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

## Exam

The date for the oral exam is 11.6.2012. We start 7.00. This refers to the time when the first candidate gets her question, her examination starts 7.30. The old set of exam questions carries over. The rules can be found there, too, and remain valid except for the dates.

## Course Log / Time Table

Here is a plan and a protocol of the course's actual progression, exercise sheets will be included. 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
15 (9.4.)* Tue 10-12 & Thu 12-14 U17

Section 7.1.

We introduced groups and abelian groups. We discussed Theorems 7.1 and 7.2, treated the permutation group SA of an arbitrary set A, and briefly repeated some facts on modular arithmetics from Chapter 2. I have put Corollary 7.3 and a more general group product than the one of Theorem 7.4 onto the exercise sheet. - 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 have 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 in both multiplicative and additive (abelian) notion. (Theorem 7.5, Corollary 7.6, Theorem 7.7, Theorem 7.8). 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.

Wed 12-14 U144 Exercises: Sheet 1.
16 (16.4.) Wed 10-12 & Thu 12-14 U17

Sections 7.2 (ctd.)

We briefly looked at Corollary 7.9.

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''. Theorem 7.19 lists some basic properties of homomorphism, proofs have been postponed until the respective first application. We proved Cayley's Theorem (Theorem 7.20) stating that every group can be considered as a group of permutations (by means of isomorphism). Corollary 7.21 is an immediate consequence.

Wed 12-14 U144 Exercises: Sheet 2.
17 (23.4.) Tue 10-12 & Thu 12-14 U17

Section 7.5.

We have introduced right congruency and its consequences (Theorems 7.22 and 7.23, Corollary 7.24, Theorem 7.25). 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 insight on 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).

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. 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 once you have determined the centralizer Z(G) then you can answer the question if G is abelian immediately.

Section 7.8.

We discussed Theorem 7.39.

Wed 12-14 U144 Exercises: Sheet 3.
18 (30.4.) Tue 10-12 & Wed 10-12 U17

Section 7.8 (ctd.)

We continued with Theorems 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} and 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.

Wed 12-14 U144 Exercises: Sheet 4.
19 (7.5.) Tue 10-12 & Thu 12-14 U17

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, proved Lemma 8.2, and finished Section 8.1 on internal products.

Section 8.2.

We started to classify the finite abelian groups (Theorem 8.7). The first step consists of proving that every finite abelian group G is the direct internal sum of its special p-subgroups (Theorem 8.5).

Wed 12-14 U144 Exercises: Sheet 5.
20 (14.5.) Tue 10-12 & Wed 10-12 U17

Section 8.2 (ctd.)

By induction we proved that every finite abelian p-group is the direct internal sum of certain cyclic subgroups (Lemma 8.6 lets the induction step work), so that Theorem 8.7 follows. The (orders of the) cyclic groups found along these lines are the elementary divisors; they are uniquely determined by G (up to order, see Theorem 8.12; proof omitted), and provide the longest factorization. By grouping these factors as in Theorem 8.10, we get the invariant factors of G, which provide the shortest factorization.

Section 8.3.

We have introduced all three Sylow theorems (8.13,8.15,8.17). As applications, we have proved Cauchy's Theorem (8.14), and Corollary 8.16.

Wed 12-14 U144 Exercises: Whatever remains from the previous sheets.
21 (21.5.) Tue 10-12 & Thu 12-14 U17

Section 8.4.

We have looked at groups whose order admits a short prime factor decomposition, and proved Cauchy's Theorem for the case of abelian groups by applying the Fundamental Theorem on finite abelian groups. We have introduced conjugacy relations on both the elements of a group and its subgroups, and proved class equations, respectively, which then led immediately to the proofs of the Sylow Theorems.

Wed 12-14 U144 Exercises: Sheet 6.

*: 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 schedule up there has been planned according to the course log of six weeks of last year's MM515, and it will be aligned based on the actual progress. 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.

Course language is english.

31.5.2012