The point of set theory isn’t just to sit around and twiddle our thumbs about
the various definitions we can heap together. It’s to create a basis on which we
can build and study interesting things. A great example of that is something
called a group. Once we’ve built up sets enough to be able to understand a set of values and an operator, we’re able to build an amazing deep and interesting construction, called a group.
Back when I started this blog, one of the first topics that I
wrote about was group theory. I was just looking back over that
series of posts, and frankly, they sort of stink. I’ve leaned a lot about
how to write for a blog in the time since then, and so I’d like to go back
and rewrite it. I’ve never reposted any of the group theory material, so
it should also be new to most readers.
As you should know from reading this blog, one of the things in math
that fascinates me the most is abstraction: taking some subject of interest,
paring it down, and reducing it to its most basic essentials, in order to
really understand it and what it means.
Group theory is part of a broader topic called abstract
algebra. Abstract algebra reduces things to the minimum concepts
of a set of values and some operations over those values. By specifying
the properties of the set and operations, you can create different
algebraic structures. Group theory is, basically, the simplest construction
of abstract algebra: one collection of values, and one operation.
What do you do in group theory? There’s a wonderful
quote, which was written by a famous writer of popularized mathematics
named James Newman. Professor Newman described group theory as:
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.
So we’re taking the idea of algebra: that is, the study of
equations representing numeric relationships, where the equations are
constructed from numbers, variables, and simple operations like
addition, subtraction, multiplication and division. What we want to do
is abstract that. So we’re going to discard the idea of numbers, and
replace it with a totally abstract notion of a collection of
values. Group theory often assumes that the collection of values is a
set, but you can also study groups built from proper classes. We’re
going pair the collection of values with a single operator, which
we’ll write “•”. Despite the fact that multiplication is often
written as a dot, it’s important to realize that here, in group theory, we’re not talking about multiplication. We’re talking about a totally abstract operator.
So, a group is formed from two things: a collection of values, and
an operator. We’ll often write that as a pair, G=(V, •). In order
to be a group, this pair must have the following properties:
- Closure: ∀a,b ∈V: (a•b)∈V. That is, applying the group operator to any pair of values in the group produces a value in the group.
- Identity: &exists;1∈V: (∀a∈V, a•1 = 1•a = a.) : there is an identity value in the group, which we’ll call “1” such that applying the group operator to a value “a” and 1 in any order results in “a”.
- Associativity: ∀a,b,c∈V: a•(b•c) = (a•b)•c.
- Inverse: ∀a∈V: (∃a^{-1}∈V : a•a^{-1}=1). For every value in the group, there’s an inverse value, such that applying the group operator to a value and its inverse gives you the identity value.
An example of a group is (Z,+): that is, the set of integers and the addition
operator. We can easily see that this is a group:
- For any two integers a and b, a+b is an integer.
- For any integer, a+0 = 0+a = a, so 0 is the identity value.
- We know addition is associative.
- For any integer a, there’s an additive inverse -a such that a+-a=0.
On the other hand, (R,×) is not a group. Why not?
- For any pair of real numbers, their product is a real number.
- For any real number r, r×1 = 1×r = r.
- Multiplication is associative.
- There is a multiplicative inverse, 1/n. But it is not true that
for all values n∈R that 1/n exists: there is no multiplicative inverse of 0.
So 0 kills the real numbers as a group. If you use R-{0} as the set of
values, then you’ve got a group.
So what does this all mean? You’ll see more in later posts. But fundamentally, this
incredibly simple idea: a bunch of values and one operator with 4 properties – this
defines the entire concept of symmetry. Everything that we understand as symmetry
is completely encapsulated in the concept of a group.
The idea of symmetry in group theory is useful for more than just abstract
reasoning. Obviously, it’s used for algebra: the basic idea of group theory was
largely developed in the study of algebra and the symmetries of algebraic solution. But it’s useful for a lot more than that. A few example, to give you a taste:
- Relativity: Much of relativity is defined it terms of groups; the invariants of relativity are
largely defined in terms of groups and symmetries. - 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.