In my last post on group theory, I screwed up a bit in presenting an example. The example was using a pentagram as an illustration of something called a permutation group. Of course, in
my attempt to simplify it so that I wouldn’t need to spend a lot of time explaining it, I messed up. Today I’ll remedy that, by explaining what permutation groups – and their more important cousins, the symmetry groups are, and then using that to describe what a group action is, and how the group-theory definition of symmetry can be applied to things that aren’t groups.
As I alluded to in the last post, permutation groups are very fundamental. You’ll see part of why
that is later in this post. But there’s also a historical reason. Group theory was developed as
a part of the algebraic study of equations. One of the main occupations of people studied algebra
up to the 19th century was finding equations to compute the roots of polynomials. So, for
example, anyone who’s taken any high school math knows the quadratic equation, which can be used
to find the roots of a quadratic polynomial.
The quadratic solution has been known for a very long time. There are records dating back to the
Babylonians that contain forms of the quadratic equation. It took a ridiculously long time to get from there to a general solution for cubics. Quartics followed very soon after cubics. But then, after the quartic solution, there was a couple of hundred years of delay with no progress. Neils Henrik Abel and Evariste Galois, both very young and very unlucky mathematicians, roughly simultaneously proved
that there was no general solution for polynomials of degree five. Galois did it by working out
symmetry properties of the solutions of polynomials – which come from the permutation groups
of those solutions – and showing that there was no possible way to get a solution because of the properties of those groups. We’ll leave it at that for now; later, I’ll come back to that, and show how you can form permutation groups from the solutions of a polynomial, and how the structure of the permutation groups can show that there are no algebraic solutions for orders greater than 4.
Getting back on topic: what is a permutation group? Given a set of objects, O, a permutation
is a one-to-one mapping from O to itself. It defines a way of re-arranging the elements of the set. So, for example, given the set of numbers {1, 2, 3}, a permutation of them is {1→2, 2→3, 3→1}. A permutation group is a set of permutations over a set, with the composition
of permutations as the group operator. So, for example, working with the set {1,2,3} again, the elements of the largest permutation group are:
{ { 1→1, 2→2, 3→3 }, { 1→1, 2→3, 3→2 }, { 1→2, 2→1, 3→3 }, { 1→2, 2→3, 3→1 }, { 1→3, 2→1, 3→2 }, { 1→3, 2→2, 3→1 } }
To see the group operation, let’s take two values from the set. Let f={1→2, 2→3, 3→1}, and let g={1→3, 2→2, 3→1}. Then the group operation of function composition will generate the result: fˆg={1→2, 2→1, 3→3}.
The identity of the group is obvious: 1O = {1→1, 2→2, 3→3}. Inverses are also obvious: just reverse the direction of the arrows: { 1→3, 2→1, 3→2 }-1 =
{ 3→1, 1→2, 2→3 }.
When you take the set of all permutations over a collection of N values, the result is the
largest possible permutation group over those values. That group is called the symmetric group
of size N, or SN. The symmetric group is fundamental: every finite group is a subgroup of a finite symmetric
group; which in turn means that every possible symmetry of every possible group is embedded in the structure of the corresponding symmetric group.
To formalize that just a tad, I’ll need to formally define a subgroup. Fortunately, that’s quite
easy. If you have a group (G,+), then a subgroup of it is a group (H,+) where H⊆G. In english,
a subgroup is a subset of the values of a group, using the same group operator, and which
satisfies the required properties of a group. So, for instance, the subgroup needs to be closed under
the group operator.
For example, if you have the group of integers, with addition of its operation, then the set of even integers in a subgroup. Any time you add any two even integers, the result is an even integer. Any time you take the inverse of an even integer, it’s still even. So it’s closed. You can work through the other properties, and it will satisfy all of them.
There’s a stronger form of subgroup, called a normal subgroup. A normal subgroup (H,+) of a
group (G,+) is a subgroup that satisfies one additional property: ∀x∈G: ∀y:∈D
x+y+x-1∈H. That looks like something that should be obviously true for all
subgroups. It isn’t. The reason that it looks obvious is that we intuitively expect the group
operator to be commutative. But our definition of groups does not require the group operator to
be commutative. There are many groups whose group operators are commutative: they’re called the
Abelian groups. But there are also many that aren’t. All subgroups of an abelian group
are normal. But there are subgroups of non-abelian groups that are not normal.
We’re almost done with the definitions. But there’s a couple easy ones that I need
before I can explain group operators.
A trivial group is a group which contains only an identity value. A simple group is basically sort-of the group-wise equivalent of a prime number: a simple group is a group whose only normal subgroups are the trivial group, and the group itself.
Ok, now we’re finally ready. As I’ve talked about before, a group defines a kind of symmetry, otherwise known as an immunity to some kind of transformation. But we don’t want to have to define groups and group operators for every set of values that we see as symmetric. What we’d like to do is capture the fundamental idea of a kind of symmetry using the simplest group that really
exhibits that kind of symmetry, and the able to use that group as the definition of
that kind of symmetry. To do that, we need to be able to describe what it means to apply the
symmetry defined by a group to some set of values. We call the transformation of a set produced
by applying a symmetric transformation defined by a group G as the group action of the group G.
Suppose we want to apply a group G as a symmetric transformation on a set A. What we can
do is take the set A, and define the symmetric group over A, SA. Then we can
define a mapping – to be more precise, a homomorphism – from the group G to SA. That
homomorphism is the action of G on the set A. To make that formal:
If (G,+) is a group, and A is a set, then the group action of G on A is a function f such that:
- ∀g,h∈G: (∀a∈A : f(g+h,a) = f(g,f(h,a)))
- ∀a∈A: f(1G,a) = a.
All of which says that if you’ve got a group defining a symmetry, and a set you want to apply a symmetric transformation to, then there’s a way o mapping from the elements of the group to the elements of the set, and you can perform the symmetric group operation through that map. The group operation is an application of the group operation through that mapping.
Every symmetric operation can be characterized by some kind of group; and using that group’s group operation, that symmetric operation can be applied to any desired set of values. So we can, for example, use the group of addition over the real numbers to define mirror symmetry on a two dimensional image.
So here’s something I’ve been trying to figure out. If you watch physicists, they talk about symmetry groups and group actions a lot. But when they talk about symmetry groups they seem to like to call them “gauge groups”, and when they talk about the group action of a symmetry group they seem to like to call it a “Yang-Mills action”. Is there a difference between a “gauge group” and a “symmetry group”? Is there a difference between a “group action” and a “Yang-Mills action”?
Thanks for the great group theory discussions. Have you, by the way, seen David Farmer’s book _Groups and Symmetry_? Maybe it was coauthored with Ted Stanford, I forget. I think your articles here are a great complement for this book, and vice-versa.
I wish my group theory instructors had been more emphatic about analogies like “a simple group is like a prime”. But I also wish that you were more clear about why only NORMAL subgroups count as factors here.
By the way, small typo: ∀x∈G: ∀y:∈D should read H instead of D.
And a question: In your categories of goodmath, badmath you have “Bad Math Education” — what would you point to as an example of Good Math Education?
Eagerly awaiting your discussion on Galois Theory and the solutions of polynomials. And btw a discussion of wreath products and Rubik’s cubes would be cool, particularly maybe an n-dimensional case. lol
In physics, a “gauge transformation” is a way of taking a set of quantities which describe a phenomenon and turning them into another set of quantities which describe the same physical phenomenon. For example, when dealing with voltages, it doesn’t matter where you set your zero point: to figure out how much power is dissipated by a resistor, you just need to know the voltage drop across the resistor, and adding the same constant to the voltage measured at both ends doesn’t change the difference between those ends. Picking your zero point is an elementary kind of choosing a gauge.
Life gets more complicated when we add magnetic fields to the situation. The electric field is minus the gradient of the scalar potential, and the magnetic field is the curl of the vector potential; gauge transformations then relate different scalar and vector potentials, all of which are equivalent in terms of the physics they describe. We can choose a gauge parameter for each point in space; for the Maxwell Equations of electromagnetism, that gauge parameter is an element of the group U(1), or in other words, a complex number of unit magnitude. Moving beyond electromagnetism brings us into theories with different gauge groups. Since groups whose multiplication operation commutes are called abelian, gauge theories involving groups whose operations are not commutative are called non-abelian gauge theories.
Whew!
The “action” in “Yang-Mills action” is meant to be understood in the sense that “action” is used in Lagrangian mechanics, which see.
[For example, if you have the group of integers, with addition of its operation, then the set of even integers in a subgroup.]
‘in’ looks like a typo here for ‘is’.
[The reason that it looks obvious is that we intuitively expect the group operator to be commutative.]
I don’t mean to argue with your intuition, but I certainly don’t expect the group operator to qualify as commutative. You may do expect such, but you speak for yourself.
[All subgroups of an abelian group are normal. But there are subgroups of non-abelian groups that are not normal.]
Proof for the first part? Examples for the second part?
[then there’s a way o mapping from the elements of the group to the elements of the set]
Read ‘of’ for ‘o’ right?
[Every symmetric operation can be characterized by some kind of group]
Do you have a formal definition for ‘symmetric operation’? Does min in min(a, b)=min(b, a) qualify as a ‘symmetric operation’?
Mark, I am about 51 or 52 years away from modern formal notation. I think I once knew much of the older parts of group theory, but no longer true. I would find it helpful if you would give a short dictionary of notation where I could find it. Easy to find would be best.
Thanks.
REH
[Proof for the first part?]
∀x∈G: ∀y:∈D x+y+x-1 = y + x + x-1 = y ∈H
The first equality is by the group being abelian, and the second is by definition of x-1.
Let me fix some typos:
∀x∈G: ∀y∈H x+y+x-1 = y + x + x-1 = y ∈H
One characterization of a normal subgroup is that every right coset is a left coset. Just to see what it looks like, find a non-normal subgroup of a group — and a right coset which is not equal to any of the left cosets.
We need to look at a non-abelian group — so the first possibility is this group of order 6:
* = Normal subgroup
Generators Subgroup
0 { } *{ A }
1 { D } { A D }
2 { E } { A E }
3 { F } { A F }
4 { B } *{ A B C }
5 { B D } *{ A B C D E F }
COSETS of subgroup generated by set: { d }
Left Cosets Right Cosets
{ A D } { A D }
{ B F } { B E }
{ C E } { C F }
The subgroup { A D } is NOT a NORMAL subgroup.
The right coset containing B and E does not coincide with any of the left cosets.
This is also the time to recite:
Q: What’s purple and works from home?
A: A non-Abelian grape. It doesn’t commute.
Also, a gauge group is an example of a continuous group (think rotations-of-a-sphere, as opposed to rotations-of-a-cube). Specifically, the gauge groups physicists tend to deal with are examples of what are called “Lie Groups” (pronounced LEE, after Sophus Lie); these are a special type of group.
Now, a group can act in many different ways on sets of objects. For example, imagine you had 3 triangular coins. The group of permutations of 3 elements can act in (at least) two ways on these coins. It can either permute the coins (taking the order A B C to the order A C B, for example), or, it can permute the corners of the coins (flipping each coin over through the axis between its SW corner and NE side, for example. This exchanges the N and SE corners of the coin.). A “Yang-Mills gauge transformation” is a group action defined in a very specific way, and it only applies to a specific type of physical theory built out of a Lie group. So “group action” is sort of a generic term, and “Yang-Mills gauge transformation” is a specific type of group action. Hope this helped.
[Let me fix some typos:
∀x∈G: ∀y∈H x+y+x-1 = y + x + x-1 = y ∈H]
Thanks. For more detail one can, of course, write
∀x∈G: ∀y∈H (x+y)+x^(-1)=(y+x)+x^(-1)=y+(x+x^(-1))=y+0=y ∈H
Also, laws in physics can be observed locally (typically stated in differential form) or globally, and symmetry laws are no exceptions. For example, the EM field conserves charges among other things due to local symmetries. A global symmetry wouldn’t depend on position in space and time, and perhaps a reference potential such as in Blake’s comment is such a conserved quantity.
I wouldn’t recognize a gauge invariance if it bit me, but I have heard that aside from usual symmetries of translation and rotation a gauge is scale invariant. Which leads to renormalization theory I guess, which I believe are descriptions of how field strengths are conserved while the metric scale changes.
Despite what earlier comments have been saying, that’s not the problem with your formula in re normal subgroups. You had the order right, but for some reason you’re writing the composition additively and the inverse multiplicatively. Since all subgroups of an abelian group are normal, you may as well just write them both multiplicatively.
Coin,
Barton Zwiebach’s First Course in String Theory (2004) is a pretty gentle introduction to lots of this stuff. I’ve heard of students braving their way through it with nothing but freshman physics and determination.
Blake, you mean with regard to the Yang-Mills stuff? I’ll try to check it out, thanks!