Chapters covered up to now: 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17 and 20.
Monday, 13 December 2010 (week 50)
This lecture we will finish matrix groups. we will introduce the concept of quaternionic structure on a vector space and of a compatible inner product. We will study the group of transformations which preserve a quaternionic structure, Gl(n,H), and group of quaternionic transformations which preserve a quaternionic inner product, Sp(n). We will finish the lecture proving that Sp(1) = S^3 = SU(2) and that there is a group homomorphism Sp(1) ---> SO(3), whose kernel is Z2. This is covered in chapters 9, 15 and 16.
Monday, 6 December 2010 (week 49)
This lecture we will studied the orthogonal matrix groups O(n) and SO(n). We saw that GL(n,R) acts on the set of metrics on vector spaces and that this action is transitive (which is just another way of saying that every metric admits an orthonormal basis and hence is isometric to R^n with the Euclidean metric) and hence used the Orbit-Stabilizer theorem to conclude that the space of metrics on R^n is given by GL(n,R)/O(n).After that we introduced the concept of a complex structure I on a vector space V and defined the complex linear transformation as the transformation in GL(V) which commute with I. For C^n, this group os denoted by GL(n,C). Again, the Orbit-Stabilizer theorem tells us that the space of complex structures on R^{2n} is given by GL(2n,R)/GL(n,C). Finally we introduced the concept of Hermitian metric on a complex vector space and defined the unitary groups U(n) and SU(n). We finished the lecture noticing that U(1) = S^1 = SO(2). This is covered in Chapters 9, 15 and 16.
Monday, 29 November 2010 (week 48)
This lecture we continued our study of the Sylow theorems by giving a proof of the Theorem. This is covered in chapter 20.
Monday, 22 November 2010 (week 47)
This lecture we saw how to use Sylow’s theorem and actions to solve exercises. We saw that given a finite group G and a subgroup H of order p^k, there is a p-Sylow subgroup of G which contains H. We also saw that if the index of a subgroup H is the smallest prime dividing the order of G, then H is normal. Finally we gave two examples of solutions to the question “Show that G is not simple”. In the first case we used a counting argument and in the second we used the action of G on its p-Sylows (for a fixed prime p) to produce a group homomorphism with nontrivial kernel. This is covered in Chapter 20.
Monday, 15 November 2010 (week 46)
This lecture we covered quotients and the homomorphism theorem(s). We showed that if H is a subgroup of G, then G/H is a group and the projection map p:G --> G/H a group homomorphism if and only if H is a normal subgroup. We saw examples where H was not a normal subgroup and how G/H failed to inherit a group structure in that case. We also saw that D6 divided by its center is isomorphic to D3. or more generally, that D2n divided by its center is isomorphic to Dn. Finally we covered the isomorphism theorem and used it to conclude that R/Z is isomorphic to the circle and Z/nZ is isomorphic to Zn. We finished seeing how the second isomorphism theorem follows from the first.This is covered in chapters 15 and 16.
Monday, 8 November 2010 (week 45)
First exam
Monday, 1 November 2010 (week 44)
In this lecture we studied more applications of Lagrange, Cauchy and Orbit-Stabilizer theorem. Those showed basic techniques which you should use to solve exercises in the exam.
Monday, 25 October 2010 (week 43)
This week we studied the Orbit-Stabilizer Theorem and Cauchy’s theorem and saw some of their applications. This material is covered in chapters 13 and 17.
Monday, 18 October 2010 (week 42)
This week we finished our study of permutation groups by showing that the alternating group is a normal subgroup of the permutation group and its order is half of the order of the permutation group. Afterwards we moved on to prove Lagrange’s theorem and study its applications. This material is covered in chapters 6 and 11.
Monday, 11 October 2010 (week 41)
In this lecture we studied permutation groups. We started introducing the cycle notation for a permutation. Then we saw the effect conjugation has on permutations and concluded that conjugation preserves the cycle structure of a permutation. We also discovered different sets of generators of the symmetric group, e.g., we proved that the transpositions generate Sn and that the set {(1 2), (1 2 3 ... n)} is also a generating set for Sn. Then we introduced an action of Sn on the set of polynomial of n variables and by picking a specific polynomial we obtained and action of Sn on a set with two elements. The kernel of this action consists of the even permutations, i.e., the permutations which can be obtained be a product of an even number of transpositions. This is the alternating group, An. This is covered in Chapter 6.
Monday, 4 October 2010 (week 40)
After introducing the concepts of homomorphism and isomorphism last lecture, we introduced the concept of group automorphism and showed that the set of group automorphisms is a group under function composition. Next we introduced the concept of group action and studied two examples of actions which one can consider whenever one is given a group G. The first was the left action of G on itself and the second was the adjoint action (i.e. conjugation) of G on itself. We also saw that the adjoint action of an element g in G was in fact a group automorphism and that the map G -- > Aut(G) given by the adjoint action is in fact a group homomorphism.
Monday, 27 September 2010 (week 39)
In this lecture we continued our study of subgroups. We defined the center of a group and saw that the center is always an Abelian subgroup. We also studied subgroups generated by a collections of elements of the group. If the subgroup generated by a collection of elements is the whole group, then that collection is a set of generators for the group. A group generated by a single element is called cyclic. We showed that a subgroup of a cyclic group is cyclic. Then we started our study of maps between groups. We defined homomorphisms and isomorphisms and showed that the kernel and the image of a group homomorphism are subgroups of the domain and codomain, respectively. We studied some examples.
Monday, 20 September 2010 (week 38)
We recalled the definition of group and went on to see a few more examples, namely the symmetries of the tetrahedron, “the integers modulo n”, and complex roots of 1. We also saw the concept of subgroup and saw that the circle, S^1, and the 3-dimensional sphere, S^3, admit a group structure.
Monday, 13 September 2010 (week 37)
We saw how groups appear as the set of symmetries of a set (maybe endowed with extra structure, for example, the structure of a vector space). Then we defined groups and proved some of their basic properties such as uniqueness of the identity element and of inverses. In the second half of the lecture we went through a bunch of examples: Z, Q, R, C and any vector space with addition as group operation are groups. So are Q\{0}, R\{0}, C\{0} and H\{0} (the quaternions) multiplication as group operation. The latter provided an example were the group operation was not commutative. We recalled from the first part that the bijections of a set for a group under composition and so do the set of invertible linear transformations of a vector space. For R^n, this group is called the “General Linear Group” and is denoted by Gl(n,R). We finished computing the group of symmetries (meaning rigid motions) of a regular hexagon.
