Watched lectures 15,16 and 17 of Abstract Algebra E-222

My first encounter with the topic of discrete groups of motions. Gross treats the subject very abstract. Have to read this in Artin first and then re-watch the videos or just wait for when the topic comes along during M336. I think that about 50% of M336 is spent on discrete groups and other entirely new stuff for me. M336 won't be as easy as I thought it was going to be. As far as E-222 is concerned, I can continue watching the other lectures since in the next lectures group actions are discussed. And soon after that the Sylow theorems. It will be interesting to see how Gross introduces both topics.

Extending C2 x C2 to Q8

What I never understood is that in books on Group
Theory Q8 is shown as a concrete group, i.e. the 
group of quaternions 
{i, j, k | i^2 = j^2 = k^2 = -1, i*j=k, j*k=i, k*i=j } 
and not as an abstract group. Well, I just 
discovered that it is fairly easy to construct Q8 
from C2 x C2 ( which is often shown in abstract 
form and concrete form: the Klein4 group ).

The group C2 x C2 has the following presentation:
<a,b | a^2 = b^2 = 1, a*b = b*a >.

The group Q8-abstract has the following presentation:
<a,b,c | a^2*c = b^2*c = 1, a*b*c = b*a >,
members of this group are:
{ 1, a, b, a*b, c, a*c, b*c, a*b*c }.

The following isomorphism can be established 
between Q8-abstract and Q8:
f: Q8-abstract -> Q8
by
{ 1 |-> 1,
  a |-> i, 
  b |-> j,
  ab |-> k,
  c |-> -1
  ac |-> -i,
  bc |-> -j,
  abc |-> -k }.

Q8 is not something like ( C2 X C2 ) : C2, where X stands for direct product and : stands for semi-direct product, but it is very likely something similar. I read briefly that there are ways to construct groups other than using the direct or semi-direct product. Will / must take some time to check this out.
The fact that Q8 can be constructed and has a fairly simple presentation predicts that there must be similar methods for -all- other finite groups.

Watched lecture 14 of Abstract Algebra E-222

Abstract babble on transformation groups in R3. We are now supposed to be ready to study some Euclidean geometry in the next lectures.

Gross: "The most important theorem in calculus is the Intermediate Value Theorem."

What is the error?

Have a look at this...



There is a clearly an error in this example. Do you see what it is?

Example taken from

Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University
February 14, 2009

( See comment for further explanation )

Watched lecture 13 of Abstract Algebra E-222

Gross provided answers to the questions of the exam the students had a few days before this lecture was held.

One question was as follows.
Q. Show that if a group has a unique element of order 2 then it is part of the center.
A. Order of element is equal to order of conjugate and because there is only one element of order 2 the following is true.
a = g a g^-1 for all g in G, or
a g = g a for all g in G,
and thus a belongs to the centre of G.

The lecture introduced subgroups of GL(n,F):
- the orthonogal group O(n,F) and
- the special linear group SL(n,F).
Where GL(n,F) consists of invertible matrices in O(n,F) this is further reduced to matrices with the property that the transposed matrix is equal to the inverse matrix. These matrices turn out to have determinants 1 or -1. ( Not true that all matrices with determinant 1 are orthogonal ). Matrices with +/- 1's on the diagonal are orthogonal as well as permutation matrices.
The elements of the Special Linear group are further reduced to those with a determinant value of 1.

A concrete orthogonal group is O(2,R) as subgroup of GL(2,R). This group consists of 2 by 2 matrices with elements from the real numbers. These matrices are linear transformations of vectors in R2.

More groups and geometry to follow.

Group Explorer 2.2

Nathan Carter's Group Explorer is now in release 2.2. He takes very good care of his program. He wrote a book about Group Theory as well: Visual Group Theory. A sort of Group Theory explained illustrated by Group Explorer. Cool.

Watched lecture 12 of Abstract Algebra E-222

The well known eigenvalues and eigenvectors stuff. Near the end Gross tells about his favourite theorem: the Cayley Hamilton theorem which says that if you plug in the matrix itself into its characteristic equation then you end up with a zero-matrix.

What is six billion ?

Six billion is 6.000.000.000, six billion dollar is the amount that Obama on behalf of the US tax-payers is lending from foreign countres in order to keep the government going. Per day. Two trillion per year, 69444 per second. How long can this go on? The US must be bankrupt, but nobody wants to say it aloud.

Personal Brain 5.5 released

TheBrain.com released version 5.5 of Personal Brain. The update is free for version 5 users. Among 150 or so ( minor, but ) new features they have added ESP(!) to your brain. That is the brain monitors what you are doing in another application via clip- and/or keyboard and moves within the Plex accordingly. Cool feature. It, indeed, 'senses' what is required of the Brain. - Btw. I am now using PB consistently for Abstract Algebra.

Rotman

How time flies. Two and a half year ago I discovered Rotman's book on Group Theory. From what I understood of it I found it a fascinating book but I knew -and hated the fact- that I simply wasn't ready for it at the time. Wasn't I? I doubt if I have been studying hard enough. I am going to give the book another try.

As Rotman put it...

To the Reader
Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.

I think that's the trick to understanding his book.

As a matter of fact I already started. As Rotman wrote ( many ) are not difficult. The following exercise is an easy one if you studied Stirling numbers, if you didn't I wouldn't call it simple.

Exercise 1.5 ( in Rotman ). If 1 < r < n, then there are (1/r) [n(n — 1)... (n — r + 1)] r-cycles in Sn.
Why doesn't he use Stirling numbers ? I am going to need to review Stirling numbers.

Now I remember, I encountered this before and decided I had to study some Discrete Math first. Well, I did. So that won't hold me up studying this book. I think all the prerequisites are 'in' now for attacking this book.

Professors

If I would have seen only Gross in the Algebra videos I would have thought about him as a good lecturer, abstract type of guy but that is his profession. The fact that, as a contrast, we also see Peter learning the profession of lecturing, it becomes clear how difficult lecturing in fact must be. I mean Peter ( the 2003 version of him I bet he is huge by now ) is already impressive but no match to Gross, what a performance that was in lecture 11.

Watched lecture 11 of Abstract Algebra E-222

Basically the same stuff as in lecture 10 with some cool examples. And finally the well known A = B A B^-1 matrix conjugation for base conversion.

GL(2,2) is the general linear group of dimension 2 over GF(2). 
Gross asked which group is isomorphic to GL(2,2) ? 
( I stopped the video and gave it a try. )

GF(2) has the following tables for addition and multiplication.
+    0 1    x    0 1    
0    0 1    0    0 0
1    1 0    1    0 1

The possible maps from F2 -> F2 have the following matrices :
0 0    1 0    0 1    1 1
0 0    0 0    0 0     0 0

0 0    1 0    0 1    1 1
0 1    0 1    0 1    0 1

0 0    1 0    0 1    1 1
1 0    1 0    1 0    1 0

0 0    1 0    0 1    1 1
1 1    1 1    1 1     1 1

Elements of GL(2,2) are the matrices which have determinant 1.
1 0    1 1    0 1    1 1    1 0    0 1
0 1    0 1    1 0     1 0    1 1    1 1

Order    1    2    2    3    2    3

We now see that GL(2,2) is generated by 
0 1    1 1
1 0     1 0
and is isomorphic to S3.


Gross however had a different ( smarter ) approach as follows.
F2 is the following set:
{ (0,0), (1,0), (0,1), (1,1) }
A linear transformation from F2 to F2 must fix (0,0) 
so the elements of GL(2,2) are the permutations of 
(1,0), (0,1) and (1,1) with group S3.

Watched lecture 10 of Abstract Algebra E-222

A lecture by Peter on Linear Algebra. I scanned parts of the video, didn't really like it. Maybe its because I think I am done with this part of Linear Algebra,( have to go through it once more for M208, I suppose ) or its simply because I am interested mainly in groups at the moment.

Anyway, a very abstract and fast talk starting with the definition of a vector space all leading up to the change of basis problem. The change of basis problem is in itself not easy and takes a lot of practive to get comfortable with. A lot of material to go through in just one lecture. It took me months to fully understand these topics. Just to compare: Strang took about 10 lectures to cover this material.
See also: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm

Improving the study process

Studying is a process which can be improved continuously. There are tons of books written with advice on how to do this. I intend to try to improve my own way of studying. I have found several books on the subject, will browse them and select one or two which I will read through. But before I do that I want to define some key statistics which I can monitor so that I can measure the effects of changes I might implement. - In order to facilitate the monitoring I started to use Personal Brain as a study tool. - ( More later. )

P.S.

I had my life to live over again, I would have made a rule to read some poetry and listen to some music at least once a week; for perhaps the parts of my brain now atrophied would have thus been kept active through use. The loss of these tastes is a loss of happiness, and may possibly be injurious to the intellect, and more probably to the moral character, by enfeebling the emotional part of our nature.
—Charles Darwin

Watched lecture 9 of Abstract Algebra E-222

( Gross is Jewish. )

Nothing much really in this lecture.
Vector spaces.
Spans.
Linear combinations.
Dimensions.

Maybe a hint to a very important notion in mathematics: all vectorspaces of dimension n are isomorphic.

Peter on Monday.

Watched lecture 8 of Abstract Algebra E-222

The following topics are discussed in this lecture:

Isomorphism Theorem

Vector spaces over an arbitrary field
- Definition field
- Examples of finite fields

Proof that Z/pZ is a field
- Added to what we know from Z/nZ as an additive subgroup of Z we must prove that each a in Z/pZ has a multiplicative inverse, so we must show that if a is not a multiple of p then there is an integer b such that a*b congruent 1 mod p.
( Actual proof is worked out in the video )

What are the finite fields beyond Z/pZ ?
- the finite fields are of order p^n where p is a prime and n>=1, so there are finite fields of order 2, 3, 4, 5, 7, 8, 9, 11, etc. ( note 6 = 2*3, 10=2*5 not of type p^n )

Definition of a vector space ( V )
- Additive abelian group
- With a map f: VxF -> V which is called the scalar multiplication
- ( All rules are written down on board. )

Examples of vector spaces
- V={0}
- V=F
- V=F2
- V=Fn
- V=F[X], vector space of all polynomials p(x) with coefficients in F

Vector subspace
- A subgroup 'stable' under scalar multiplication

Vector space homomorphisms
- Linear transformations ( as we knew it ) are explained as group homomorphisms stable under scalar multiplication
- So for T: V-> W we can define the Kernel of T as a subspace of V and the image of T as a subspace of W. We can also define the quotient space V/W analog to the quotient group

Book of beauty: Visual Symmetry

Visual Symmetry by Magdolna and Istvan Margittai is a beautiful book with hundreds of pictures about symmetry. Aftrer reading this book you will look at the world through different eyes, forever. This book is also an excellent conpanion to M336 imo.

Watched lecture 7 of Abstract Algebra E-222

Gross formulated the following question: Can we put a group structure on the set of cosets {aH} for a subgroup H in G? He subsequently based the entire lecture on answering this ( simple ) question. The answer is ( of course ) yes if H is normal in G.

At the end, briefly for students with an interest in Algebraic Topology, Gross mentioned sequences like 1 -> H -> G -> G' -> 1.  With examples 1 -> Z3 -> Z6 -> Z2 -> 1 and 1 -> A3 -> S3 -> {1,-1} -> 1 which show that by knowing  Z3 ~ A3 and Z2 ~ {1,-1} does not say anything about the resultgroup.

Basicly a long abstract theoretical discussion about factorgroups, with basicly zero examples. How will group theory develop in people exposed to such lectures? I am not sure if I want to think about that.

The prototypical mathematician ( is NOT ).

A picture of the prototypical Linux type of person. As one gains experience in Linux one tends to start looking like the Ultimate Nerd. I wish I could understand why. Thank G_d, there are all sorts of mathematicians, maybe it's because mathematics in itself is so rich in subjects that there is no such thing as the prototypical mathematician. - I read somewhere that it is rather not done to say " I am a mathematician", the proper thing is " I studied math ", or something like that. One becomes a mathematician not before others ( in the field ) are calling you one. - I agree there is a difference between simply having done some math courses ( even if they add up to a B.Sc. or M.Sc ) and practicing mathematics at the research level.

Galois Theory

Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of  a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.

I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.

Update: the field we work in is Q.

Watched lecture 6 of Abstract Algebra E-222

( A lecture by Peter again ).
Arithmetic congruent mod n.
Addition
Multiplication
How a congruent b mod n is in fact an equivalence relation.
And thus induces a partition of the integers.
Cosets are nZ, 1+nZ, 2+nZ, ... (n-1)+nZ
Addition can be defined on these cosets and then they have a group structure.
The map Z -> nZ is then a homomorphism with 0 as kernel.

( Around min 35 or so I lost interest... I fast forwarded watching minutes here and there, just to make sure there was not introduced anything I did not know already. I hope I am not losing interest in the series all together. We'll see. )

Watched lecture 5 of Abstract Algebra E-222

Defines the equivalence relation on a set as a partition in disjoint subsets whose union is the set.
Properties of an equivalence relation:
- reflexive: a~a
- symmetric: a~b <=> b~a
- transitive: a~b and b~c => a~c.

A homomorphism f: G->H with kernel K which is a normal subgroup of G implies an equivalence relation on G where K is one of the equivalence classes. The other equivalence classes have the form aK = { ak; k in K, for some a in G}. aK is also called a left coset of K. ( Gross writes complete proof of this proposition on board. )
A bit of mathematical history about Lagrange ( born in Italy! ) who writes a letter to Euler at age 17 containing some very sophisticated mathematics. Euler immediately recognizes the genius of Lagrange and arranges further education for Lagrange who until that time learned his math through self-study.
(The famous) Theorem of Lagrange.
If G is a finite group and H is a subgroup of G then the order of H divides the order ( size ) of G.
More propositions are discussed.
- Groups of order p are simple.
- Groups of order p^2 are abelian.
- An is simple for n>=5.
- Any finite, non-abelian group has even order.

( Next lecture Peter. )

Watched lecture 4 of Abstract Algebra E-222

Definition of homomorphism.
Proof that e is mapped to e by any homomorphism.
Proof that inverses are mapped to inverses by any homomorphism.
Definition of Image.
Definition of Kernel.
Properties of the Kernel.
- subgroup;
- normal subgroup.
Any normal subgroup is the kernel of a homomorphism.
Example homomorphism.
f: GL(n,R) -> R_x
f(A) = det(A)
f has as kernel the matrices with det=1, also called SL(n,R). ( Special linear group )
Example homomorphism.
f: Sn->GL(n,R)
f(p)=Ap ( permutation matrix associated with p )
f( (1,2,3) ) = {{0,0,1}, {1,0,0}, {0,1,0} }
Definition center of G.
Example homomorphism G-> Aut(G) i.e. Klein4 -> S3

Watched lecture 3 of Abstract Algebra E-222

( Off-topic: Since I am ' in between jobs ', which happens if you are a freelance IT professional and there in an economic crisis, I should be studying new Oracle features or something like that. Instead I watched another Algebra lecture, well the day is still young. )

Watched lecture 3 of Abstract Algebra E-222. ( A lecture by Peter, Gross's assistant, if he isn't a professor yet, he will be soon, I suppose. )

Review of lectures 1 and 2.
- Groups. And examples of groups GL(n,R), Sn, Z+.
- Subgroups. Cyclic subgroups.
- ( Hom(Rn, Rn) has the structure of a vectorspace. )
- All subgroups of Zn are of the form bZ. ( Emphasis on importance of proof of this proposition.)
- ( Studying the subgroup structure of a group is in general very difficult. )
- Example of a cyclic subgroup of GL(2,R). The group generated by {{1 1}, {0,1}} is {{1 n}, {0,1}} n in Z.

Example of an isomorphism.
G1 = {i, -1, -i, 1}
G2 = {(1,2,3,4}, (1,3),(2,4), (1,4,3,2), ()}
G1 and G2 are isomorphic by i -> (1,2,3,4)
( Permutations are here in cyclic notation which are not introduced in the course yet. )

Example of an isomorphism.
G1 = {R,+}
G2 = {R\{0},*}
G1 and G2 are isomorphic by f: G1->G2; x |-> e^x
Proof:
f(x+y)=e^(x+y)=e^x * e^y = f(x)*f(y).

Klein4 group.
V={() , (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)} as a subgroup of S4.
V={ {{1 0}, {0,1}}, {{-1 0}, {0,1}}, {{1 0}, {0,-1}}, {{-1 0}, {0,-1}} as a subgroup of GL(2,R).

Definitions.
-Automorphism.
-Homomorphism.
-Image ( of a homomorphism)

Next lecture Gross on images of homomorphisms ( and more ).

( Thank you, Peter. )

MS221 course result ...

... will be available by the 18th december 2009. Why does that take so long. And even after the expiry dates of registering for new courses?

Watched lecture 2 of Abstract Algebra E222

Topics

Example of group: GL(n,R), invertible nXn matrices with elements a_ij taken from R.

Definition of a group
- operation is closed
- operation is associative
- group has identity element
- elements have inverses

Definition of S(n), group of all bijective maps f: S->S, the symmetry group of n elements, with as operation the composition of maps

Definition of a subgroup
- closed
- group has identity element
- elements have inverses

Examples
- S1
- S2
- S3. This example was very messy. Instead of correctly naming the elements e, s, s2, t, st, s2t he named them e, t, t', s, s' and s'', not clearly emphasizing that s' was in fact st and so on. Here he could have nicely drawn a Cayley Table but he didn't.

Definition of transposition as the exchange of only two elements. ( Very important concept )

Example of all the subgroups of Z.
- bZ, all multiples of an integer including {0}, so {+/-2, +/-4, ...}, {+/-3, +/-6,... } are all subgroups.

Proof that all subgroups of Z are of form bZ.
- bZ is a subgroup
- any subgroup is of type bZ
This last part was really excellent, he used the Euclidian division algorithm to complete this part of the proof.

Definition of cyclic subgroup. The smallest subgroup containing an element g. This is the collection {g,g^2,g^3,...}. This subgroup can be either finite or infinite.

Definition of the order of an element g of a group. The smallest positive integer e such that g^m=e.

Next time his colleague / assistent will be lecturing and he will start with Lagranges theorem about that the order of a subgroup is a divisor of the order of a group. So I'll expect cosets will be introduced as well.


Note:

Two surprising, remarkable comments from Benedict Gross of which I am not sure I agree:
- " You cannot learn too much Linear Algebra "
( I agree Linear Algebra is important and fun but imho it will never be able to grasp deep theorems in say Number Theory. I am aware of the importance of Linear Algebra in Group Theory especially Representation Theory )

- " I do not recommend writing out multiplication tables "
( Playing with Cayley Tables gave me definitely more insight in the structure of many groups, if you have Mathematica or GAP producing a Cayley table is not difficult. I doubt if Gross has actual experience with either of the two, or he hides it carefully until later. )

It is Artin...

... at least for as lang as I watch these Abstract Algebra video lectures.

( While browsing through some old blog entries and I stumbled upon this entry about Lang or Artin. )