upper right. Similarly, is the flip across the horizontal line through the center of the square, and is the flip about the diagonal going from upper left to lower right. Geometrically, we see that we have inverses: flips are obviously their own inverses, and the inverse of a rotation is given by going the rest of the way around: . Associativity is a little more annoying to prove, as is showing that we don’t get any more elements upon further multiplication, but all of that can be worked out with a bit of work in the privacy of one’s own bedroom. Ignoring those few details, we can see that is indeed a group!
One last thing: while the operation of the trivial group and the operation of addition in were both commutative, group operations are not commutative in general: to see this, simply look at and see what happens to the square in the case of the motion and in the case of the motion . Here’s a tip: they’re not the same!