Teaching
The book we will be using as reference for this course is Armstrong’s “Groups and Symmetry”
The lectures take place in the Minnaert room 211 from 13:00 to 15:00. The exercise classes take place in Ruppert rooms 102, 103, 104 and 109.
The final mark for this course is given by a complicated formula: Each quiz taken contributes with 4% of the final mark and the exam contributes with the rest, so, for example, if you solved three quizzes your final mark would be given by the
(quiz average) * 0.12 + (exam mark) * 0.88.
The exceptions to the rule are:
1) if you took all the 6 quizzes, the lowest quiz mark will be disregarded when computing the final mark and the quizzes will contribute with 20% of the final mark;
2) if your exam mark is higher than your quiz average, then your final mark is just the exam mark.
The final mark is rounded to the nearest half integer.
For those re-sitting the exam, the final mark is computed in the same way, but with the second exam mark in place of the exam mark.
If you feel that this does not make sense or that I miscomputed your final mark, please let me know.
Exercise sheets
Hand-in sheets
Group theory
Monday, 15 December 2008 (week 51)
This week we will prove a classification theorem for finitely generated Abelian groups. This material is covered in chapter 21 in Armstrong. There are no exercises for this subject. You should instead finish the exercises on Sylow groups from previous weeks.
Monday, 8 December 2008 (week 50)
This week Andre will take over the lecture. He will use Sylow’s Theorems to get information about the structure of finite groups, i. e., he will solve exercises using Sylow’s Theorems. Still solving all the problems in chapter 20.
Monday, 1 December 2008 (week 49)
This week we will study Sylow Theorems, a great improvement on Cauchy’s theorem. One of the applications being the classification of groups or order 12. This material is covered in chapter 20 in Armstrong. For this and next week’s exercises, you should try and solve all the problems in chapter 20.
Monday, 24 November 2008 (week 48)
This week we will see more on quotients. We will also study the homomorphism theorem, which states that given a group homomorphism f:G --> H, the quotient G/ker(f) is isomorphic to the image of f. This material is covered in Armstrong in chapter 16. Exercises for this week are from chapter 16: 1, 2, 3, 4, 7, 8, 11, 12.
Monday, 17 November 2008 (week 47)
This week we will study quotients of groups. More precisely, we will see that given a group G and a subgroup H, the product on G induces a product on G/H, the space of left cosets of H, precisely when H is a normal subgroup of G. This shows once again why normal subgroups are so important in group theory. This material is covered in Armstrong chapter 15. Exercises for this week are exercises 2, 3, 5, 6, 7, 8, 9, 13, 14, from chapter 15.
Monday, 10 November 2008 (week 46)
This lecture I made some comments about what to check when asked whether two given groups are isomorphic. Then I moved on to matrix groups, covered in chapter 9 in Armstrong. This week there is no exercise sheet. You should instead try and finish the previous exercise sheets including the extra exercises set to complement to book.
Monday, 27 October 2008 (week 44)
This lecture was given by Dr. Cranic. He recalled the effect of conjugation on permutations, showing that the cycle structure of a permutation is preserved by conjugation. Then he studied the alternating groups and sketched the proof that A4 is the only alternating group which has proper, nontrivial, normal subgroups. This material is covered in chapters 6, 8 and 14. After this, you can solve the exercise sheet 7.
Monday, 20 October 2008 (week 43)
This lecture I will talk about permutations (Armstrong chapter 6). I will show that several different sets generate the group of permutations. I will also show that there is a natural map from the group of permutations to Z_2 called parity and whose kernel are the even permutations. The even permutations are generated by the 3-cycles. We will also see that any group can be realized as a subgroup of a group of permutations. Finally, we will see the effect of conjugation on a permutation. After this you can try and solve the sixth exercise sheet.
Monday, 13 October 2008 (week 42)
In this lecture I recalled systematically properties of conjugation, including the facts that conjugation is an action of G on itself, that inner automorphisms are a normal subgroup of the group of automorphisms and that the adjoint map sending each element in G to the corresponding inner automorphism is a group map. Then I used the orbit stabilizer theorem to prove that the center of any group of order is a power of a prime is nontrivial. In particular a group of order p^2 is either Z_p x Z_p of Z_{p^2}. This material is covered in chapters 14 and 17. Now you can try and solve the following problems from Armstrong: 4.8, 12.10, 14.1, 14.2, 14.3, 14.7 and 14.10.
Monday, 6 October 2008 (week 41)
In this lecture I recalled again Lagrange’s Theorem, Cauchy’s theorem and the Orbit-Stabilizer theorem and then will gave examples of how to use these results to obtain results about structure of finite groups. One such result was a classification of finite groups of order less than 12. This week there was no exercise sheet. You should instead try and finish the exercises from previous weeks.
Monday, 29 September 2008 (week 40)
In this lecture I recalled the Orbit-Stabilizer Theorem, gave examples of action of groups and applications of these to produce results about structure of finite groups. This material is covered in chapter 17 in Armstrong. This week exercise sheet are the following exercises from chapter 17: 2, 4, 5, 6, 7, 9, 10, 12, 13 and 14.
Monday, 22 September 2008 (week 39)
In this lecture I presented proofs of the two most important results of this first part of the course: Lagrange’s theorem and Cauchy’s theorem. This material is covered in chapters 11, 12 and 13 in Armstrong. In the course of the proof of Cauchy’s theorem I also used the orbit--stabilizer theorem, a short proof of which can be found here. After this, you can solve the exercise sheet 3.
Monday, 15 September 2008 (week 38)
In this lecture I introduced the dihedral groups, one of which we met in the first lecture. Then I introduced the concepts of subgroups, normal subgroups, homomorphisms and isomorphisms. This is covered in chapters 4, 5, 7 and 10 in Armstrong. After covering this material, you can solve the exercise sheet 2.
Monday, 8 September 2008 (week 37)
In this lecture I defined groups and proved some of their basic properties such as uniqueness of the identity element and of inverses. Most of the lecture was used to provide examples of groups. The most familiar examples include Z, Q, R, C and vector spaces in general with addition as group operation. Also, Q\{0}, R\{0}, C\{0}, the unitary circle and nth roots of 1 provide further examples. These groups are all Abelian. More interesting examples where given by considering rotational symmetries of a tetrahedron and symmetries of hexagon. This is covered in chapters 1, 2 and 3 in Armstrong. Now, you can have a go at the first exercise sheet.
Retake marks and final course marks are out.
All of those wanting to collect their exams, hand-in sheets and quizzes please come to my office on Tuesday 20th of January from 14:00 to 16:00.
For those who failed, the re-take will be on monday 16th of march on Educ Alpha at 14:00 and the same advice given before on how to prepare for the exam still holds for the re-take: solve the exercise sheets and the hand-in sheets then try and solve the mock exam 1 and mock exam 2. You should also take a look at the questions of the exam.
The chapters from Armstrong whose content will be examined are 1 to 20, except 18 and 19. You can use results from chapter 21, covered in the last lecture, but there will be no questions to test your knowledge on that material.
Here are hints on how to solve or on how to find solutions to mock exam 1, mock exam 2 and the actual exam.
