Good Math/Bad Math

Sunday, April 16, 2006

Fun Stuff: Group Theory (Corrected)

One of the things that you can do in math that's really interesting is to play games with abstracting things. What I mean by that is that you can look at things that are part of everyday math, and strip them down to their bare essentials, and see what happens. Some fascinating theories have come out of this, and they're not just theoretical math - they have concrete, practical applications: mathematical structure is part of the world, and playing with abstraction is one way of seeing that.

In Algebra, one of the abstraction games that people like to play is called Group Theory, and I'm going to write a bunch of posts talking about it, because there's so many fascinating things hiding inside of it. Today, I'm just going to give you a very brief introduction; later in the week, I'll build on that.

Group theory comes from looking at the idea of multiplication, and asking: what are the real essential properties of multiplication? If I get rid of numbers, so that I'm just talking about multiplication as a bare operation, what are its essential properties?

What you wind up with is something called a group. A group Gis a set of values, along with a binary "multiplication" operation, which meets a set of properties, all of which should be familiar from working with multiplication on real numbers:
Closure
Closure says that for any two values a and b are in G, then their product "a * b" must be in G. In other words, there is no way that you can multiply any two values in G, and end up with a value that isn't in G.

Associativity
If a, b, and c are in G, then (a*b)*c = a*(b*c).

Identity
There is an element, "1" in G, such that for all a in G, a * 1 = a, and 1 * a = a.

Inverse
G contains an inverse (reciprocal) for each of its members. The inverse of an element "a" in G is written "a^-1"; and for all a, a * a^-1 = 1 and a^-1 * a = 1.
What this comes down to is that a group is a structured set of values that has an interesting kind of symmetry expressed in its multiplication operation.

An interesting to note is that group theory is an abstraction of a set with a multiplication operation; but the Real numbers with the normal real multiplication operation are not a group (because 0 breaks the identity and inverse properties); on the other hand, real numbers using the addition operation are a group!

A very famous mathematician, James Newman, who was particularly famous for writing books that explained mathematics to laymen, describe group theory in a wonderful way:

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.

What kinds of things can be expressed as groups?
  1. Music: There's a way of looking at music theory using groups: natural "operations" that occur in music and chords, like inversion, transposition, etc., all exhibit group symmetries.
  2. Chemistry: you can determine the polarity of a molecule by using group theory to identify the symmetries in the structure of the molecule.
  3. Physics: group theory is apparently used in special relativity, although I don't pretend to understand how.
(Update note: in the original version of this, I forgot that Group Theory does not assume that multiplication is commutative; I've corrected it since the error was pointed out in the comments.)

13 Comments:

  • Group Theory is also very important in particle physics.

    By Anonymous Stephen, at 10:07 PM  

  • Particle physics, nuclear physics, statistical mechanics, condensed matter, you name it...

    Though we more often refer to Lie algebras, where not only do all of the elements form a group, but also the so-called "generators" of the Lie algebra form their own group.

    By Blogger A, at 11:23 PM  

  • What I find interesting is that you can do same things with addition:

    Closure
    Associativity: a+(b+c) = (a+b)+c
    Neutral element: a+0 = a
    Inverse: a + -a = 0

    By Blogger oku, at 2:04 AM  

  • Take two observers, Joe and Moe, who are watching some mechanical system -- say a star system with planets and moons spinning around -- and testing if the motions they see follow Newton's laws. To describe motion mathematically, they use coordinates, which might be the Cartesian x-y-z system: locate an object at a particular time by specifying how far it is above some plane, how far in front and how far to the left (with negative numbers to describe positions in the opposite directions).

    Joe and Moe do not have to use the same coordinate systems to agree upon the laws of motion. If Moe is standing 10 million kilometers to Joe's left, say, then Moe's coordinates for each planet will be offset by that amount, but Moe will still see the motion as obeying Newton's laws. Likewise if Moe is turned with respect to Joe: a point which lies on Joe's x-axis will to Moe be some mixture of x and y, but the physical laws will be the same.

    In Newtonian mechanics, it is even true that Joe and Moe can be in motion relative to each other. Joe believes he is standing still and sees Moe moving past with a constant speed in a straight line; Moe believes he is standing still and sees Joe going past in a straight line. Uniform motion in a straight line is indistinguishable from a state of rest, or put another way, there is no "gold standard" absolute state of rest from which all speeds should be measured. (Think of riding in an airplane on a smooth flight: when can you tell you're not sitting on the ground without looking outside?)

    Einstein's revision of Newtonian mechanics says that the same principle holds true for all physical laws. Even if Joe and Moe try to determine who is "really" moving by measuring the speed of a light beam, they will measure the same speed, 300,000 km/s. One would expect that if Joe measures light going at 300,000 km/s and sees Moe flying past at 100,000 km/s, Moe will measure the light traveling at 200,000 km/s, but such is not the case.

    This all comes down to group theory because we can consider these transformations -- translation by a fixed distance, rotation by an angle, movement at a uniform velocity -- as symmetry operations on the physical laws. Just like a vase or a starfish is symmetrical if we can rotate it around and it looks the same as it did before, a physical law is symmetrical if we can change the coordinates we use and it still takes the same form.

    The magic buzzword to give a search engine is "Lorentz group".

    By Blogger Blake Stacey, at 7:22 AM  

  • The music theory applications center around mod-12 arithmetic. There are 12 notes in the chromatic scale & rather than refer to them as C, C# etc. they can be numbered 0 through 11. The two basic operations are then:

    Transposition by n semitones: x + n (mod 12)

    Inversion: 12 - X (mod 12)

    All possible chords with n notes can be classified by sets where the notes are placed in the form that minimizes through the operations of transposition and inversion the sum of the intervals. If two chords can be made identical by either transposition or inversion they are considered identical. By this classification, for instance, there are 12 possible 3 note chords (interestingly enough, a major and minor triad are the same entity under this system)

    By Anonymous steve, at 10:37 AM  

  • Hmm, it's been a while since I studied group theory, but I think two of your properties are incomplete.

    Identity should require a * 1 and 1 * a to *both* equal a.

    Inverse should require a * a^-1 and a^-1 * a to *both* equal 1.

    This is necessary because not all groups are commutative (in other words, a * b = b * a is not always true), unlike real number addition and multiplication.


    Certain kinds of functions from a set to itself form a group under composition (defined by f o g(x) = f(g(x)) for all x); in particular, permutations, symmetries and self-isomorphisms. So group theory comes up in linear algebra, topology, real analysis - all kinds of places, really.

    By Anonymous Chris, at 1:41 PM  

  • I recently found my copy of Cotton's "Chemical Applications of Group Theory". Fun stuff-had not thought about most of these problems in years.

    By Blogger Ruth, at 2:28 PM  

  • To mention an example of how group theory is used in particle physics, Paul Dirac discovered the existence of the positron through manipulation of complex valued matricies in the general linear group, GL.

    By Blogger eulerfx, at 2:49 PM  

  • chris:

    Thanks for the correction, you're absolutely right. I forgot that group theory does not require commutativity. It's been a while since I thought about this; I think I was thinking of Abelian groups instead of just plain groups.

    By Blogger MarkCC, at 3:09 PM  

  • Groups are also used in cryptography... Nothing exciting like in physics, though.

    By Blogger Mr. Turtle, at 7:32 PM  

  • Well, you don't exactly *need* to have all four rules explicitly stated:

    Say you have a * 1 = a and a^-1 * a = 1.

    Then

    a * a = (a * 1) * (a * 1)
    a * a = a * (1 * (a * 1))
    a * a = a * (1 * a)

    a^-1 * a * a = a^-1 * a * (1 * a)

    a = 1 * a


    Say b is such that b * a^-1 = 1

    b * (a^-1 * a) = b * 1 = b
    (b * a^-1) * a = 1 * a = a

    So a = b

    Thus, you only need 2 of the rules (one for inverse, one for identity) to get all four. It just takes a little work with the proofs.

    Commutativity (or its absence) is an issue, but not when dealing strictly with the inverse and identity.

    By Anonymous Anonymous, at 7:50 PM  

  • The interesting thing about the application of group theory to space time is that the conservation laws of physics can be shown to be equivalent to the invarience of physics under space time transformations. For example
    1. conservation of angular momentum can be shown to be equivalent to the statement that the laws of physics are invarient under static rotations (the rotation group in three dimensions);
    2. conservation of linear momentum can be shown to be ewuivalent to the statement that the laws of physics are invarient ulnder static space translations (the translation group in three dimensions);
    3. conservation of energy can be shown to be equivalent to the statement that the laws of physics are invarient under static time translations (the one dimensional translation group).

    By Anonymous Anonymous, at 8:23 PM  

  • A correction to the correction to the correction. Recursion and meta-reasoning, yay! Also apologies, as I'm responsible for the incorrect correction to the correction.

    Given three of the relationships (for identity and inversion), it is possible to prove the fourth.

    Say you have both inversion relationships and a * 1 = a

    a * a^-1 * a = a * 1 = a
    a * a^-1 * a = 1 * a

    So 1 * a = a, the remaining identity relationship. Having both inversion relationships gives 1*a = a*1, so having one identity relationship is equivalent to the other in this case.


    Say you have both identity relationships and one inversion relationship (here, a^-1 * a = 1)

    Let b satisfy b * a^-1 = 1, which is possible since a^-1 exists.

    b * a^-1 * a = b * 1 = b
    b * a^-1 * a = 1 * a = a

    Thus a = b, and we have a * a^-1 = 1.


    Having only the identity relationships is not sufficient, as can be seen using multiplication: 0 satisfies identity (0*1 = 1*0 = 0), but not inversion (there is no x such that 0*x = 1).

    Having only the inversion relationships is not sufficient, which is again illustrated with multiplication: define "1" as being equal to 0, and a^-1 = 0 for all a. Then a^-1 * a = "1", since 0*x = 0, and x*0 = 0. We do not, however, have x * 0 = x, which is the identity requirement.

    Having one of each also does not work, though examples are harder to create. Essentially, the counter-counterexample I used fails because the first set of equations boils down to 1 * a = 1 * 1 * a, which is not the same as a = 1 * a.

    At best, the prior example provides inversion and identity for all values that are of the form 1*x, but says nothing about values that cannot be expressed this way.

    By Anonymous Anonymous, at 6:16 PM  

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