Group Theory: What is symmetry? Why do I care?

When I started talking about group theory on monday, I said that ultimately, groups are a way of capturing symmetry: multiplication over the real numbers (if you omit zero) has an amazing symmetry about it, and group theory abstracts away the details of multiplication to home in on that symmetry concept.

The problem that a lot of non-mathematicians have when you talk about Groups is that it's kind of hard to grasp what that symmetry is, what it means, and why they should care. The list of applications that commenters came up with is impressive, and is a pretty compelling argument to me for why this stuff is interesting; but it's important to grasp the intuition behind the symmetry of groups before you can really get group theory.

So - what does symmetry mean?

Let's start by looking at multiplication. Think about multiplication with non-zero real numbers. Forget about the fact that we can add numbers - just focus on multiplication.

If I were to have you take the real numbers, and come up with some funny way of writing them, so that I don't know which numbers are which, and then you ask me to tell you which number is which, only by using multiplication and testing equality, would I be able to do it?

Well, I could certainly tell which number was 1: it's the only number where anytime I pick some other number and multiply it by one, I get the other number.

What else could I work out? Given enough time, could I figure out which number was two? Well, two is interesting in its way, because if I could figure out which numbers were integers, then I could recognize two because every other integer is a multiple of it - so if I tested enough numbers, I could at least make a pretty good guess which number was two - that is, if I could figure out which numbers were integers. But I can't do that - there's no way for me to tell the difference between an integer and any other real number.

Now, here's where it gets a bit subtle: Suppose I were to give you the integers from 1 to 1000, and their reciprocals. So you've got a bunch of numbers which you know are 1, 2, 1/2, 3, 1/3, 4, 1/4, 5, 1/5, 6, 1/6, etc, but you don't know which is which. Now, could you tell me which of them was the number 2?

The answer is no. You could narrow it down to being one of *two* numbers, and you could tell that those numbers were each others inverses - but you couldn't tell which was which. You can't tell apart 1/2 and 2 - because the group is symmetric around 1.

Think about it: how could you tell the difference between "x*y=z" where x=1/2, y=1/3, and z=1/6, and "x*y=z" where x=2, y=3, and z=6?

Or to be visual for a moment: look at this graph of the equation y=1/x in the first quadrant (that is, the part that we normally draw on the upper right, where x and y are both positive).



Without the axes being labeled, which is X and which is Y?

It doesn't matter - you can call either one X and the other one Y, and that's still the right graph. What that means goes beyond that one graph: any statement that you can make about numbers using only multiplication is also true if you replace every value with its reciprocal.

For a more concrete and practical example of this kind of symmetry: the symmetry property works in electronics. You can analyze a circuit two different ways: you can look at any circuit, work out what it does, what current, voltage, amperage, resistance, etc., exists at what point in the circuit; and you can do that under the assumption that the positive charges are moving, or that the negative charges are moving - and the answers will come out exactly the same. In fact, for years (and maybe even still now), electrical engineers were trained to do the computations with the assumption that it was the positive charge that moved! (It's a historical error".)

Symmetry is a fascinating property, which allows you to recognize similarities, and discover ways in which properties can't change even though they've been twisted around somehow.

Remember that great quote from Newman? "The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing." Group theory lets you see the similarities between different things, or the ways in which things can't be different, by expressing the fundamental symmetries.