Group Theory - Exercise

Just had the pleasure of watching an episode of Poirot. What we share with Poirot is the need to exercise our gray cells...

If $G$ is a finite group of order $n$, prove that $G$ is isomorphic to a subgroup

of the alternating group $A_{n+2}$'

From 'Groups and Symmetry (Springer UTM) by M. A. Armstrong'

Enjoy!

M208 Group Theory video

To give you an idea what the OU videos ( delivered on DVD of course ) are like, I made some pictures.

This video is about the Counting Theorem. Applying this theorem enables you to answer questions like: "How many different dodecahedrons are there up to rotation if a face can have any of five colours?". - The OU videos are excellent. The counting theorem is one of the more difficult theorems in group theory. I don't think there is a better way of explaining it than they did in this video. - You can't compare an OU video with a lecture in class. The lecturer simply doesn't have the resources that are used in preparing a video like this. - Just think of the hundreds of abstract algebra lecturers over the world repeating the same talk they did last year. And of the same lesser quality than if they would work together and prepare videos like this one. I am sure we are heading to that direction. Just look at all the educational content that is already available for free on internet. Teachers don't have to be afraid of losing their jobs. They can create new materials, help students in small groups and do research.

Group Theory tutorial using Mathematica

Today I got particularly excited about a book. The book is meant for physicists who need a LOT more group theory than what is covered in M208. The book covers Group Theory from M208 + M336 and beyond (i.e. Group Representations, Character Tables, Lie Groups). All this math is presented in tutorial form with a massive amount of examples programmed in Mathematica. ( Most introductory group theory books seem written in a weekend. Aren't they basically a reshuffle of the same set of definitions and theorems? You don't have to do original research to write a math book. ) This book however is extremely original and must have tons of work in it.

Details:
William Martin McClain
Symmetry Theory in Molecular Physics with Mathematica
'A new kind of tutorial book'
ISBN 978-0-387-73469-9 e-ISBN 978-0-387-73470-5
706 pages

Did you know...? - Religious symbols

In mathematics the Greek letters theta ($\theta$), phi ($\phi$) and lambda ($\lambda$) are used to describe mathematical objects. They don't have a fixed meaning though. The universe of mathematical objects is too large to establish a one-to-one mapping to the Greek alphabet. Even $\pi$ doesn't always reflect the ratio of circumference and diameter of the circle, in number theory $\pi(x)$ means the number of primes less than or equal than x, in group theory $\pi(x)$ is sometimes used as a symbol for a permutation.

Did you know that the Greek letters theta ($\theta$), phi ($\phi$) and lambda ($\lambda$) have a sacred meaning in the Scientology religion? In Scientology the symbol for the life static is the Greek letter theta ($\theta$), the symbol for the physical universe ( aka MEST in Scientoloy-speak: matter, energy, space and time) is the Greek letter phi ($\phi$) and living organisms are represented by the Greek letter lambda.

Summary:
- $\theta$ = Life Static;
- $\phi$ = Physical Universe or MEST;
- $\lambda$ = Living organism.

Math and physicists (2)

A while back I asked why physicists know so much math. In this video Prof. Ramamurti Shankar from Yale University teaches differential equations.

We solve DEs by guessing.

I like to watch science documentaries, since I have seen all Horizon and other channel's docs on physics by now ( some of them even twice or more ), I watch introductory physics stuff instead. Prof. Shankar lectures in an entertaining fashion.

Introductory Physics video.

M208 - What's missing ?

Suppose you want to do some 3D graphics programming. Think of anything between writing low level code in the game-industry and developing educational flash movies, JavaFX or whatever is buzzing at the moment. Math wise you need to be very good in Linear Algebra. You must be able to implement the elements of the group O(3). In MS221 we learned everything about O(2) and SO(2). MS221 means you can do anything in R2. I expected that M208 covered everything in R3. This is not true, unfortunately. M208 does not cover the rotations around a line for example.

Rotating around a line takes a translation a rotation around one of the x,y or z-axis, another rotation around one of the x,y or z-axis and another translation. The two angles of rotation must be calculated first btw. - The exact strategy has to be learned from some mathematics book about the subject.
Rotations in R3 become very simple if you perceive R3 as part of R4 which can be mapped to the space of quaternions. A rotation then becomes simply a calculation of three quaternions.

Remark 1. Quaternions are buzzing. They did so between 1850-1900 or so when they became rather obscure. Things changed: nowadays programming languages with a 3D graphics API have standard classes for quaternions. Quaternions are things we ( as mathematicians ) must know about. That's why they are painfully absent in M208, the course where they belong most imho. Quaternions should be standard knowledge especially since they are very easy to learn. They fit in nicely in the sections on complex numbers.

Remark 2. Mathematics books require a different reading and study protocol than Open University course packs. I like the course packs. They are very efficient. Are fun to read, etc. They do their job. Transferring knowledge from the pack to the brain of the student. If you are only used to OU course packs picking up knowledge from books ( not even mentioning articles ) is very difficult. Imho at least one topic should be learned from a book in standard mathematics format, i.e. in ( densed ) Definition - Theorem - Proof - Example - Exercise format.

You must have guessed that I have been studying the quaternions. They are very interesting indeed. The quaternions aren't final though. Expect the octonions too. After that sedonions come in the picture, I believe that these numbers can be divided by zero. ( Wouldn't that be interesting?!

Example: Hopf Fibration article.

Result M208 - TMA04

Result M208 - TMA04, 76%. I am not sure if I would have deserved more than 76%. Real Analysis AA was extremely boring. AB is less boring, btw. - I suppose that on universities the stuff in the various MST121, MS221 and M208 books is grouped and lectured by topic. My guess is four 30 point courses called Calculus I, Linear Algebra, Group Theory and Discrete Mathematics is the equivalent at a university. If you don''t do well in one subject all your results are dragged down considerably at the OU. At a university your Calculus grade would be less for example. - Next Action: Group Theory B. A topic I can relate with.

GoldenRatio Phi encoded in a crop circle ?

I watched a National Geographic documentary on crop circles. As you can imagine the NGC is very skeptical about crop circles so I had to see what the hard core croppies had to say about the documentary. While searching for comments I found another theory about some mathematical constant encoded in a crop circle. This time phi.

In the documentary, one of the crop circle makers ( 'hoaxers' as they are known in the research community ) says they need the researchers as an audience for their work since they consider crop cirle making as a form of art. The makers call themselves 'artists'. ( Most 'circles' are of spectacular beauty imho. ) Vice versa the researchers need a continuous stream of new circles to explore. No one denies that some circles are hoaxed, the skeptics believe that all circles are hoaxed. Fact is that a large proportion of tourism in the area where the circles pop up is directly related to the circles. A large proportion of crop circles appear in a small area in the UK.

Images from http://www.cropcircleconnectorforum.com

Did you know ... ? - Cauchy

Augustin-Louis Cauchy (1789–1857) was a French mathematician who is generally regarded as being the founder of mathematical analysis, including the theory of complex functions. Cauchy emerged as one of the most prolific mathematicians of all time. He authored at least 789 mathematical papers, and his collected works fill 27 volumes, this is on a par with Cayley and second only to Euler. It is said that more theorems, concepts, and methods bear Cauchy’s name than any other mathematician.
From Matrix Analysis by C. Meyer

Results Poll-3

Calculator: 9
Spreadsheet: 2
MathCad: 6
Mathematica: 1
Maple: 1
Other: 3
11 voters, more than one answer allowed.
Since the use of MathCad is mandatory for MST121 and MS221 I am not surprised that MathCad is the most used mathematics package. Besides that calculators are still popular. - Obviously the high-end packages are not popular in our population. I do use Mathematica a lot and all I can say is this: If you love mathematics than you deserve Mathematica. I thought about this, "deserve" is definitely the right word.

M208 GTB

Bookset 5 of M208 is about Group Theory again. The main topics are conjugacy, homomorphisms and group actions with counting. I started studying the material as I want to start working on the TMA if at all possible this week. I did make time for reading about some other, although closely related, topics though: matrix groups. There are some beautiful maps $f: \mathbb{C} \rightarrow GL_2(\mathbb{R})$ mapping complex numbers to equivalent real matrices. - I am close to getting the click / cognition about the essence of Lie Groups. More about that another time.

Sense of urgency required

If Google changed their logo for the world-cup final I am allowed to go off-topic. At least my country is actually playing in the final. Everyone in this tiny overpopulated country has gone absolutely berserk. And it seems it stays like that until at least tuesday wether the Dutch win or not. - I don't understand why it should affect me if the Dutch win. - Four years is a long time. Mathematics wise I made progress since 2006. I sincerely hope to say the same in 2014. It will depend on my ability to create a certain sense of urgency.

Cayley's Table

Although Evariste Galois laid open the route to Group Theory his paper was basically about the theory of algebraic equations. The first paper on Group Theory ( with a reference to Galois, of course ) was written by Arthur Cayley and published in the Philosophical Magazine, vol. VII. (1854), pp.40-47 and included in the Collected Papers of Cayley Vol. II. The article is called "On the theory of groups, as depending on the symbolic equation $\theta^n=1.$ If you have studied the M208 GTA books on Group Theory the paper is very accessible. My suggestion is that you read it. If you do you will certainly understand why group tables ( Cayley Tables ) are named after Cayley.

( Although we can learn all the required mathematics for B31 straight from the OU books I still want to know how the guys who created all that miraculous stuff originally wrote it down. Besides that there will come a day that we have to read original papers anyway. )

Fibonacci computational system

Exercise.

2.15. (Fibonacci computational system) Prove that each positive integer admits a unique representation in the form $a_1f_1 + a_2f_2 + \cdots$, where $f_n$ are the Fibonacci numbers, each of the numbers $a_i$ is either $0$ or $1$, the number of ones in the representation is finite, and no two subsequent elements of the sequence $a_i$ are equal to $1$ simultaneously. For example, the first few representations are $1 = f_1$, $2 = f_2$, $3 = f_3$, $4 = f_3 + f_1$, $5 = f_4$, $6 = f_4 + f_1$, $7 = f_4 + f_2$. (Pay attention to the fact that the number $f_0 = 1$ is not used in this computational system, so that the Fibonacci sequence starts with $1,2,3,5,8,\cdots$). Invent algorithms for converting numbers from the Fibonacci system to the decimal positional number system and back, and algorithms for adding and multiplying numbers written in the Fibonacci sequence.

( "Lectures on Generating Functions by S.K. Lando, AMS 2003" )

I like problems like this. Haven't solved it yet, I am working on it.

Off and on I study the book "A course in enumeration - GTM238, by M. Aigner. Springer 2007", a book for graduate students so I need a lot of extra reading to understand the book. A good companion for Chapter 1 is the well known Concrete Mathematics by Ron Graham, Don Knuth and Patashnik. For chapters 2 and 3, mostly about generating functions, I use the book as mentioned above by Lando. - Generating functions are among the mathematical objects I love most. The power of GFs are only limited by our own imagination.

LU Factorization

$\left(
\begin{array}{ccc}
1 & 2 & 3 \\
2 & 6 & 10 \\
3 & 10 & 12
\end{array}
\right) =
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 2 & 1
\end{array}
\right) \cdot
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
0 & 2 & 4 \\
0 & 0 & -5
\end{array}
\right)
$
or
$A=L.U$ where $A$ is a non-singular matrix and $L,U$ are respectively lower- and upper-triangular matrices.

Every non-singular matrix can be factorized in the product of a lower- and upper-triangular matrix. ( The factorization itself is trivial. )

How come physicists know so much about mathematics?

Although I am somewhat of a Science Fiction fan ( 2001: A Space Odyssey? Yes!, Star Wars? No! - Moon? Yes!, Avatar? Not really. ), I am illiterate in the world of science which is largely based on quantum mechanics. By choosing the right modules ( i.e. MST209, MS326 ) in my B31 track I'll be able to read and study quantum mechanics texts soon though.
I often wondered why a B.Sc. in physics has learned so much more about mathematics than what's in an undergraduate mathematics program. Is mathematics simple, easy, even relaxing compared to physics? It certainly seems that way. Well, seemed, for me anyway. I think I understand it now.
Undergraduate physics books are almost self containing with regards to mathematics. Mathematics is the language of physics enabling them to solve all sorts of problems. ( I am not surprised that innovations in physics were carried by the introduction of new mathematics to the field. Since they aren't trained in developing new mathematics they are also constrained by it. ) Large quantities of math are pumped into their brains without proof.
This video illustrates my point. It is lecture 3 from the MIT OpenCourseWare Physics I course.

This lecture is about vectors and how to add, subtract, decompose and multiply vectors. Decomposing vectors in 2 (or 3) dimensions is a key concept that will be used throughout the course.

M208 - TMA04

Just finished M208-TMA04. Real Analysis. More difficult than I thought but I managed to complete the TMA.