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Group Theory – Fall 2014 GROUP THEORY – FALL 2014 THOMAS CREUTZIG ABSTRACT. These are lecture notes in progressfor Math 328 – Group Theory. We will focus on finite groups. The highlights of the course are Sylow’s Theory and the classification of finitely generated abelian groups. The lecture concludes with applications of group theory. December 1, 2014. 1 2 T CREUTZIG CONTENTS 1. Introduction 3 2. Basic Definitions and Properties 4 3. Subgroups 7 4. Cosets 9 5. Normal subgroups 11 6. Homomorphisms 12 7. Exercises and Vocabulary 15 8. The Symmetric Group 17 9. Isomorphism Theorems 22 10. Group Actions 26 11. Sylow’s theory 30 12. Solvable Groups 37 13. Abelian Groups 42 14. Applications 46 15. Practice Questions 51 References 53 MATH 328 3 1. INTRODUCTION Groups are a set endowed with an operation satisfying certain properties. They are in some sense the most fundamental and most important objects in algebra. This course is an an intro- duction to this topic. Prerequisites are a good knowledge of linear algebra. There are many excellent books on algebra, and they usually start with a few chapters on group theory. I will mainly follow the book by Jantzen, Schwermer [JS]. Other good references are the book by Artin [A] and by Goodman [G]. In my opinion [JS] is an excellent resource, unfortunately it is not available in English. The book by Goodman is available for free at http://www.math.uiowa.edu/ goodman/algebrabook.dir/algebrabook.html. The author asks for a (voluntary) donation to a charity organization. I will also use some results of online notes of Keith Conrad [Co] and Wolf Holzmann [H]. If you are interested in algebra and you plan to continue in this direction, then surely [A] is a good (but expensive) investment. However, this course will be entirely based on these lecture notes and I donot require you to buy any book. All exercises and exams will be based on material covered in these notes. Groups are fundamental in both nature and mathematics. They appear as automorphism group or symmetry group of a complicated systems. One very well-known example is quantum field theory or gauge theory. These theories often have a compact Lie group as symmetry, meaning that the fields describing fundamental particles like electrons, quarks or gluons carry an action of some Lie group. For the electron this group is called U(1) and for quarks and gluons it is called SU(3). MaPh 451 treats questions in that direction [Cr]. More examples are crystals, molecular structures and other symmetric materials. Since this is an algebra course, let me start with a relevant example. An algebra example are roots of unity. Take ξ = exp(2πi/p) a p-th root of unity for some prime number p. Every p-th root of unity is of the form ξ n for some (non-unique) positive integer n. Multiplying two such p-th roots of unity, we get another one ξ n · ξ m = ξ n+m. There is one relation, namely that thep-th power of a p-th root of unity is one, (ξ n)p = 1. So that the set of p-th roots of unity is 1,ξ ,ξ 2,...,ξ p−1 . This set is closed under multiplication, it has a multiplica tive identity namely the number 1, and every element ξ n has an inverse ξ p−n. This means that multiplication endows the set of p-th roots of unity with a group structure, called the (additive) group Z/pZ. Question for you: Why do you think it is called additive? Now, consider the map n mn σm : ξ 7→ ξ for positive integer m not divisible by p. Since ξ is a p-th root of unity, we have n (m+p)n mn mp mn n σm+p(ξ ) = ξ = ξ ξ = ξ = σm(ξ ) we see that the maps σm and σm+p coincide, so that the set of such maps is represented by σ1,σ2,...,σp−1 . 4 T CREUTZIG If we compose two such maps, we get n nm rmn n σr ◦ σm : ξ 7→ σr(ξ ) = ξ = σrm(ξ ). So that we get the composition law of maps σr ◦ σm = σrm and we remember that we compute ”modulo p” meaning that we have the relations σm+p = σm. The set of these maps forms another group, called the group of multiplicative units of Z/pZ and denoted by (Z/pZ)∗. Groups of this type play a central role in cyclotomic fields and field extensions. 2. BASIC DEFINITIONS AND PROPERTIES Let us start with basic definitions, some examples and proofs of properties. Definition 1. An assignment on a non empty set G is a map ◦ : G × G → G, (a,b) 7→ a ◦ b that assigns to each pair (a,b) of elements of the set G another element a ◦ b of G. A group G is a set G together with an assignment ◦ such that the following three properties hold: (1) The assignment ◦ is associative, that means that for every three elements a,b,c in G the identity (a ◦ b) ◦ c = a ◦ (b ◦ c) is true. (2) There is a unique element e in G such that e ◦ a = a ◦ e = a for all a in G. The element e is called the neutral element. (3) For every a in G there exists another element b in G with the property that a ◦ b = b ◦ a = e. The element b is called the inverse of a in G. A group G is called commutative or abelian if for all a,b in G a ◦ b = b ◦ a is true. The order of a group G is defined as the cardinality of the set G. Here are some examples that you probably know: Example 1. (1) The sets Z,Q,R,C of the integers, rational numbers, real numbers and complex num- bers are commutative groups with assignment the usual addition of numbers. The neu- tral element is 0 and the inverse of a is −a. (2) The sets Q∗ = Q \{0},R∗ = R \{0},C∗ = C \{0} are commutative groups with as- 1 signment the usual multiplication. The neutral element is 1 and the inverse to a is a . (3) Let M be a non-empty set, then the set SM of bijective maps M → M is a group with assignment the concatanation of maps. This group is called the symmetric group of M. MATH 328 5 The neutral element is the idenitity map. If M = {1,2,...,n} then we write SM = Sn and calls Sn the symmetric group of degree n. Its order is n!. (4) Let k be a field and V a k-vector space. Then the set GL(V ) of k-linear bijective maps from V to V is a group with assignment the concatanation of maps. The identity map is the neutral element. (5) Let k be a field. Theset Mn(k) of n × n matrices with entries in k has an assignment given by matrix multiplication. The subset GLn(k) := {A in Mn(k) |det(A) 6= 0 } of invertible matrices is a group. It is called the general linear group of degree n over k. (6) Let M be a non-empty set and G a group. Then the set of maps Map(M,G) from M to G is a group with assignment defined by f ◦ g : M → G, m 7→ f (m)g(m). Exercise 1. Are the groups of the previous example point (3) commutative? Give a counter example or prove your answer. Note, that the answer might depend on M. Solution. If M is a set consisting only of one or two elements, then SM is commutative. The reason is that a one element group consists of only the neutral element, while a two element group (2! = 2) consists of the neutral element e and one additional element a. Since a must have an inverse and e cannot be the inverse of a, the inverse of a must be a itself. Hence the group operations are e ◦ e = e, e ◦ a = a ◦ e = a, a ◦ a = e, so the group is commutative. If M has more than two elements, then choose three distinct elements m1,m2,m3 in M and consider a map σ that has the property m1 to itself and m2 to m3 and consider another map µ with the property that it maps m1 to m2. Then σ ◦ µ maps m1 to m3, while µ ◦ σ maps m1 to m2. Hence these maps donot commute and so SM cannot be commutative. We can relax the conditions in the definition of a group and still get a group. This goes as follows. Theorem 1. Let G be a non-empty set with an assignment ◦ : G × G → G, such that (1) The assignment ◦ is associative. (2) There is a left neutral element e in G, satisfying e ◦ a = a for all a in G. (3) For every a in G there exists another element b in G with the property that b ◦ a = e. The element b is called the left inverse of a in G. Then G is a group with neutral element e and the left inverse of any element a is also its inverse. Proof. Consider an arbitrary a in G. We want to show that e is the neutral element, that means we have to show that a ◦ e = a. Let b be the left inverse of a and c the left inverse of b. Then by associativity a = e ◦ a =(c ◦ b) ◦ a = c ◦ (b ◦ a) = c ◦ e.
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