Lecture 23 of Abstract Algebra E-222

Another lecture by Peter.
Summary.
- Sylow Theorems
- Groups of order p*q
- Groups of order p^2 * q
- S5
Topic of the lecture: A5.
- Prove the following proposition: " If G is a simple group of order 60 then G is isomorphic to A5."



And more A5 stuff.

Jokes about math and mathematicians

The stereotype mathematician in jokes is a male writing some incomprehensible formula on a blackboard and not being able to communicate the meaning of it to his audience. There are however jokes where the mathematician is viewed more favourable. I found two of them on the site from Simon Singh. ( Simon Singh is an English author, journalist and TV producer, specialising in science and mathematics. He wrote a book on Fermat's Last Theorem. As I mentioned Singh collects jokes, you'll find his jokes on his website. )

An assemblage of the most gifted minds in the world were all posed  the following question:"What is 2 + 2 ?" 

The engineer whips out his calculator, taps away at it for a while and finally announces "3.99".

The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02". 

The mathematician cogitates for a while, oblivious to the rest of the world, then announces: "I don't know what the answer is, but I can prove an answer exists!".  

The philosopher strokes his chin for several days, finally asking:  "But what do you mean by 2 + 2?" 

Finally the accountant closes all the doors and windows,  looks around carefully then asks "What do you want the answer to be?"

( By Helen Arney )

An astronomer, a physicist and a mathematician were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field.

"How interesting," observed the astronomer, "all Scottish sheep are black!"

To which the physicist responded, "No, no! Some Scottish sheep are black!"

The mathematician gazed heavenward in supplication, and then intoned,  "In Scotland there exists at least one field, containing at least one sheep,  at least one side of which is black."

( By Stephen Oman )

Math cartoon

Cartoons about math are usually not funny.

Don't you agree?

Symmetry is everywhere

Another pic from the beautiful book, I wrote earlier about, Visual Symmetry.

( 23rd Street, New York City )

Mathematica now supports handwriting recognition

In this blog-entry I wrote that I couldn't wait for handwriting recognition in Mathematica. Well, it's here. That is it works on my laptop. I have been very careful with installing / upgrading Windows 7. I have been burned too many times by ( Microsoft ) software. I first did a fresh install on a PC I have ready and available for test purposes. ( Although I have addressed it as the 'Database Server', have to finish the apps first. But that's off-topic here. ) Anyway. There were problems, of course. A driver issue of the built-in wireless network card. Some more, I can't remember. I fixed all problems eventually. Then it was time for one of my 'production' PC's: the laptop. Considering what Windows 7 adds to Vista, which is not much ( really, except handwriting recognition in Mathematica, it is nothing really ) I decided I should go for an upgrade. The idea of re-installing all my apps was too much.

I had a copy of Windows 7 Ultimate. UK version. That did not work as an upgrade of a Dutch version of Vista. Yakh. That delayed the upgrade several weeks. When I finally got hold of a Dutch 7 Ultimate I continued. MAJOR PROBLEMS, this time. The Wacom Driver ( I have a tablet PC ) did not work anymore, screen rotation portrait / landscape did not work, the HP specific buttons did not work anymore. I had to download Windows 7 drivers for most of the hardware. There is however no Wacom tablet driver for Windows 7. Bye, bye tablet. I thought for a day anyway. While browsing some forum I read a story of someone who experienced the same problems and was so kind to document what he did. He just -reinstalled the Wacom Vista drivers-. And that worked. There were several other issues but I managed to solve all of them. The entire upgrade took me about 12 ( Twelve ) hours. That includes the waiting and searching for problem solutions.

The good thing is that I can now use handwriting to enter mathematics into Mathematica. And that is extremely cool. It takes practice though, as with everything. Still discovering the how-to's. The tablet features of Windows 7 did got a major upgrade compared to Vista. ( And Vista was already much better than XP in that respect. ) - For now I decided to leave my desktop AS IS: Vista. An upgrade from Vista to Windows 7 is nice but it is not a must. Unless there is this specific feature you are after, like me with math handwriting recognition.

One more thing: initially handwriting did not work as described in the Mathematica documentation. After re-installing Mathematica it did however work.

Oxford numbers ( math mystery movie )

Definitely NOT a family Christmas movie: too much sex and nudity. Don't know who wrote the script, probably written by a dreamer, a wannabee of some kind. Young Ph.D. student from Arizona moves to Cambridge as an overseas student. He managed to get a room at the house of one of the people ( an older woman living together with her young attractive cello playing daughter, hint. ) who together with Alan Turing (-himself-) cracked the enigma. In the first scene he walks into the room. And says "that's an enigma", "No, merely a copy, the real one is at the museum", she says. More math stuff. The story plays in 1993 when a certain Wilkins announces that he cracked the 300-year old Bormat Theorem. It turned out that Wilkins stole the idea from a student. ( Maybe the entire reason for the movie was to leak the fact that Andrew Wiles stole the idea for cracking Fermat's theorem from a Ph.D. student. It wouldn't be the first time. Highly speculative of course. ) Anyway, all the girls the Ph.D. student meet fall instaneously for his charmes and seduce him. How surreal. Nerds and math students are instant turn-offs for women, like nude men wearing white socks, sort-of. The Oxford Murders is a paradox in many ways. It's a murder mystery. It's custom not to give away the murderer so I won't. The paradox is that it's a very bad movie, but still so much better than a Beautiful Mind ( except for Jennifer Connolly of course ) which was an award winning movie. There is -some- math in the movie. Fibonacci sequence, just as in PI, which is still the best math movie by far, if you ask me.

Mathematics in Movies ( 2 )

June, 2008 I wrote about mathematics in movies. As a comment on that post I received a link to this page with movies that are somehow linked to mathematics.
I got two math movies today. Fermat's Room and the Oxford Numbers. Both from 2008, maybe that explains why neither are on the page I mentioned. Haven't seen them yet though. Something for Christmas, perhaps.
I am not sure, but I think a movie about Ramanujan will be released in 2010. A must see. Ramanujan and Hardy in Cambridge. - Haven't watched Numb3rs in a while. I saved two seasons, for 'someday'. Well, maybe that day should be soon. Although Charlie is completely surreal, I mean he is an expert in -every- subject and has time to lecture, write books and be a full-time consultant to the FBI as well. Secret: I adopted one of his habits: I often use a noise-reduction headphone just to create silence around me, not to listen to music. It's great, you should try it while studying.

Lecture 22 of Abstract Algebra E-222

Richard Taylor ( once Ph.D. student of Andrew Wiles, the Fermat Theorem guy ) gave a lecture about the Symmetric group.
He demonstrated a short way of calculating the conjugate of a permutation and he talked a bit about S5, including the number of conjugacy classes it has. Each partition of n in Sn represents a conjugacy class. So for 5 it is:
-5 : 24
-41 : 30
-32 : 20
-311 : 20
-221 : 15
-211 : 10
-11111 : 1
Total 120=5!
The Stirling numbers of the first kind are used to calculate these numbers, which incidentally produce a Pascal triangle type-of matrix.

Lecture 21 of Abstract Algebra E-222

Again the Sylow theorems and a summary of the proofs.
Then finite groups of certain orders were classified.
Order p, p prime. Cyclic groups of order p.
Order p * q, p > 2, p < q. Cyclic of order p*q and if p / (q-1): Cq : Cp.
Order 2p, cyclic of order 2p and the Dihedral groups Dp.
Order p^2*q. Started with order 12:
- C12
- C6 X C2
- A4
- D6
- C4 : C3.
I can add that
- A4 = ( C2 X C2 ) : C3.

Gross sofar never talked about GAP, Mathematica, Magma or Maple. Group theory can get very complicated without the aid of a mathematics package. The groups were classified solely on the basis of the Sylow theorems in this lecture. The semi-direct product has not been lectured yet. For those new I would say these lectures were real hard. Unnecessarily hard in my opinion. The next lecture is about the Symmetry group by a famous number theory guy ( forgot his name ).

Anyway, I discovered an interesting fact about S6 today, the symmetry group on 6 letters. It is the only symmetric group whose automorphism group is not eqaul to the group itself. - Now, why would that be? Seems like some deep fact to me. - This is were mathematics gets a grip on you. You must know why.

Linear Algebra ( Books for 2010 )

Professor Gross ( Harvard ) said in one of the abstract algebra ( e-222 ) lectures that " you can't learn too much linear algebra ". I always liked linear algebra but I thought I knew most of it. I could not have been more ignorant. I discovered a book today 'Advanced Linear Algebra' by Steven Roman, a book from the Springer Graduate Texts in Mathematics series. It contains two parts, part 1: basic linear algebra contains ten chapters called:
- 1. Vector spaces
- 2. Linear transformations
- 3. Isomorphism theorems
- 4. Modules I
- 5. Modules II
- 6. Modules over a PID
- 7. The structure of a linear operator
- 8. Eigenvalues and eigenvectors
- 9. Real and complex inner product spaces
- 10. Structure theory for normal operators
Part 2: topics, contains another 9 chapters.
I think I am ready for part 1, since M208 contains linear algebra as well, this book definitely comes on my 2010 list. I just decided I am going to make a list for the math books I want to study in 2010 besides M208, MT365.

I was actually studying a book on Group Representation Theory. That's a topic which relies heavily on linear algebra. When I was studying Maschke's Theorem I realized I had to review my linear algebra, especially inner product spaces. And then I found Roman's book. My understanding of vector spaces (1) is ok, linear transformations (2) as well, and if not, M208 has lots of stuff on that. I studied the isomorphism theorems (3) in group theory, I think they are more or less the same in linear algebra. I have some notion about modules (4,5,6): like vector spaces but with scalars from a ring instead of from a field. Have to study them deeper, I suppose. Maybe the CG Modules ( vector spaces where the vectors can be multiplied with group elements as well as scalars ) I studied are a sort of modules, have to check it out. Looking forward to learn more about structure of linear operators (7) and (10) as well. The stuff in Eigenvalues and eigenvectors
(8) and Real and complex inner product spaces (9) is familiar but most likely goes much deeper here.

From a first browse through the book I can say that I like the style. Clearly written and enough examples. I hate books that don't have examples. I think that authors who don't include examples in their books are - A) too lazy, or B) not really willing to communicate their knowledge, or C) sadistic. In all cases bad people. - The book is packed with exercises but alas for the self-study student: no answers. I haven't made my mind up about authors who do provide exercises but keep the answers to themselves. Fortunately I found some problem books on advanced linear algebra as well. More on that another time, perhaps.

Assessment strategy M208, MT365.

For MT365: one exam, and the OCAS consists of 4 TMA's and 4 CMA's. I wasn't expecting the CMA's. The TMA's count for 15% a piece and the CMA's 10% a piece.

For M208: one exam, and the OCAS consists of 7 (not 4) TMA's.

Had a brief look at some past exams from both courses. Both exams seem doable, passable. Most of the MT365 is entirely new to me, so I must be careful not to underestimate this one. Rating the course at level 3 must have had a reason.

For the moment I'll stop preparing, pre-reading for M208, MT365, will do other math stuff.

Registered for MT365

Just committed myself to MT365. - I had doubts if I should go for 90 or not. Had to think about it for a while. I still think 60 is the best to do but the group theory part in M208 can't be difficult for me. I can spend that time entirely on MT365.

What's in MT365? ( Although it's a level 3 course there aren't much hard pre-requisites, just a certain mathematical maturity whatever that is. )

The course is divided into three related areas: graphs, networks and design. The Introduction introduces two themes of the course, combinatorics and mathematical modelling, and illustrates them with examples from the three areas.

Graphs 1: Graphs and digraphs discusses graphs and digraphs in general, and describes the use of graph theory in genetics, ecology and music, and of digraphs in the social sciences. We discuss Eulerian and Hamiltonian graphs and related problems; one of these is the well-known Königsberg bridges problem.

Networks 1: Network flows is concerned with the problem of finding the maximum amount of a commodity (gas, water, passengers) that can pass between two points of a network in a given time. We give an algorithm for solving this problem, and discuss important variations that frequently arise in practice.

Design 1: Geometric design, concerned with geometric configurations, discusses two-dimensional patterns such as tiling patterns, and the construction and properties of regular and semi-regular tilings, and of polyominoes and polyhedra.

Graphs 2: Trees Trees are graphs occurring in areas such as branching processes, decision procedures and the representation of molecules. After discussing their mathematical properties we look at their applications, such as the minimum connector problem and the travelling salesman problem.

Networks 2: Optimal paths How does an engineering manager plan a complex project encompassing many activities? This application of graph theory is called ‘critical path planning’. It is one of the class of problems in which the shortest or longest paths in a graph or digraph must be found.

Design 2: Kinematic design The mechanical design of table lamps, robot manipulators, car suspension systems, space-frame structures and other artefacts depends on the interconnection of systems of rigid bodies. This unit discusses the contribution of combinatorial ideas to this area of engineering design.

Graphs 3: Planarity and colouring When can a graph be drawn in the plane without crossings? Is it possible to colour the countries of any map with just four colours so that neighbouring countries have different colours? These are two of several apparently unrelated problems considered in this unit.

Networks 3: Assignment and transportation If there are ten applicants for ten jobs and each is suitable for only a few jobs, is it possible to fill all the jobs? If a manufacturer supplies several warehouses with a product made in several factories, how can the warehouses be supplied at the least cost? These problems of the system-management type are answered in this unit.

Design 3: Design of codes Redundant information in a communication system can be used to overcome problems of imperfect reception. This section discusses the properties of certain codes that arise in practice, in particular cyclic codes and Hamming codes, and some codes used in space probes.

Graphs 4: Graphs and computing describes some important uses of graphs in computer science, such as depth-first and breadth-first search, quad trees, and the knapsack and travelling salesman problems.

Networks 4: Physical networks Graph theory provides a unifying method for studying the current through an electrical network or water flow through pipes. This unit describes the graphical representation of such networks.

Design 4: Block designs If an agricultural research station wants to test different varieties of a crop, how can a field be designed to minimise bias due to variations in the soil? The answer lies in block designs. This unit explains the concepts of balanced and resolvable designs and gives methods for constructing block designs.

Conclusion In this unit, many of the ideas and problems encountered in the course are reviewed, showing how they can be generalised and extended, and the progress made in finding solutions is discussed.

Registered for M208

Just committed myself to M208.

What's in M208?

Introduction Real Functions and Graphs is a reminder of the principles underlying the sketching of graphs of functions and other curves. Mathematical Language covers the writing of pure mathematics and some of the methods used to construct proofs. Number Systems looks at the systems of numbers most widely used in mathematics: the integers, rational numbers, real numbers, complex numbers and modular or ‘clock’ arithmetics.

Group Theory (A) Symmetry studies the symmetries of plane figures and solids, including the five ‘Platonic solids’, and leads to the definition of a group. Groups and Subgroups introduces the idea of a cyclic group, using a geometric viewpoint, as well as isomorphisms between groups. Permutations introduces permutations, the cycle decomposition of permutations, odd and even permutations, and the notion of conjugacy. Cosets and Lagrange’s Theorem is about ‘blocking’ a group table, and leads to the notions of normal subgroup and quotient group.

Linear Algebra Vectors and Conics is an introduction to vectors and to the properties of conic sections. Linear Equations and Matrices explains why simultaneous equations may have different numbers of solutions, and also explains the use of matrices. Vector Spaces generalises the plane and three-dimensional space, providing a common structure for studying seemingly different problems. Linear Transformations is about mappings between vector spaces that preserve many geometric and algebraic properties. Eigenvectors leads to the diagonal representation of a linear transformation, and applications to conics and quadric surfaces.

Analysis (A) Numbers deals with real numbers as decimals, rational and irrational numbers, and goes on to show how to manipulate inequalities between real numbers. Sequences explains the ‘null sequence’ approach, used to make rigorous the idea of convergence of sequences, leading to the definitions of pi and e. Series covers the convergence of series of real numbers and the use of series to define the exponential function. Continuity describes the sequential definition of continuity, some key properties of continuous functions, and their applications.

Group Theory (B) Conjugacy looks at conjugate elements and conjugate subgroups, and returns to the idea of normal subgroups in this context. Homomorphisms is a generalisation of isomorphisms, which leads to a greater understanding of normal subgroups. Group Actions is a way of relating groups to geometry, which can be used to count the number of different ways a symmetric object can be coloured.

Analysis (B) Limits introduces the epsilon-delta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives l’Hôpital’s rule for evaluating limits. Integration explains the fundamental theorem of calculus, the Maclaurin integral test and Stirling’s formula. Power Series is about finding power series representations of functions, their properties and applications.

Real matrix representations of quaternions

The quaternions were invented in 1843 by Hamilton. There are three quaternions: i, j and k where i is the well known imaginary number i. The rules for the multiplication of quaternions are as follows :

i * j = k, j * i = -k
j * k = i, k * j = -i
k * i = j, i * k = -j
i^2 = j^2 = k^2 = -1.

The quaternion group has eight members { 1, -1, i, j, k, -i, -j, -k } and is non-abelian.

The most common presentation of Q8 is < a,b | a^4 = 1, a^2 = b^2, b^-1*a*b = a^-1 >.

Using only real numbers the quaternions can be represented as 4-by-4 matrices as follows.

MS221: Grade 2 pass ( Cont. )

( From my Open University home page )

All in all, it has been a few hours since I know the result. I am happy with it, of course I am. I am fully motivated for the courses next year, that's what counts now. I can't wait to make another TMA actually.

Should register asap. Registration for the 2010 courses closes the 16th.

MS221: Grade 2 pass

Initial reaction: happiness
2nd reaction: relief, not the feared grade 4 pass
3rd reaction: disappointment only 3pnts below distinction
4th reaction: doesn't mean a thing because 90% passed at eirher d, 2, 3 or 4
5th reaction: jealousy, 20% or so did get a distinction
Just recording what I felt when I read the pass.

I MUST DO BETTER NEXT TIME.

Time to sign up for M208, MT365.

( I had 82/100 on the examinable component, that's really bad. Yakh. )

Playing games with groups

Doing mathematics is a pleasurable activity. It can be very much like playing a game. With the help of GAP as a juror, teacher there are zillions of games you can do by yourself. Some ideas:
- finding all groups ( of a specific order );
- finding the presentation formulas;
- finding the automorphism group;
- finding the character table;
- drawing the Cayley graph;
Once you start on this you get a sort of natural need for the use of either the Mathematica Abstract Algebra package, or even GAP ( or similar ), that is a mindset which makes learning the language rather easy.

Aut( Q8 ) = S4

The automorphism group of the quaternion group is the symmetric group on four letters. Sounds fascinating to me. But once I figured it out it's rather easy.

The elements of Q8 are:
1:1, 2:-1, 3:i, 4:j, 5:k, 6:-i, 7:-j, 8:-k.
The generators are:
3:i, 4:j.

Then there are 24 isomorphisms, since there are 6 possible generators i, j, k, -i, -j, and -k but after 1 has been chosen, 4 remain ( choose i then -i can't be chosen anymore ).

So up to here I have shown that Aut(Q8)=24, not yet that it is isomorphic to S4. ( That's for later. )

Extensions of a group

I want to understand the groups of order 16, in order to construct them I have to understand the extension problem. So I am currently studying chapter 7 of Rotman.

The groups of order 16 are:
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
Q16
QD16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)

At this moment I can construct
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4

Work to do on:
Q16
QD16
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)

C13 symmetry.

The presence of the number 13, a Fibonacci number, is not a coincidence in this picture.

(Picture from the book Visual Symmetry.) - Btw. that book could be a terrific Christmas present. Although interest in mathematics is not required, appreciation of beauty is.

Breaking news ( I missed ): TeXnicCenter 2.0 alpha released

Last October TeXnicCenter 2.0 ( alpha 1 ) was released. I just read about it and immediately had to make a note of it in my blog. How could I have missed it? October had a lot of things going on for me, including the MS221 exam ( results expected on the 18th ), suppose that must be the reason. Since I am not working actively on TMA's at the moment I don't use TXC a lot right now.

To the point: the new version. It is really great news that the developers took the time to upgrade TXC. Although near perfect already new technologies in IDEs became standard in recent years which aren't in TXC 1. Cold folding is one of them and it has been implemented in TXC 2 according to the website. Cold-folding is really a must-have nowadays. I recently blew TXC and started to use an Eclipse based TeX IDE. After only one TMA I returned to TXC which simply works best for me.

I will install and test the version of TeXniCCenter 2.0 alpha and report my findings here.

TexNicCenter

Watched lecture 20 of Abstract Algebra E-222

The entire lecture was about proving the Sylow Theorems. The Sylow Theorems are easy to remember ( because they are so useful ) the proofs aren't. More on the Sylow Theorems here ( MathWorld ).

Change of plan ?

Alert! Battlestations. Change of plan.

Since I have to register for the 2010 courses before the 14th this month I have to make up my mind fast. I thought to be sure about 2010 but watching the E-222 videos changed my mind. It was my plan to do MST209 and M336 next year, thus leaving M208 for 2011. ( Ideal would be M208 + MST209 but is just not advised to take on a load like that. If you slip a few weeks for whatever reason it is impossible to catch up. So M208 + MST209 is out of the question. )

Both M208 and M336 contain Group Theory. M208 introduces the basic concepts: groups, subgroups, cosets, Lagrange's theorem, normal subgroups, and quotient groups in one module. A second module covers conjugacy classes, homomorphisms and group actions. All stuff I understand fairly well. That's why I though M336 was an option although M208 is a prerequisite. M336 reviews the M208 stuff and then covers counting with the aid of group actions ( necklace problem, I suppose ). The theory of abelian groups is covered fairly deeply just as the Sylow Theorems. These topics are half of the course. The other half is about geometry using group theory. The solids in two and three dimensions ( what I have just seen in the E222 videos ), tilings, frieze patterns, lattices and the wallpaper patterns.

My conclusion is that M336 in 2010 and M208 in 2011 is not an option. M208 + M336 is an option though. M208 in 2010 and M336 in 2011 is not an option because M336 doesn't run in 2011, so M208/2010, M336/2012 is the second option.

Recapping...
MST209 + M336 - NOT

1. M208 + M336 - Option
2. M208 ( only ) - Option
3. M208 + MT365 - Option
Both M208, MT365 have MST121, MS221 as prerequisite, so that fits. ( That is if I have a pass for MS221 ).

Just MST209 is an option as well, I suppose, although in that case I might jeopardize the overall plan.

Watched lecture 19 of Abstract Algebra E-222

A superb lecture about the proof of the proposition that A5 is a simple group. I got new insights from this lecture on conjugacy classes and constructing group theory proofs.

Watched lecture 18 of Abstract Algebra E-222

Topics.
- Group actions.
- Counting formula.
- Conjugation action.
- Conjugacy classes.
- Class equation.
- Symmetry group of the Tetrahedron (A4),Icosahedron (A5).
- Nice proof of the fact that a group of order p^n always has a non-trivial centre.
- At the time this video was recorded mathematicians conjectured that shape of the universe was SO(3)/A5.
( For me not a particularly simple lecture, but am still following. )