Algebra I Randall R. Holmes Auburn University Copyright (cid:13)c 2008 by Randall R. Holmes Last revision: November 28, 2016 This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. Contents 0 Introduction 7 1 Definition of group and examples 9 1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Examples: Z, Q, R, C . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Nonexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Example: Integers modulo n . . . . . . . . . . . . . . . . . . . 10 1.5 Example: Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Example: General linear group . . . . . . . . . . . . . . . . . 10 1.7 Example: Symmetric group . . . . . . . . . . . . . . . . . . . 10 1.8 Example: Circle group . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Example: Group of nth roots of unity . . . . . . . . . . . . . 11 1 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Elementary notions 12 2.1 Multiplicative notation . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Abelian group . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Generalized associativity . . . . . . . . . . . . . . . . . . . . . 12 2.4 Unique identity, unique inverse . . . . . . . . . . . . . . . . . 13 2.5 Left and right cancellation . . . . . . . . . . . . . . . . . . . . 13 2.6 Properties of inverse . . . . . . . . . . . . . . . . . . . . . . . 13 2.7 Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.8 Power of element . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.9 Order of group, order of group element . . . . . . . . . . . . . 15 2.10 Direct product of two groups . . . . . . . . . . . . . . . . . . 15 2.11 Operation table . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.12 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.13 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Subgroup 20 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Proper subgroup, trivial subgroup . . . . . . . . . . . . . . . 20 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Subgroups of the symmetric group . . . . . . . . . . . . . . . 20 3.5 Subgroups of the general linear group . . . . . . . . . . . . . 21 3.6 Centralizer of a set; Center . . . . . . . . . . . . . . . . . . . 21 3.7 Intersection of subgroups is a subgroup . . . . . . . . . . . . . 21 1 3.8 Subgroup generated by a subset . . . . . . . . . . . . . . . . . 21 3.9 Cyclic subgroup generated by an element . . . . . . . . . . . 22 3 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Generators of a group 24 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 Example: General linear group . . . . . . . . . . . . . . . . . 24 4.3 Example: Dihedral group . . . . . . . . . . . . . . . . . . . . 25 4.4 Cyclic group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Coset 29 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.4 Equality of cosets . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.5 Congruence modulo a subgroup . . . . . . . . . . . . . . . . . 30 5.6 Cosets partition the group . . . . . . . . . . . . . . . . . . . . 31 5.7 Cosets have same cardinality . . . . . . . . . . . . . . . . . . 31 5.8 Index of subgroup . . . . . . . . . . . . . . . . . . . . . . . . 32 5.9 Lagrange’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 32 5.10 Corollaries of Lagrange’s theorem . . . . . . . . . . . . . . . . 33 5 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 Normal Subgroup 35 6.1 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Example: Conjugation and change of basis. . . . . . . . . . . 35 6.3 Definition of normal subgroup . . . . . . . . . . . . . . . . . . 36 6.4 Normalizer of a set . . . . . . . . . . . . . . . . . . . . . . . . 37 6.5 Product of subgroup and normal subgroup . . . . . . . . . . . 37 6.6 Index two subgroup is normal . . . . . . . . . . . . . . . . . . 38 6.7 Normality is not transitive . . . . . . . . . . . . . . . . . . . . 38 6 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 Quotient Group 40 7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.2 Example: Z/3Z . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.3 Quotient by commutator subgroup . . . . . . . . . . . . . . . 41 7.4 Simple group . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2 8 Homomorphism 43 8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.3 Elementary properties . . . . . . . . . . . . . . . . . . . . . . 44 8.4 Kernel and image . . . . . . . . . . . . . . . . . . . . . . . . . 44 8.5 Kernel same thing as normal subgroup . . . . . . . . . . . . . 45 8.6 Homomorphism is injective iff kernel is trivial . . . . . . . . . 45 8.7 Fundamental Homomorphism Theorem. . . . . . . . . . . . . 46 8 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 9 Isomorphism Theorems 48 9.1 First Isomorphism Theorem . . . . . . . . . . . . . . . . . . . 48 9.2 Example: Classification of cyclic groups . . . . . . . . . . . . 48 9.3 Quotient same thing as homomorphic image . . . . . . . . . . 49 9.4 Second Isomorphism Theorem . . . . . . . . . . . . . . . . . . 49 9.5 Third Isomorphism Theorem . . . . . . . . . . . . . . . . . . 50 9.6 Correspondence Theorem . . . . . . . . . . . . . . . . . . . . 50 9.7 Quotient is simple iff normal subgroup is maximal . . . . . . 52 9.8 Group with operator domain . . . . . . . . . . . . . . . . . . 53 9 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 10 Composition series 55 10.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10.2 Example: Z . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 10.3 Zassenhaus butterfly lemma . . . . . . . . . . . . . . . . . . . 56 10.4 Schreier refinement theorem . . . . . . . . . . . . . . . . . . . 56 10.5 Jordan-Holder theorem . . . . . . . . . . . . . . . . . . . . . . 57 10.6 Example: Fundamental theorem of arithmetic . . . . . . . . . 57 10.7 Classification of finite simple groups . . . . . . . . . . . . . . 58 10.8 Extension problem . . . . . . . . . . . . . . . . . . . . . . . . 59 10 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 11 Symmetric group of degree n 61 11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.2 Cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 11.3 Permutation is product of disjoint cycles . . . . . . . . . . . . 62 11.4 Permutation is product of transpositions . . . . . . . . . . . . 65 11.5 Even permutation, odd permutation . . . . . . . . . . . . . . 66 11.6 Alternating group . . . . . . . . . . . . . . . . . . . . . . . . 67 11.7 Conjugacy classes in the symmetric group . . . . . . . . . . . 68 3 11 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12 Group action 70 12.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.2 Example: Natural action of symmetric group . . . . . . . . . 70 12.3 Example: Left translation . . . . . . . . . . . . . . . . . . . . 70 12.4 Example: Conjugation . . . . . . . . . . . . . . . . . . . . . . 70 12.5 Permutation representation . . . . . . . . . . . . . . . . . . . 71 12.6 Cayley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 71 12.7 Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 12.8 Example: Orbits of a permutation . . . . . . . . . . . . . . . 73 12.9 Stabilizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.10Example: Coset is orbit . . . . . . . . . . . . . . . . . . . . . 74 12.11Class equation . . . . . . . . . . . . . . . . . . . . . . . . . . 74 12 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 13 p-Group 76 13.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 13.2 Fixed points of p-group action . . . . . . . . . . . . . . . . . 76 13.3 Center of nontrivial p-group is nontrivial . . . . . . . . . . . . 76 13.4 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 77 13.5 Element characterization of p-group . . . . . . . . . . . . . . 78 13.6 Normalizer of p-subgroup . . . . . . . . . . . . . . . . . . . . 78 13 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 14 Sylow theorems 81 14.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 14.2 Sylow existence theorem . . . . . . . . . . . . . . . . . . . . . 81 14.3 Sylow conjugacy theorem . . . . . . . . . . . . . . . . . . . . 82 14.4 Sylow number theorem . . . . . . . . . . . . . . . . . . . . . . 83 14.5 Example: Group of order 28 not simple . . . . . . . . . . . . 83 14.6 Alternating group is simple . . . . . . . . . . . . . . . . . . . 84 14 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 15 Category 86 15.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 15.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 15.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 15.4 Motivation for definition of product . . . . . . . . . . . . . . 89 15.5 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4 15.6 Coproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 15.7 Example: Coproduct of sets is disjoint union . . . . . . . . . 92 15 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 16 Direct product and direct sum 93 16.1 Definition: Direct product . . . . . . . . . . . . . . . . . . . . 93 16.2 Definition: Direct sum . . . . . . . . . . . . . . . . . . . . . . 94 16.3 Internal direct product/sum . . . . . . . . . . . . . . . . . . . 95 16.4 Internal direct product is isomorphic to direct product . . . . 96 16.5 Example: Vector space . . . . . . . . . . . . . . . . . . . . . . 97 16 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 17 Group decomposition 98 17.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 17.2 Example: Normal Sylow p-subgroups . . . . . . . . . . . . . . 98 17.3 Example: Finite cyclic group . . . . . . . . . . . . . . . . . . 99 17.4 Krull-Remak-Schmidt theorem . . . . . . . . . . . . . . . . . 100 17.5 Fundamental theorem of finite abelian groups . . . . . . . . . 100 17.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 18 Solvable group 103 18.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 18.2 Example: Dihedral group is solvable . . . . . . . . . . . . . . 103 18.3 Derived series . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 18.4 Subgroup and homomorphic image . . . . . . . . . . . . . . . 104 18.5 S not solvable for n ≥ 5. . . . . . . . . . . . . . . . . . . . . 105 n 18.6 A group of odd order is solvable. . . . . . . . . . . . . . . . . 106 18 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 19 Nilpotent group 107 19.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 19.2 Upper central series . . . . . . . . . . . . . . . . . . . . . . . 107 19.3 Lower central series . . . . . . . . . . . . . . . . . . . . . . . . 107 19.4 Upper/lower central series characterization of nilpotent . . . 108 19.5 Nilpotent group is solvable. . . . . . . . . . . . . . . . . . . . 109 19.6 Finite p-group is nilpotent . . . . . . . . . . . . . . . . . . . . 109 19.7 Subgroup and homomorphic image . . . . . . . . . . . . . . . 110 19.8 Product is nilpotent iff factors are . . . . . . . . . . . . . . . 110 19.9 Finite group nilpotent iff product of p-groups . . . . . . . . . 111 19 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 20 Free object of concrete category 113 20.1 Motivation for definition of free object . . . . . . . . . . . . . 113 20.2 Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 20.3 Examples of functors . . . . . . . . . . . . . . . . . . . . . . . 115 20.4 Free object . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 20.5 Free group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 20.6 Initial/Terminal object . . . . . . . . . . . . . . . . . . . . . . 118 20 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 21 Group presentation 122 21.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 21.2 Generators and relations . . . . . . . . . . . . . . . . . . . . . 122 21.3 Presentation of a group . . . . . . . . . . . . . . . . . . . . . 123 21.4 Von Dyck’s theorem . . . . . . . . . . . . . . . . . . . . . . . 123 21 – Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A Writing proofs 125 A.1 Strings of relations . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 If P, then Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.3 P if and only if Q . . . . . . . . . . . . . . . . . . . . . . . . 125 A.4 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.5 Showing “there exists” . . . . . . . . . . . . . . . . . . . . . . 126 A.6 Showing “for every” . . . . . . . . . . . . . . . . . . . . . . . 127 A.7 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . 127 A.8 Contrapositive . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.9 Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 A.10Variable scope . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6 0 Introduction Thiscourseisprimarilyacourseingrouptheory. Wewillstartfromscratch, even giving the definition and elementary properties of a group. However, the material in the undergraduate group theory course will be reviewed rather quickly and at times using an approach more sophisticated than that used in the first course. After the review, more advanced topics in group theory will be covered. If time permits, topics from module theory will be included. The course contains an introduction to category theory, which is playing anever-increasingroleinmathematicaldiscourse. Idonotintendtosuggest that category theory is a major focus of the course (it is not), but I do want totakeamomenttosayalittlebitaboutwhatitisincaseyouareunfamiliar with the idea. Your experience with mathematical constructs to date has most likely been confined to an inspection of internal structure. For instance, in group theory you have studied things like the order of an element, the cyclic sub- group generated by an element, the cosets of a subgroup, and so forth. In topology (or analysis) you have studied things like limit points, interiors of sets, boundaries of sets, least upper bounds, and so forth. In category theory, one looks at the bigger picture. Taking the case of groups,forexample,thegroupsthemselvesbecometheelementsasonesteps backandviewsthecollectionofallgroupsasanewmathematicalconstruct, a “category.” Information is obtained by studying the structure preserving maps (homomorphisms) running between the groups. The standard visu- alization is that of a directed graph, which is roughly an array of points together with various arrows joining the points. The points represent the groups and an arrow from one point to another represents a homomorphism between the corresponding groups. As a simple example of this new way of thinking, take the trivial group. Usingtheinternalviewpoint,onecharacterizesthetrivialgroupasthegroup having a single element (one has to look inside the group to see that it has only one element). In category theory, all groups look alike (they are all points) except for the array of arrows (homomorphisms) going out of them and coming into them. So how is one even to recognize the trivial group in the vast collection of all groups? A little reflection reveals that it is the only group with a single arrow going to each other group. (I have taken the 7 liberty of identifying isomorphic groups here.) One can form other categories as well, like the category of all rings, or the category of all topological spaces, or the category of all differentiable manifolds. Then one can step back even further and view these categories as points themselves with structure-preserving maps (functors) represented by arrows between them. For instance, each pointed topological space gives rise to a certain group, namely, its fundamental group at the distinguished point. Thiscorrespondencedefinesafunctorfromthecategoryofallpointed topological spaces to the category of all groups. By seeing how a mathematical construct interacts with other mathe- matical constructs through functors one gains insights beyond those made possible by an isolated study. 8 1 Definition of group and examples 1.1 Definition A group is a pair (G,∗), where G is a set and ∗ is a binary operation on G satisfying the following: (i) x∗(y∗z) = (x∗y)∗z for all x,y,z ∈ G, (ii) there exists e ∈ G such that x∗e = x and e∗x = x for all x ∈ G, (iii) for each x ∈ G there exists y ∈ G such that x∗y = e and y∗x = e. If (G,∗) is a group, we say that G is a group under ∗ (or just that G is a group, when the binary operation is clear from the context). Part (i) says that ∗ is associative. An element e satisfying (ii) is an identity element. An element y satisfying (iii) is an inverse of x. 1.2 Examples: Z, Q, R, C • Z,Q,R,andCareallgroupsunderaddition. Ineachcase,anidentity element is 0 and an inverse of x is −x. • Q×, R×, and C× are all groups under multiplication. (The symbol × signifies that the element 0 is omitted.) In each case, an identity element is 1 and an inverse of x is 1/x. 1.3 Nonexamples • Z, Q, R, and C are not groups under multiplication; the number 1 is an identity element (and the only candidate for such), but the element 0 has no inverse. • Z×, Q×, R×, and C× are not groups under addition. In fact, they are not even closed under addition since, for instance, they contain 1 and −1 but not the sum 1+(−1) = 0. • Z× is not a group under multiplication; the number 1 is an identity element (and the only candidate for such), but 2 has no inverse. 9