Some Applications of Group Theory, promoted from comments

A bunch of people posted some really great descriptions of applications of group theory in the comments; I thought they were good enough that they deserved to be promoted to the front page.

Groups and Relativity

Blake Stacey wrote an excellent comment explaining how group theory applies in relativity.

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".

In a very similar vein, an anonymous poster (email me if you want me to put your name here!) commented on how group theory within the framework of relativity explains the basic laws of conservation:

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).

Other Applications in Physics

Another nifty example of how group theory has been used in physics; eulerfx explains:

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.

Music Theory

Steve added some detail to my comment about how group theory can be applied to music theory:

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)

Big thanks to you guys, and all of the other commenters!