Video Lectures about Lie Groups

"... A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. ..." ( Wolfram Site )

The (self-) study of Lie Group theory is hard. I found two aids that helped me going somewhat in the subject. A book called Naive Lie Theory by John Stillwell and a series of weblectures by Erik van den Ban of the University of Utrecht. On van den Ban's homepage under Lecture Notes you'll find a Lie Group's prerequisites pdf with explanations of manifolds, tangent maps etc. which appear frequently in texts about Lie Theory.

E8 structure visualized

A beautiful ( Norwegian ) theorem

#mathematics# #norway#

Theorem:

Every abelian group is the direct product of its Sylow subgroups.

Perhaps it is not the theorem in itself I like so much but what this theorem illustrates about the nature of mathematics. Most laymen think of mathematics as the scribbles of physicists they see in science documentaries, i.e. partial differential equations, stuff they call 'formulas'. So in that sense the theorem above may not even be recognized as mathematics, let alone beautiful mathematics.

Mathematics starts with a very precise, razor blade sharp, use of the tool that differentiates us humans from the rest of nature: language. Einstein once said “If you can't explain it to a six year old, you don't understand it yourself.” (*). He must have meant the "root of your knowledge tree", I suppose. Because the beauty of the theorem lies in what it represents: a large graph of concepts with - ( abelian ) group, direct product and ( Sylow ) subgroup - in the center. To anyone 'owning' these concepts the particular relation between an abelian group and its Sylow subgroups can be described in one sentence with no room whatsoever for misinterpretation. The construction of all that knowledge is the collective work of thousands and thousands of mathematicians before us.

P.S.
(*) The simplest way to explain a group is ( as far as I know ) "A collection of movements with no visible effects ( = symmetries )".

Sylow

Both Abel and Sylow were Norwegians. So was Lie, another giant, a special branch in group theory is named after him: Lie Group Theory. It is amazing that a small country like Norway ( measured in population ) can have such an impact.

Explorations beyond M336: the permutohedron

#maths #openuniversity

M336 is a two track level 3 Open University Mathematics Course with geometry track covering frieze- and wallpaper patterns, tilings and polyhedra, and a group theory track covering the Correspondence Theorem, the Sylow Theorems and the classification of Abelian groups. - When you are doing a course you are not only learning the course materials but it also broadens your view on the field. Well, I have seen quite a few new and ( fascinating ) topics lately.

Two short ones in this post and more to follow.

If you are into mathematics I bet that you have seen Inception, not that it is a mathematics movie per se but it is the type of movie math geeks love, I am sure. Anyway, do you remember the scene where Cobb and Ariadne walk on the Seine boulevard where she turns a mirror around and suddenly you see an infinite number of images. There is a name for the symmetry group of that pattern, it is a Dihedral Group with symbol $D_{\infty}$. Part of Group Theory is dedicated to studying that sort of groups, they are called Coxeter groups.

It took a while before I could dream the names of the five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron and isocahedron. But these are just the tip of the iceberg. There are enough familiar objects I don't know the name of. But there fascinating objects I never even heard of. Like the Permuatohedron for example: it is the n-dimensional generalization of a hexagon.

Permutohedron

Exploring new territory in mathematics can be quite fascinating. A library ( brick and / or online ) is a good place to start.

Guessing the Frieze Group

There are probably as many Frieze Group ( and Wallpaper Group ) 'guesser programs' as there are programming languages. I chose a Frieze Group guesser that has been written in Mathematica.

http://demonstrations.wolfram.com/GuessingTheFriezeGroup/

If you don't have Mathematica, there is a free Mathematica Player. There are also Student and Home Editions ( with all functionality ) at a much reduced price.

Open University upgrades student websites

Today the Open University will implement several changes on the student websites. The look and feel of the course websites have already been changed. I definitely like the changes. Anyway, M336 opened. M336 is an advanced group theory course about tilings, frieze and wallpaper groups, the classification of finite abelian groups and the sylow theorems.

This is a mini-talk about Frieze groups I found on YouTube:

Generating a list of permutation matrices with Mathematica

The following function defined by

f[n_] := Map[# // MatrixForm &, 
  Map[SparseArray[{i_, i_} \Rightarrow 1, {n, n}][[#]] &, 
   Permutations[Array[# &, n]]]]

maps $n$ to $S_n$, the symmetric group of $n$ elements, and displays its elements as a list of permutation matrices.

Mathematica keeps surprising me.

The function above is basically a one line function in Mathematica. ( Mathematica is a functional programming language. Every Mathematica instruction is a function, including IF and WHILE. Compare Microsoft Excel. All data in Mathematica is essentially a list. )

M381 Mathematical Logic is about the Universal Register Machine, the URM. It is a theoretical computer with only four instructions. Z(n): zero Register n, S(n) add 1 to Register n, C(m,n) copy register m to n and J(m,n,p) if the contents of Registers m and n are equal then jump to the instruction at line p. - The function above is computable and thus can be implemented in the form of an URM program. I don't know how many registers and instructionlines would be involved but it will be in the hundreds if not thousands. Just think of programming the one line function in a ( higher ) programming language like Java without using any third party imports.

P.S.
All Mathematica output can be copied as LaTeX and pasted in your LaTeX IDE, unfortunately MathJax leaves the handling of tables and matrices to HTML ( not entirely logical because you are publishing to the web and not to a PDF document ). That is the reason I included a printscreen image of a Mathematica notebook in this post.

Marcus du Sautoy on Symmetry ( Explained )

If you watched du Sautoy's talk on symmetry I posted yesterday and your background in Group Theory matches MS221 or M208 you might have lost him when he was talking about the symmetries found in the Alhambra. Let me explain.

These symmetry groups are called the Wallpaper Groups and they are part of the Open University Course M336 Groups and Geometry. Although there are only 17 wallpaper groups ( mathematical proof in M336 ) recognizing them in nature or in a piece of art is difficult. Well, it takes practice. This beautiful applet ( Java required on your PC ) lets you experiment with the wallpaper groups. Some theoretical background can be found here.

Enjoy!

[Video] - Marcus du Sautoy on Symmetry

A talk on Symmetry by Marcus du Sautoy.



Marcus du Sautoy is also the narrator of the beautiful BBC - Open University co-production "The story of maths" which is a 4 hour documentary ( in 4 parts ) accompanying the OU course TM190 The story of maths. ( Level 1 / 10 points / 1 CMA only ).

Orbit-Stabilizer Theorem

Let $G$ be a group, $X$ be a set, with $g \in G$, $x \in X$.

The number of elements in the orbit of $x$ is equal to the index of the stabilizer of $x$ in $G$:
$|\text{Orb}(x)| = [G: \text{Stab}(x)]$ ( Orbit-Stabilizer Theorem )

The total number of orbits is equal to the number of elements in $x$ fixed under an action of $g$, summed for all elements in G and finally divided by the size of $G$:
$|\text{Orb}| = \frac{1}{G} \sum_{g \in G} \text{Fix}(g)$ ( Counting Theorem )

Two theorems in which $\text{Orb}$ occurs but with a distinct different meaning.

Making study notes (2)

What I have in mind is a set of notes similar to ( but from memory and in my own words ) this set from prof. Kiyoshi Igusa on Group Theory. This is a clear 60-page summary of an undergraduate Group Theory course. It can't replace a book but it will help when you are reviewing. It gives you yet another view on the topic.

Producing PDF documents like this is very simple when you use LaTeX with a good LaTeX IDE. The work on presentation and lay-out issues is zero. This is why most mathematics books look more or less the same. Which is a good thing imo, just think of future generations accessing the vast amounts of new mathematics created in our time.

Making study notes

I have been puzzling for quite a while on how to best make notes in a way that they are productive, i.e. help in getting a better understanding of the topic, as well as reusable.

A while back I came to the idea of building a linked repository of definitions and theorems in a Personal Brain database ( plex ). Although I will be using PB whenever I can, building a repository is not what it will be. As an activity of understanding the topic it may be effective but not efficient. - I have read, seen and heard it all over the place: what really makes you understand mathematics is 1: doing exercises, 2: teaching mathematics. Although I am not a teacher, thus have no students, I can pretend as such and prepare lectures. The lecture notes and slides become the end product, they are reusable and the math presented is well understood by me. It's an idea, worth trying.

I have been a tiny bit, actually. It suddenly daunts that you can choose the format of the theorem, fonts, Greek letters, symbols used, etc. But most of all, I can present it in the most beautiful and clear way I think is possible. - ( I am currently writing a technical note on Group Actions ). M208 stuff, I know, but it will return in M336.

Quaternion group as a Freegroup

The Quaternion group can be defined as follows $\{(a,b) : a^4=e,b^4=e,a^2=b^2,ab=ba^3\}$. Let's be practical and set a=i, b=j and let's implement this in Mathematica.

The following Mathematica code

$r:=\{ \text{iiii} \to \text{""} , \text{jjjj} \to \text{""} ,\text{ii} \to \text{jj},\text{ij} \to \text{jiii}\}$

$f[\text{x_}]:=\text{StringReplace}[x,r]$

$\text{NestList}[f,\text{"ijij"},5]$

yields:

$\{\text{ijij},\text{jiiijiii},\text{jjjjiiijji},\text{jjjiiiji},\text{jjjjjjiiii},\text{jj}\}$


Let me explain. The variable $r$ is a list which contains four ( production ) rules, i.e.:
$\text{iiii} \to \text{""}$ - Says that $i^4 = 1$.
$\text{jjjj} \to \text{""}$ - Says that $j^4 = 1$.
$\text{ii} \to \text{jj}$ - Says that $i^2 = j^2$. And $i^2=-1$ as we know.
$\text{ij} \to \text{jiii}$ - Says that $ij = jiii$. Or $ij=-ji$.

The command $f[\text{x_}]:=\text{StringReplace}[x,r]$ takes a string as input and applies the production rules once from left to right. This command can be repeated until the input string no longer changes. In the case of $"ijij"$ it took $5$ times and the input and resp. outputs were as follows.

$\{\text{ijij}$,
$\text{jiiijiii}$,
$\text{jjjjiiijji}$,
$\text{jjjiiiji}$,
$\text{jjjjjjiiii}$,
$\text{jj}\}$

As you see $(ij)^2=k^2$ is correctly evaluated to $j^2=-1$.

From here on it's fairly easy to generate all elements from the quaternion group by string concatenation and applying the production rules.

From AbstractAlgebra for Mathematica to GAP.

A few notes about the Mathematica video's I created on YouTube.

AbstractAlgebra is a Mathematica add-on package. It is open source, has been written entirely in the Mathematica programming language by two mathematics professors Hibbard and Levasseur. The download url is http://www.central.edu/EAAM/Downloads/AAPackage.asp.

The functions I used like FormGroupoid, GenerateGroupoidByRelations and FormMorphoid are defined in a Mathematica package called Master in the AbstractAlgebra directory. The group I created as G1=FormGroupoid[Range[0,11],Mod[#1+#2,12]&] is by default available as Z[12] ( when SwitchStructureTo[Group] has been used ), in fact all groups I used are available as defaults. I wanted to demonstrate that you can define any group you want, in any case much more than the familiar 'textbook example groups'.

The AbstractAlgebra package is meant as a tool to visualize the often abstract concepts in Group Theory and other topics in Abstract Algebra. There is a book available from which you can learn Abstract Algebra with Labs and Exercises in Mathematica with the Abstract Algebra package. Other algebraic structures which can be created are Rings, Polynomials, Polynomials over Rings, Galois Fields, Permutations, Permutation Cycles and more.

The package AbstractAlgebra is NOT a tool for Computational Group Theory. Group Theory is alive as it is being actively researched. A state-of-the-art tool for Computational Group Theory is GAP. ( I might give some GAP demo's soon. )

Group Theory and Mathematica - 2

I have uploaded another Mathematica demo video.

For this and other math videos go to my YouTube channel.

Or watch here.



In "Group Theory and Mathematica-2" I show how to create groups by using generators and relations and how to create and visualize homomorphisms and kernels of homomorphisms.

Group Theory and Mathematica

My first video with audio...

Automorphism Groups #3

Introduction

In this post I will explain the concept of an Automorphism Group. We will make a list of the automorphism groups of all 24 groups of order less than or equal to 12 and (to our surprise) we will see that one of these groups has as much as 168 elements and that different groups can share the same automorphism group. Finally, we will make a strategy that can be of help in finding automorphism groups in general. So far we have come across direct products of groups when we studied groups of type C2 X C2 or C2 X C2 X C2. The study of automorphism groups prepares us for the study of another type of group product, the semi-direct product. ( Which I might discuss in detail in a future post. ) In this post I consider all groups to be finite. I'll try to use as much examples as I can at first and formalize later in final wrap up. Let's begin!

Generating sets

" In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. " As so very often is the case, simple things can be hard to catch in words. Let's simplify this a bit to "If [the only subgroup of G containing X = G] then [X generates G]." So it is all about -the only subgroup of G containing X-. But we don't know X! Time for examples.

List of subgroups of C3 (cyclic group of 3 elements).
1
1, a, a^2.
The sets which are only contained in C3 are: {a}, {a^2} and {a, a^2}. Note that we can reduce this list to {a}, {a^2}. ( We will do so immediately in the next examples.) Verify that C3 has two different generating sets. C3=(a)=(a^2).

List of subgroups of D3 (dihedral group of an equilateral triangle or 3-gon).
1
1, a, a^2.
1, b
1, ba
1, ba^2
1, a, a^2, b, ba, ba^2
The sets which are only contained in D3 are: {a,b}, {a,ba}, {a,ba^2}, {a^2,b}, {a^2,ba} and {a^2, ba^2}. Verify that D3 has six different generating sets.

Automorphisms

An isomorphism from a group G to a group H is a map which is surjective, injective and preserves the group operation. An automorphism is basically an isomorphism from a group to itself. Let's ilustrate this with some examples.

Let's establish an isomorphism first.

This is a CayleyTable from a group of order 8, we call the group G.


| 0 1 2 3 4 5 6 7
- | - - - - - - - -
0 | 0 1 2 3 4 5 6 7
1 | 1 0 3 2 5 4 7 6
2 | 2 3 0 1 6 7 4 5
3 | 3 2 1 0 7 6 5 4
4 | 4 5 6 7 0 1 2 3
5 | 5 4 7 6 1 0 3 2
6 | 6 7 4 5 2 3 0 1
7 | 7 6 5 4 3 2 1 0


This is another CayleyTable from a group of order 8, we call the group H.


| a b c e b**a c**a c**b c**b**a
- | - - - - - - - -
a | e b**a c**a a b c c**b**a c**b
b | b**a e c**b b a c**b**a c c**a
c | c**a c**b e c c**b**a a b b**a
e | a b c e b**a c**a c**b c**b**a
b**a | b a c**b**a b**a e c**b c**a c
c**a | c c**b**a a c**a c**b e b**a b
c**b | c**b**a c b c**b c**a b**a e a
c**b**a | c**b c**a b**a c**b**a c b a e


$\begin{array}{cccccccccc}
& | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
- & | & - & - & - & - & - & - & - & - \\
a & | & e & b\text{**}a & c\text{**}a & a & b & c & c\text{**}b\text{**}a & c\text{**}b \\
b & | & b\text{**}a & e & c\text{**}b & b & a & c\text{**}b\text{**}a & c & c\text{**}a \\
c & | & c\text{**}a & c\text{**}b & e & c & c\text{**}b\text{**}a & a & b & b\text{**}a \\
e & | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
b\text{**}a & | & b & a & c\text{**}b\text{**}a & b\text{**}a & e & c\text{**}b & c\text{**}a & c \\
c\text{**}a & | & c & c\text{**}b\text{**}a & a & c\text{**}a & c\text{**}b & e & b\text{**}a & b \\
c\text{**}b & | & c\text{**}b\text{**}a & c & b & c\text{**}b & c\text{**}a & b\text{**}a & e & a \\
c\text{**}b\text{**}a & | & c\text{**}b & c\text{**}a & b\text{**}a & c\text{**}b\text{**}a & c & b & a & e
\end{array}$

( Got the table in TeX but as you can see, blogger format is too small... )

We will investigate if G,H are isomorphic and ( if so ) then define an isomorphism f: G-> H.
( Post in progress, thus more later... )

Group Theory - Exercise ( 24/9-'10 )

Which group is represented by the following representation :
$(a,b|a^5=1,b^2=1,(a \circ b)^3=1)$.

Although I don't expect a question like this on the M208 or MS221 exams on Group Theory due to the ugly '5-min-to-think' constraint, candidates for M208 ( and possibly MS221 ) are well prepared to solve it.

Sophus Lie - Lie Groups

From the book Groups and Symmetries, From Finite Groups to Lie Groups By Yvette Kosmann-Schwarzbach ( and translated by Stephanie Frank Singer ).

Once  you understand the concept of a 'group' you can hardly imagine that there were days that you didn't understand groups or worse: that you were completely ignorant about them. That's basically how important groups are, they are as fundamental to mathematics as numbers or graphs. - Although Lie Groups aren't part of M208 or courses I planned for next year I am studying them. I am making progress but it's slow. I am no longer in complete darkness but there is a lot of mist.

New Youtube Video

I made a tiny Mathematica program which demonstrates all the 23+1 rotations of the Cube. If you haven't done Group Theory yet: the Cube has four diagonals, which can be permuted in 4! = 24 = 23+1 ways. The 23+1 rotations create just these permutations. - I joined Facebook ( at last ), found an M208 study group but with disappointing little discussion. Here is the video:

Group Theory - Exercise - Continued

I have been working on the problem that I published last week....

S3
Element - Order - Permutation Sign - Transpositions
() - 1 - +1 - ()
(1,2,3) - 3 - +1 - (1,2)(1,3)
(1,3,2) - 3 - +1 - (1,3)(1,2)
(1,2) - 2 - -1 - (1,2)
(1.3) - 2 - -1 - (1,3)
(2,3) - 2 - -1 - (1,2)

Now take the following subgroup of A5:
() - 1 - +1 - ()
(3,4,5) - 3 - +1 - (3,4)(4,5)
(3,5,4) - 3 - +1 - (3,5)(3,4)
(1,2)(4,5) - 2 -  +1 - (1,2)(4,5)
(1,2)(3,4) - 2 -  +1 - (1,2)(3,4)
(1,2)(3,5) - 2 -  +1 - (1,2)(3,5)
This is a group with S3 structure but conisting entirely of even permutations.

If any group of n elements is a subgroup of A(n+2) it must have an isomorphic copy consisting of all positive permutations.