Sunday, November 1, 2009
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. )
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. )
( While browsing through some old blog entries and I stumbled upon this entry about Lang or Artin. )
Saturday, October 31, 2009
Education in the US
MIT has it's OpenCourseWare program. Except for a few courses on video ( Linear Algebra and Differential Equations ) the entire ' openness ' is nothing more than publishing the syllabi and some handwritten lecture notes of students. - What a hoax. What is the big deal?
Today, I watched the first videolecture of Harvard's E-222 course on Abstract Algebra. The lectures are about the book Algebra by M. Artin. ( Well known to me, I self-studied it in 2007 ). Anyway Benedict Gross, the lecturer, was talking about other Harvard courses in numbers. I tried to look up the topics of these courses and then I found out that a login is required for that.
That's it! There is secrecy about the actual content of the mathematics programmes of the various universities. That is called capitalism. Competition among universities. It is also a way to hoax-up the so-called 'quality' of the educational material. They are asking ludicrous fees for reading out loud Artin's books. It's a system to introduce classes in society.
Everyone who read Ramanujan's life story probably agrees with me that Mathematics should be free and accessible to everyone on the planet wether studying in Cambridge, through the Open University or at home self-studying a copy of Artin's book downloaded from internet.
What I like about the US is, that it is home of Mathematica -AND- Sage, of Bush/Obama -AND- Ron Paul, and a zillion other contradictions.
Today, I watched the first videolecture of Harvard's E-222 course on Abstract Algebra. The lectures are about the book Algebra by M. Artin. ( Well known to me, I self-studied it in 2007 ). Anyway Benedict Gross, the lecturer, was talking about other Harvard courses in numbers. I tried to look up the topics of these courses and then I found out that a login is required for that.
That's it! There is secrecy about the actual content of the mathematics programmes of the various universities. That is called capitalism. Competition among universities. It is also a way to hoax-up the so-called 'quality' of the educational material. They are asking ludicrous fees for reading out loud Artin's books. It's a system to introduce classes in society.
Everyone who read Ramanujan's life story probably agrees with me that Mathematics should be free and accessible to everyone on the planet wether studying in Cambridge, through the Open University or at home self-studying a copy of Artin's book downloaded from internet.
What I like about the US is, that it is home of Mathematica -AND- Sage, of Bush/Obama -AND- Ron Paul, and a zillion other contradictions.
When closure is sufficient for a subset to be a subgroup.
While browsing Group Theory I ( Suzuki ) I noticed the following proposition.
If G is a finite group and S is a subset of G then closure in S suffices for S to be a subgroup.
Proof:
S is a subgroup if for all a,b,c in S
( i ) ab in S - closure
( ii ) a(bc)= (ab)c - associativity
( iii ) e in S - has identity
( iv ) a^(-1) in S - has inverse
let's prove them one by one:
( i) is proposed to be true ;
( ii ) is true for G and thus true for S ;
( iii ) since G is finite there is an integer n such that a^n = e thus e in S
( iv) since a a^(n-1) = e all a have an inverse.
[]
When in exercises the word 'finite' is added to group like 'G is a finite group ' we know that for G the group axioms are true like (i) to (iv) above AND that there is an integer n such that for all g in G g^n = e.
If G is a finite group and S is a subset of G then closure in S suffices for S to be a subgroup.
Proof:
S is a subgroup if for all a,b,c in S
( i ) ab in S - closure
( ii ) a(bc)= (ab)c - associativity
( iii ) e in S - has identity
( iv ) a^(-1) in S - has inverse
let's prove them one by one:
( i) is proposed to be true ;
( ii ) is true for G and thus true for S ;
( iii ) since G is finite there is an integer n such that a^n = e thus e in S
( iv) since a a^(n-1) = e all a have an inverse.
[]
When in exercises the word 'finite' is added to group like 'G is a finite group ' we know that for G the group axioms are true like (i) to (iv) above AND that there is an integer n such that for all g in G g^n = e.
Friday, October 30, 2009
Michio Suzuki
Michio Suzuki (1926-1998) was one of the 20th century pioneers of modern Group Theory. When I found his books Group Theory I and Group Theory II my first thought was that I had to wait a while before any books of him become accessible to me. I was pleasantly surprised that, unlike say Lang, his writings about Group Theory are written to explain Group Theory to the uninitiated, instead of documenting Group Theory for the experts. At least, that's my expression. So these are my preferred books on Group Theory for now.
Links:
- In memoriam M. Suzuki
- Suzuki, Group Theory I
Links:
- In memoriam M. Suzuki
- Suzuki, Group Theory I
Sunday, October 25, 2009
Skiena's CSE 547
Discrete Math Lectures on Video
10 years old - 1999 was clearly a small bandwidth area.
Nice to have anyway.
10 years old - 1999 was clearly a small bandwidth area.
Nice to have anyway.
Video lectures Abstract Algebra
I found a set of video lectures on Abstract Algebra.
MATH E-222 Abstract Algebra - http://www.extension.harvard.edu/openlearning/math222/Enjoy!
( Update 1-feb/'11: )
You might also like:
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis
MATH E-222 Abstract Algebra - http://www.extension.harvard.edu/openlearning/math222/Enjoy!
( Update 1-feb/'11: )
You might also like:
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis
Saturday, October 24, 2009
M336 - Groups and Geometry
M336 starts in feb 2010. 4TMA's and 1 examination, like MS221. The M336 course covers two related topics: groups and geometry and is delivered in 16 well known OU type of books including exercises, solutions, summaries and so on. The group theory-stream consists of the following:
Axioms and examples
Subgroups
Generating subgroups
Cyclic groups
Group actions
Group axioms
Subgroups and cosets
Normal subgroups and quotient groups
Isomorphisms and homomorphisms
Generators and relations
Equivalent colourings
Group actions
The counting lemma
The cycle index
Polya's enumeration formula.
Direct products
Abelian groups and groups of small orders
Cyclic groups
Subgroups and quotient groups of cyclic groups
Direct products of cyclic groups
Finitely presented abelian groups
The reduction algorithm
Existence and uniqueness of torsion coefficients and rank
Finitely generated abelian groups
Finite abelian groups
Subgroups of abelian groups
Permutation groups
Conjugacy - p-groups
Sylow p-subgroups
Sylow's first and second theorems
Sylow's third theorem
Applications of the Sylow theorems
Subgroups of prime power order
Review
Groups of order 2p
Groups of order 12
Where now?
Axioms and examples
Subgroups
Generating subgroups
Cyclic groups
Group actions
Group axioms
Subgroups and cosets
Normal subgroups and quotient groups
Isomorphisms and homomorphisms
Generators and relations
Equivalent colourings
Group actions
The counting lemma
The cycle index
Polya's enumeration formula.
Direct products
Abelian groups and groups of small orders
Cyclic groups
Subgroups and quotient groups of cyclic groups
Direct products of cyclic groups
Finitely presented abelian groups
The reduction algorithm
Existence and uniqueness of torsion coefficients and rank
Finitely generated abelian groups
Finite abelian groups
Subgroups of abelian groups
Permutation groups
Conjugacy - p-groups
Sylow p-subgroups
Sylow's first and second theorems
Sylow's third theorem
Applications of the Sylow theorems
Subgroups of prime power order
Review
Groups of order 2p
Groups of order 12
Where now?
Thursday, October 22, 2009
Plan for 2010.
OK people,
Just decided to do MST 209 first instead of M208. As far as possible from the Open University website I compared both courses. My main conclusion is that the pay off in "skills" is much higher from MST 209. Why start on the theoretic Real Analysis if your can't solve the basic differential equations? And MST 209 has to be done anyway. I can still add Groups and Symmetry to my schedule for next year.
Yep. As far as MST 209 is concerned I made up my mind.
Just decided to do MST 209 first instead of M208. As far as possible from the Open University website I compared both courses. My main conclusion is that the pay off in "skills" is much higher from MST 209. Why start on the theoretic Real Analysis if your can't solve the basic differential equations? And MST 209 has to be done anyway. I can still add Groups and Symmetry to my schedule for next year.
Yep. As far as MST 209 is concerned I made up my mind.
Tuesday, October 20, 2009
About MS221 exam (3)
A commenter asked if the TMA results count in any way.
Well, yes and no.
Your final result is the lowest of the two!
So a 100 for TMA's and a 40 for exam = 40.
so a 40 for TMA's and a 100 for exam = 40.
At the end of the day only the exam counts, that is if your TMA's were 85+.
Well, yes and no.
Your final result is the lowest of the two!
So a 100 for TMA's and a 40 for exam = 40.
so a 40 for TMA's and a 100 for exam = 40.
At the end of the day only the exam counts, that is if your TMA's were 85+.
Intermezzo
No courses for two / three months. How shall I spend the free study time, if any? I am currently self-studying the book Introduction to Analytic Number Theory by Tom M. Apostol, this book is used in Analytic Number Theory I ( M823 ) and Analytic Number Theory II ( M829 ). Lots of new subjects. I also have a problem book with exercises about the subject so that keeps me going.
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