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510A Lecture Notes 2 Contents 510A Lecture Notes 2 Contents I Group theory5 1 Groups 7 1.1 Lecture 1: Basic notions...............................................8 1.2 Lecture 2: Symmetries and group actions...................................... 12 1.3 Lecture 3: Automorphisms, conjugation and normality............................... 15 1.4 Lecture 4: More on normal subgroups, and generators............................... 17 1.5 Lecture 5: Free groups................................................ 21 1.6 Lecture 6: Orders of elements............................................ 26 2 Abelian groups 27 2.1 Lecture 6 bis: Free and finite abelian groups.................................... 27 2.2 Lecture 7: Mostly finitely generated groups..................................... 30 2.3 Lecture 8: Infinitely generated groups continued: torsion and divisibility...................... 33 2.4 Lecture 9: Divisibility and Injectivity........................................ 37 3 Symmetric, alternating and general linear groups 41 3.1 Lecture 9 bis: symmetric groups........................................... 41 3.2 Lecture 10: Conjugacy in symmetric groups.................................... 43 3.3 Lecture 11: Alternating groups and general linear groups over finite fields..................... 47 4 Sylow’s theorems and applications 53 4.1 Lecture 12: The Sylow theorems.......................................... 54 4.2 Lecture 13: More on Sylow’s theorems and applications.............................. 57 4.3 Lecture 14: Group extensions............................................ 60 5 Extensions and cohomology 65 5.1 Lecture 15: Extensions continued.......................................... 66 5.2 Lecture 16: Extensions and group cohomology................................... 71 5.3 Lecture 17: Solvability and nilpotence....................................... 75 5.4 Lecture 18: Nilpotence and solvability continued.................................. 78 5.5 Lecture 19: the Jordan–Holder¨ theorem....................................... 81 6 More on simple groups 85 6.1 Lecture 20: Group actions and simplicity...................................... 86 6.2 Lecture 21: Simplicity of some group actions.................................... 90 II Fields and Galois theory 95 7 Basic theory of field extensions 97 7.1 Lecture 22: Preliminaries on Rings and Fields................................... 98 7.2 Lecture 23: Extensions, automorphisms and algebraicity.............................. 101 7.3 Lecture 24: Algebraic extensions and splitting................................... 104 7.4 Lecture 25: Splitting fields............................................. 106 7.5 Lecture 26: Finite fields............................................... 108 3 CONTENTS 4 8 Separable extensions 111 8.1 Lecture 27: The primitive element theorem and separability............................ 111 8.2 Lecture 28: Separability............................................... 114 8.3 Lecture 29: Separability, embeddings and the primitive element theorem...................... 116 8.4 Lecture 30: Separability and embeddings...................................... 118 9 Galois extension and the Galois correspondence 121 9.1 Lecture 31: Separability and Galois extensions................................... 122 9.2 Lecture 32: Normal and Galois extensions..................................... 124 9.3 Lecture 33: The Galois correspondence and examples............................... 127 10 Irreducibility and cyclotomic extensions 131 10.1 Lecture 34: More on irreducibility......................................... 132 10.2 Lecture 35: Cyclotomic extensions and abelian Galois groups........................... 134 10.3 Lecture 36: Cyclotomic and abelian extensions................................... 136 11 Classical applications 139 11.1 Lecture 36 bis: Solvability in radicals........................................ 139 11.2 Lecture 37: Classical applications continued.................................... 141 11.3 Lecture 38: Classical applications ctd........................................ 143 11.4 Lecture 39: Classical applications continued.................................... 145 12 Transcendental extensions 147 12.1 Transcendental extensions.............................................. 147 12.2 Transcendental extensions ctd............................................ 150 12.3 Luroth’s¨ theorem.................................................. 152 A Additional material 155 A.1 Category theory................................................... 156 A.2 Group cohomology and Schreier’s theory of extensions............................... 166 Part I Group theory 5 Chapter 1 Groups 7 1.1 Lecture 1: Basic notions 8 1.1 Lecture 1: Basic notions Before getting started, I wanted to make some comments about prerequisites. Here are some things that you should be familiar with. • Sets - I assume you have some working idea of the notion of a set and functions between them. Furthermore, I assume you know about basic operations that may be performed on sets: unions, intersections, Cartesian products, etc. • Groups - I assume you have seen the notion of a group before and know a few examples (cyclic groups, symmetric groups) • Arithmetic - I assume you know about basic facts from elementary number theory, e.g., gcd, divisi- bilty results for integers, etc. 1.1.1 Groups and homomorphisms I will present the definition of a group in a rather formal way. Definition 1.1.1.1. A quadruple (G; ·; (−)−1; e) consisting of a set G, a function · : G × G ! G (called multiplication), a function (−)−1 : G ! G (called inversion) and a distinguished element e 2 G is called a group, if the following diagrams commute: i) (Associativity) ·×id G × G × G / G × G id×· · · G × G / G: ii) (Units) e×id id×e G / G × G and G / G × G id id · · " # G; G: iii) (Inverses) (−)−1×id id×(−)−1 G / G × G and G / G × G e e · · " " G; G; where the function G ! G denoted e is the constant function with value e 2 G. If (G; ·; (−)−1; e) and (G0; ·; (−)−1; e0) are two groups, then a function f : G ! G0 is a group homomor- phism if f preserves multiplication, i.e., the diagram · G × G / G f×f f · G0 × G0 / G0 commutes. As mentioned in Appendix A.1.1, groups and group homomorphisms form a category Grp (objects are groups, morphisms are group homomorphisms). 9 1.1 Lecture 1: Basic notions Remark 1.1.1.2. The definition of group we have given is not “minimal”: the notion of group can be specified with less data. You can, instead, specify only a function m : G × G ! G corresponding, in the notation above, to (x; y) 7! xy−1 and reformulate the axioms of a group in terms of this function. Remark 1.1.1.3. For notational and linguistic convenience, we will speak of the group G instead of always carting around all the baggage of the quadruple. Nevertheless, it is important to remember that a group structure is additional data on a given set. Group objects in a cateogry The above definition of group has the benefit of being easily transplanted to other settings: it can be used in any context where one can make sense of “products”. For example, we may speak of group objects in a category C (see Definition A.1.1.1 for the definition of a category and basic properties. Example 1.1.1.4 (Topological groups). If X is a topological group, then we can talk about what it means to have a group structure compatible with the topology: repeat Definition 1.1.1.1 and simply replace G by a topological space X, e by a point of X, require that · : X × X ! X is a continuous function (we give X × X the product topology!) and require that ()−1 be a continuous map X ! X and impose all the axioms in the definition. Example 1.1.1.5 (Lie groups). Similarly, a “Lie group” is a manifold with compatible group structure: repeat Definition 1.1.1.1 and replace the set G by a smooth manifold M, e by a point of M, assume · is a smooth map of manifolds, and assume inversion is a smooth map of manifolds, once more assuming all additional axioms are satisfied. Example 1.1.1.6. In greatest generality, we may speak of “group objects in a category C ”. From the point of view of the above examples, we think of objects of our category equipped with a group structure compatible with whatever structure is present on objects. For example, a topological group is a group object in the category of groups. A Lie group is a group object in the category of smooth manifolds. Homomorphisms and isomorphisms Example 1.1.1.7. Any 1 element set has a group structure. If feg is the single element, multiplication is the unique function feg × feg 7! feg and inversion is given by the identity map on the set. We will write 1 for this group. This group structure is unique in a strong sense: given any other 1 element set, there is a unique bijection between the two 1 element sets and this bijection “preserves” all the additional structure. Moreover, if G is any group, there is a unique group homomorphism 1 ! G that sends 1 to e. Definition 1.1.1.8. A group homomorphism f : G ! G0 is an isomorphism (resp. monomorphism, epimorphism) if f is a bijection (resp. injection, surjection). A group homomorphism f : G ! G0 is called trivial, if the image of f is the identity e0 2 G0. Remark 1.1.1.9. Definition 1.1.1.8 is precisely the definition of isomorphism in a category (see Definition A.1.1.8) specialized to the category Grp. Lemma 1.1.1.10. If f : G ! G0 is an isomorphism of groups, then the inverse function f −1 is also an isomorphism. Proof. The inverse is a function f −1 : G0 ! G. We need to first show that f −1(e0) = e. To see this, write e0 = f(e). In that case, f −1(f(e))
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