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:
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:
What kinds of things can be expressed as groups?
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.
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?
- 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.
- Chemistry: you can determine the polarity of a molecule by using group theory to identify the symmetries in the structure of the molecule.
- Physics: group theory is apparently used in special relativity, although I don't pretend to understand how.
posted by MarkCC at
8:07 PM
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10 Comments:
Group Theory is also very important in particle physics.
By
Anonymous, 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
Anonymous, at 11:23 PM
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, 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, at 1:41 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
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
MarkCC, at 3:09 PM
Groups are also used in cryptography... Nothing exciting like in physics, though.
By
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, 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, 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, at 6:16 PM
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