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Where Do We Start?

We're going to start in Geometry, and investigate the Symmetries of 2-Dimensional Regular Polygons.
Then we'll move on to other areas, and hope to show how they all relate via the Theory of Groups
No! Don't leave yet! I'm hoping to make it simple enough for everyone to understand, and I plan to avoid using Greek symbols, Theorems, and Proofs!
Having said that, I do acknowledge that I will be using (and defining) a fair amount of mathematical terminology. Please bear with me; Mathematicians do this in order to be more precise in their communications.

February 2009

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The "Deer in the Headlights" Syndrome

It is my belief that many (most? all?) people who say they have never understood (or could ever understand) mathematics have been subject to two personal experiences in their lives:

Poor teaching of mathematics. in particular, insufficient relationship to the application of mathematics to their daily lives.
A 'deer in the headlights' reaction to mathematics whenever they experienced it.

I do not plan to "run you over" today!

February 2009

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2-Dimensional Regular Polygons

So, we start by looking at 2-Dimensional Regular Polygons.
First, what's a regular polygon in 2-dimensions, anyway?

Answer: A regular polygon is a polygon which is equiangular and equilateral (all sides have the same length)
A polygon is a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.

February 2009

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2-Dimensional Regular Polygons

Some interesting properties of polygons include:

A polygon which has a circumscribed circle, upon which all its vertices lie, is called a cyclic polygon.
Every regular polygon has a circumscribed circle.
Every regular polygon also has an inscribed circle that touches each of its sides (a.k.a. edges).

February 2009

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2-Dimensional Regular Polygons

A regular polygon with n vertices (and therefore n sides/edges) is n-way rotationally symmetric.
That means that, if you rotate that polygon around its center by 360 degrees/n (or any integer multiple thereof), the shape remains the same.
Mathematicians say that the shape has rotational symmetry of order n, which is represented by the cyclic group named C_n.

February 2009

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Symmetries of an Equilateral Triangle

So, what is this cyclic group C_n ?
Let's work out the details for pretty much the simplest case:

The Equilateral Triangle

Now, we'll have you folks do some work…

February 2009

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Symmetries of an Equilateral Triangle

Everybody done?

Once we get here, you should have already constructed representations for:

The cyclic group, C3
and
The dihedral group, D3

February 2009

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Properties of C3 and D3

Note the following about these groups:

C3 is a subgroup of D3 (we'll understand that better in a little while)
Within C3, the order of operations doesn't matter (in all cases). �In the jargon of Group Theory, this is known as an Abelian group.
Within D3, however, the order of operations matters in some cases. �In the jargon of Group Theory, this is a non-Abelian group.

February 2009

a ● b = b ● a ?

(for all a, b)

Yes: Abelian

No: Non-Abelian

Niels Henrik Abel

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Sets

So, we've constructed a couple of groups, but what exactly are groups, as defined in mathematics?
First, we have to learn some more math jargon:

A mathematical set is a collection of distinct objects
In mathematics, two objects are called distinct if they are not equal (whatever that might mean, in context)
The order in which the elements of a set are listed is irrelevant
A set may have zero or more objects, or elements. The number of elements in a set is its cardinality (or size).
If the set has an infinite number of elements, it is called an infinite set; otherwise it is called a finite set.

February 2009

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Groups

In mathematics, a group is an algebraic structure consisting of

a set, together with
an operation that combines any two of its elements to form a third element

Notice that we are talking about an algebraic structure, so the concept is originally based on the conventional algebra of numbers

February 2009

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Definition of a Group

Here's a more specific definition of a group:

A group is a set, G, together with an operation "•" that combines any two elements a and b of the group, to form another element denoted a • b.

Note that a = b may be true (i.e. a and b are identical)
Note also that the operation can be any operation; addition is only one such possibility

To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:

Closure: For all a, b in G, the result of the operation a • b is also in G.
Associativity: For all a, b and c in G, the equation �(a • b) • c = a • (b • c) holds.
Identity element: There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds.
Inverse element: For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

February 2009

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The Simplest Groups

The least complicated group is one that has an operation of 'do nothing', and a single element.
Given the group axioms, this single element must be the identity element, e, and that element must also be its own inverse.
This group is not very interesting.

February 2009

{e}

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The Simplest Groups

The next group in terms of complexity is one that has two elements, e and a.
To satisfy the group axioms:

a ● a must belong to this group, so either:

a ● a = a

a ● a = e

However, a ● a = a, implies that a = e, which violates the uniqueness of the elements. �Thus a ● a = e .

You may think that such a group isn't very useful, but as we'll see a little later, you'd be wrong!
An example of such a group is {1, -1} under ordinary arithmetic multiplication:

1 is e; -1 is a;
Each is its own inverse
1 ● 1 = 1; -1 ● -1 = 1; �1 ● -1 = -1; -1 ● 1 = -1

February 2009

{e, a}

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Example: The Group of Integers

Here's an example:

The group, Z, of all positive and negative integers (including 0) under addition (the operation)

Here are the rules for this group:

For any two integers a and b, the sum a + b is also an integer (i.e., an element belonging to the same group) �This property is known as closure under addition.
For all integers a, b and c, �(a + b) + c = a + (b + c)�a property known as associativity
If a is any integer, then �0 + a = a + 0 = a�Zero is called the identity element of addition
For every integer a, there is an integer b such that �a + b = b + a = 0�The integer b is called the inverse element of the integer a and is denoted −a.

Z is an infinite group, because it has an infinite number of elements.

February 2009

…-4, -3, -2, -1, 0, 1, 2, 3, 4…

The conventional mathematical name for this particular group is Z (from zahl, the German word for number)

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Example: Symmetry Group of a Square

The symmetries (i.e., rotations and reflections) of a square form a group called a dihedral group denoted D4. The following symmetries occur:

February 2009

The identity operation leaving everything unchanged, denoted id;
Rotations of the square by 90° right, 180° right, and 270° right, denoted by r1, r2 and r3, respectively;
Reflections about the vertical and horizontal middle line (fh and fv), or through the two diagonals (fd and fc).

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Example: Symmetry Group of a Square

Any two symmetries a and b can be composed; i.e., applied one after another. The result of performing first a and then b is conventionally written symbolically from right to left as

b • a -- "apply the symmetry b after performing the symmetry a"

The group table on the left lists the results of all such compositions possible for this group.

The elements id, r1, r2, and r3 form a subgroup, which is highlighted in pink (upper left region).
The green and yellow areas denote the left and right cosets of this subgroup, respectively. We won't worry about what that means…

February 2009

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Example: Symmetry Group of a Square

What makes the pink area a subgroup of D4 ?
Does this subgroup satisfy the group axioms by itself?
What do you think the name of this subgroup might be?
Is it Abelian? How can you immediately tell?
Is the D4 group Abelian? Again, how can you tell?

February 2009

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Permutation Groups

A permutation group is a group whose elements are permutations of a given set, and whose group operation is the composition of permutations in that group.
For example, consider the set of letters {A,B,C}. These letters may be rearranged into different orderings (see left). �For n items, the number of permutations is:

n! ( n factorial = n·(n-1)·(n-2)·…·3·2·1 )
When n = 3, n! = 3·2·1 = 6 .

Does this look familiar?
Do you notice anything different about the first three permutations, compared with the last three?

February 2009

ABC

BCA

CAB

ACB

BAC

CBA

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Another Example: Modular Arithmetic

Modular arithmetic (sometimes known as clock arithmetic) was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
If we use a conventional 12-hour clock, and if the time is now 7:00, then 8 hours later it will be 3:00.

7 + 8 = 15, but then we must then divide the 15 by 12 and take the remainder (aka modulus) to obtain the answer

Can you think of other applications of modular arithmetic?

February 2009

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Another Example: Modular Arithmetic

In modular arithmetic, we are no longer dealing with an infinite set of numbers, as we are in conventional integer arithmetic.
If we consider arithmetic modulo n, we will have remainders within the range 0..(n-1), or a total of n possible values.
For example, modulo-5 arithmetic will have a range of possible values 0 through 4.

February 2009

{0,1,2,3,4}

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Another Example: Modular Arithmetic

So, let's see what the group table looks like for modulo 4 addition:

and:

modulo-5 addition:

Do you see a pattern?
What can you say about�this group?

February 2009

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Another Example: Modular Arithmetic

Now, let's see what the group table looks like for modulo 4 multiplication :

and:

modulo-5 multiplication :

Do you see a pattern?
What can you say about�this group?

February 2009

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Another Example: Modular Arithmetic

It's harder to discern a pattern in the mod-n multiplication table. Here's what the multiplication table looks like for mod 13:

February 2009

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Another Example: Modular Arithmetic

What about division?
We know that, in ordinary arithmetic, if

ac = bc

then we can divide by c on both sides, to get:

a = b

unless c = 0.

However, in modular arithmetic this sometimes works, and sometimes doesn't.
It turns out that division works all the time (except of course for dividing by zero) only if the modulus is a prime number.

February 2009

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Another Example: Modular Arithmetic

Remember when we were talking about the group with two elements, we said:

"You may think that such a group isn't very useful, but as we'll see a little later, you'd be wrong!"

Well, here's where you can see one use for this seemingly too-simple group: the case of mod-2.
To the left are the tables for mod-2 addition and multiplication.
If you compare these two tables with Boolean arithmetic, where you are dealing with only two values: true (T) and false (F), and consider 0 to be equivalent to F, and 1 to be equivalent to T, these tables show identical values to the two Boolean operations:

Addition -> XOR (eXclusive OR)
Multiplication -> AND

This is critical to the proper working of modern-day computers.

February 2009

XOR

AND

Incidentally, 2 is a prime number. Surprised?

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Continuous vs. Discrete Groups

We previously looked at the symmetry and group properties of regular polygons in 2 dimensions, with n vertices/edges. We saw that these produced groups S3, S4, etc. and D3, D4, etc., which were finite groups.
What if we took n to ∞ (infinity)?�What would our polygon then be? How many elements would our group contain?
Up to this point, each of our geometrical transformations have been discrete – that is, there were only a limited number of them that could be performed.
When we take n to infinity, the number of transformations that can be performed also becomes infinite. In the case of rotations, for example, any infinitesimal rotation can be performed. These are termed continuous transformations, and the resulting groups are known as continuous groups.

February 2009

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Continuous vs. Discrete Groups

The group of all rotations about a fixed point in 2-dimensional space is called the special orthogonal group, SO(2).
Adding in all the possible reflections about any axis of symmetry leads to a larger group, O(2).
Orthogonal means that the transformations are rigid motions of the plane.
Special means that the rotations do not flip the plane over.

February 2009

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Continuous vs. Discrete Groups

Both SO(2) and O(2) are infinite, reflecting the fact that the circle has infinitely many rotational symmetries and infinitely many reflectional symmetries
SO(2) is the simplest example of what mathematicians call a Lie group, which have become incredibly important in mathematics, and in physics.
They are named in honor of the Norwegian mathematician, Sophus Lie (pronounced 'Lee'), who largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.

February 2009

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Lie Groups

One of Lie's insights was to realize that the natural algebraic operation on the Lie algebra was the difference, AB – BA, which is known as the commutator.
For groups like SO(2) (and like Cn) the commutator is zero (they are Abelian)
However, for groups like SO(3), the rotation group in three dimensions, the commutator is non-zero unless the axes of rotation of A and B are either the same or at right angles.
Thus, the geometry of the group shows up in the commutators.

February 2009

AB - BA

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Lie Groups

Lie groups may be thought of as smoothly varying families of symmetries. Examples of such symmetries include rotations about an axis.
What must be understood is the nature of 'small' transformations, e.g., rotations through tiny angles, that link nearby transformations.
The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups").

February 2009

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Lie Groups

Lie groups are used in many areas:

Euclidean space with ordinary vector addition as the group operation becomes an n-dimensional Abelian Lie group
The special unitary group, SU(n), is critical to modern-day physical theories of the universe
The Lorentz group and the Poincare group are the groups of linear and affine isometries of the Minkowski space (interpreted as the space-time of Einstein's special relativity). They are Lie groups of dimensions 6 and 10.
The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the Standard Model in particle physics. The dimensions of the factors correspond to the 1 photon + 3 vector bosons + 8 gluons of the standard model.

February 2009

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Summary

As you can see, groups have become extremely important in mathematics, chemistry, biology (DNA, and the structure of proteins, for example), and particularly in physics (about which we'll learn more details later)
I hope you have a little more understanding of what group theory is, and how it might be applicable to all these endeavors.

February 2009