# Groups and Vector Spaces [Grupper og Vektorrum, MM515]

## Exam

The date for the oral exam is 16.6.2011, in U42 and U43. The interrogations are from 8.30 to 11.30 and from 12.00 to 15.00. Please show up half an our earlier for drawing a topic, that is: first candidate 8.00, seventh candidate 11.30 etc. Here is a list of exam topics [6.6.2011].

## A Course in Combinatorics

Here is is the course page!

Here are the slides of my presentation at the Pizza meeting from 5.5.2011. The last page indicates how you could proceed with your master studies if you are interested in writing a thesis in graph theory.

## Course Log / Time Table

Here is a protocol of the course's actual progression, including exercise sheets. All numerical citations except those to exercises refer to the 2nd edition of Hungerford's book mentioned below. Instead of looking up the few terms imported from ring theory (or on basic properties of the integers) in the book, you might want to use my Rings and Number Theory slides.

Week Events Content
14 (4.4.)* Mon 14-16 & Wed 12-14 U14

Section 7.1.

We discussed Theorems 7.1 and 7.2 plus Corollary 7.3, treated the permutation group SA of an arbitrary set A, and briefly repeated some facts on modular arithmetics from Chapter 2. I have put a more general group product than the one of Theorem 7.4 onto the exercise sheet. - Again recall that in the context of groups, many ordinary algebraic transformations you would apply to the reals or the complex numbers are simply not permitted by the axioms (for example, you cannot exchange the order of factors in general), or need a detailled proof (see the proof that the set of units of a ring is closed under ring multiplication). We discussed the Dihedral group Dn of degree n (for the explicit version given in the lecture, see also Exercise 1.11).

Section 7.2.

We have treated some basic properties of groups and introdued the order of an element of a group, including all conventions and rules concerned with the nth power of an element. (Theorem 7.5, Corollary 7.6, Theorem 7.7).

Fri 8-10 U24 Exercises: Sheet 1.
15 (11.4.) Mon 14-16 & Wed 12-14 U14

We looked at Theorem 7.8 and Corollar 7.9. The third part of Theorem 7.8, that the kth power of an element a of finite order is 1 if and only if the order of a divides k, is one of the most frequently used elementary facts in what comes up. Do not mix up the notion |a| for the order of a with absolute value notion of the reals or anything alike. Most books on the topic use other notions here (for example ord(a)). Try to find the typo in the book proof of the last part of 7.8.

Section 7.3.

We have introduced subgroups and proved two sufficient subgroup conditions, Theorems 7.10 and 7.11. Be aware that a subgroup of a group does not occur by taking a subset of the group and then make it a group by constructing some arbitrary group operation. The operation is always inherited here, not to be constructed. Therefore, not all possible subsets constitute subgroups of a group. We looked at the subgroup formed by those elements of the group which commute with every other element (Theorem 7.12), and discussed several basic properties of cyclic groups (Theorems 7.13, 7.14, 7.15, and 7.16). We have seen how arbitrary collections of group members generate subgroups (Theorem 7.17).

Section 7.4.

We discussed group homomorphisms and isomorphisms. Theorem 7.18 characterizes the finite groups ``up to isomorphism''.

Fri 8-10 U24 Exercises: Sheet 2.
16 (18.4.) Mon 14-16 U14 Bjarne took over the lecture of this week! He proved Cayley's Theorem (Theorem 7.20) stating that every group can be considered as a group of permutations (by means of isomorphism). Half way of the proof is prepared in Theorem 7.19, and Corollary 7.21 is an immediate consequence. He then introduced congruency, and, among the basic facts mentioned in the book, he proved Theorem 7.22. The main Theorem here is Lagrange's Theorem (Theorem 7.26) that the order of a subgroup of a finite group G divides the order of G. This immediately implies that the order of any element of such a G divides the order of G (Corollary 7.27). These statements provide already some structural inside into how a group of a given order looks like (up to isomorphism). For example, up to isomorphism, there is only one group whose order is a given prime number (Theorem 7.28), and there are just two groups of order 4 and 6, respectively. (Theorems 7.29 and 7.30). One of the biggest achievements in modern mathematics is the classification of finite simple groups, and we will touch a little bit of this exciting topic later in the course.

17 (25.4.) Wed 12-14 U14

Section 7.6.

We discussed the left counterpart of (right) congruence modulo a subgroup and right cosets (Theorems 7.31 and 7.32), and introduced normal subgroups. Theorem 7.34 contains various characterizations of being a normal subgroup. The most important feature of normal subgroups is resembled in Theorems 7.33 and 7.35: Right congruence modulo a normal subgroup N is a congruence relation with respect to group multiplication. Therefore, it is possible to...

Section 7.7.

...define a group operation on the right cosets of N, so that the product of two right cosets A,B is the right coset containing any product of a member of A and a member of B, the quotient group of the given group with respect to N. As one could expect, several features of the group inherit to this group (Theorem 7.36), and it gives yet another way of constructing new groups from given ones.

Fri 8-10 U24 Exercises: Sheet 3.
18 (2.5.) Mon 14-16 & Wed 12-14 U14

We discussed two examples on how properties of G, N, and G/N are related, Theorems 7.37 and 7.38. The second one is a good example of a bad example; it says that once the quotient G/Z(G) is cyclic then G is abelian - but if you know the centralizer Z(G) then you can answer the question if G is abelian immediately.

Section 7.8.

We discussed Theorems 7.39, 7.40, 7.41, and the First and Third Isomorphism Theorem (7.42 and 7.43), to get a characterization of the subgroups and normal subgrous of G/N in terms of the subgroups and normal subgroups of G (Theorem 7.44). We determined the simple abelian groups (Theorem 7.45).

Section 7.9.

We introduced cycle notation for permutations of {1,...,n}.

Fri 8-10 U24 Exercises: Sheet 4.
19 (9.5.) Mon 14-16 & Wed 12-14 U14 We discussed some ways of factorizing a permutation into cycles or transpositions (Theorem 7.46, Theorem 7.47, and Corollary 7.48). A permutation is even if it is the product of an even number of permutations and odd if it is the product of an odd number of transpositions. From Corollary 7.48 we know that every permutation is even or odd, and Theorem 7.50 tells us that a permutation cannot be even and odd. This follows immediately from the fact that the identity permutation is not odd (Lemma 7.49). Theorem 7.51 tells us that the set An of even permutations of {1,...,n} forms a normal subgroup of Sn of order |Sn|/2 = n!/2.

Section 7.10.

We proved Theorem 7.52 that An is simple if n is larger than 4 (via Lemma 7.53 and 7.54).

Section 8.1.

We discussed how to consider the factors of a product G of groups as normal subgroups and proved Lemma 8.2.

Fri 8-10 U24 Exercises: Sheet 5.
20 (16.5.) Mon 14-16 & Wed 12-14 U14

Monday: We finished Section 8.1 on internal products

Section 8.2.

We started to classify the finite abelian groups. The first step consisted of proving that every finite abelian group is the direct internal sum of its special p-subgroups (Theorem 8.5). The next step will be to further decompose p-groups.

Tue 8-10 U14 & Thu 8-10 U17 Exercises: Sheet 6.
21 (23.5.) Mon 14-16 & Wed 12-14 U14 TBA
Tue 8-10 U14 & Fri 8-10 U24 TBA

*: The date (dd.mm.) indicates monday of the respective week.

## Contents

This is an introductory course to abstract groups plus a little bit of theory of vector spaces.

## Literature

The course covers the material of some sections of the book

Abstract Algebra: An Introduction by T. W. Hungerford, 2nd edition, Saunders College Publishing (1997).

The plan is to deal with all of Chapter 7 and, if time permits, parts of Chapters 8, 10, and 16. The course relies on the same book as the course on Rings and Number Theory [Ringe og Talteori, MM510] of the 2nd quarter (2nd year).

Further books are in the MM515 slot in IMADA's library.

14.6.2011