Exercise sheets and other homework

mock exam 2

exercise sheet 12

exercise sheet 11

exercise sheet 10

exercise sheet 9

exercise sheet 8

exam 1

mock exam 1

exercise sheet 7

exercise sheet 6

exercise sheet 5

exercise sheet 4

exercise sheet 3

exercise sheet 2

exercise sheet 1

Group theory

Announcements

Quiz dates: quizzes will take place on 29/09, 13/10, 27/10, 17/11, 01/12 and 15/12.

Here is  a solution to the first exam.

Here is a solution to the second mock exam.

Final marks after retake are available now.

The 2nd Exam will take place on the 18th of  January from 14:00 to 17:00 in the Educatorium Gamma.

Chapters covered up to now: 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17 and 20.

Monday, January 18, 2010 (week 3)

2nd Exam today. Chapters included: 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17 and 20.

Monday, December 14, 2009 (week 51)

We studied matrix groups further. We introduced the unitary, special unitary and symplectic groups. We finished the lecture with the proof that Sp(1) = SU(2) and Sp(1)/Z2 = SO(3).

Monday, December 7, 2009 (week 50)

Today we started our study of matrix groups. This material is covered in chapter 9.

Monday, November 30, 2009 (week 49)

In this lecture we solved some more exercises using Sylow’s theorems and went through part of the proof of Sylow’s theorems. Please read the rest of the proof in the book, as we will not finish that in lectures.

Monday, November 23, 2009 (week 48)

In this lecture we studied applications of Sylow’s theorems. The exercise sheet for this and next week are all the exercises from chapter 20.

Monday, November 16, 2009 (week 47)

In this lecture we studied the homomorphism theorem and saw some of its applications. This material is covered in Chapter 16.

Monday, November 9, 2009 (week 46)

In this lecture we studied quotient groups and saw some examples. We took a particular look at the sequence of groups obtained by the operation “quotient by the center”. This material is covered in Chapter 15.

Monday, November 2, 2009 (week 45)

1st Exam today. Chapters included: 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14 and 17.

Monday, October 26, 2009 (week 44)

In this lecture we saw how to use Cauchy’s, Lagrange’s and the Orbit--Stabilizer theorem to obtain results about structure of groups. Precisely, as “exercises”  we saw that

1. 1)A group of order 4n+2 has a subgroup of order 2n+1 and hence is not simple for n > 0;
   

2. 2)A group of order pn for some prime p has nontrivial center;
   

3. 3)A group of order p2 is isomorphic to either Zp x Zp or Zp2.
   

Finally we used these results and results from last class to classify all the groups of order less than 12.

This is material from chapters 11, 13 and 17.

Monday, October 19, 2009 (week 43)

In the first half of this lecture we solved some exercises on the question “isomorphic or not”, saw more applications of Lagrange’s theorem and covered some loose ends from previous chapters. Notably, we proved that if a subgroup has index two, then it is normal. We also proved some basic statements about simple groups. In the second half proved Cauchy’s theorem and, as an application of Cauchy and Lagrange, we showed that if a group G has order 2p, with p a prime greater than 2, then G is isomorphic to either Z2p or Dp. This is covered in chapter 13.

Monday, October 12, 2009 (week 42)

In this lecture we dealt with products and then moved on to learn Lagrange’s theorem and its twin statement for group actions: the Orbit--Stabilizer theorem. Then we covered some basic applications of Lagrange’s theorem. This is covered in chapters 10, 11 and 17.

Monday, October 5, 2009 (week 41)

In this lecture we learnt a bit more about actions, including the definitions of the orbit of a point and the stabilizer of a point. Then we studied the alternating group (group of even permutations) which was initially introduced as the stabilizer of a certain polynomial and then we saw that it consisted of the permutations which can be written as a product of an even number of transpositions. We finished the lecture with some important definitions: group homomorphisms and group automorphisms and normal subgroups. We saw that the kernel of a group homomorphism is always a normal subgroup of the domain. This is covered in chapter 6, and is highly related to material in chapter 7.

Monday, September 28, 2009 (week 40)

In the first half of the lecture we 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 behaved well with respect to products on G, i.e.,

Adg(h1h2) = Adg(h1)Adg(h2)

later in this course we will see that this means Adg is a group homomorphism.

In the second part of the lecture we studied permutation groups. 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. The material about the symmetric group os covered in chapter 6 in Armstrong.

Monday, September 21, 2009 (week 39)

In this lecture we were introduced to the groups of symmetries of a regular n-gon, the dihedral group, and studied further 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 finished showing that a subgroup of a cyclic group is cyclic. Up to now we have covered chapters 1 to 5 of Armstrong.

Monday, 14 September 2009 (week 38)

In this lecture I recalled the notion of groups together with a few basic examples which you might have seen before, such as Z, Q, R, C and vector spaces with addition as group operation. Also, Q\{0}, R\{0}, C\{0} and H\{0} (the quaternions) with multiplication as group operation. We came across the concept of subgroups and showed that the spheres S^1 and S^3 are subgroups of C\{0\} and of H\{0}. We also saw that de set of integers modulo n, Z_n, has two natural operations on it, addition and multiplication and that (Z_n,+) is an Abelian group, while (Z_n\{0},.) is not a group in general. In the exercise sheet, you should prove that the latter is a group iff n is prime.

Monday, 7 September 2009 (week 37)

This lecture was given by Prof. Andre Henriques. He 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. He showed how groups appear as the set of symmetries of a set (maybe endowed with extra structure) and computed such groups of symmetry for some planar figures as well as the cube and the tetrahedron. The concepts of order of an element and order of the group were introduced in this lecture. In this Lecture you  also encountered the notion of Abelian, or commutative, group.