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
Tuesday, November 10, 2009
Book of beauty: Visual Symmetry
Saturday, November 7, 2009
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.
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.
Friday, November 6, 2009
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.
Wednesday, November 4, 2009
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. )
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. )
Tuesday, November 3, 2009
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. )
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. )
Monday, November 2, 2009
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
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. )
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. )
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. )
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)
