Milne - Group Theory.. - Free

More documents

Recommendations

Info

i h Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Finitely presented groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Exercises 5–12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Isomorphism Theorems. Extensions. 21 Th eorems concerning omomorph isms . . . . . . . . . . . . . . . . . . . . 21 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Automorph isms of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Extensions of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 The Hölder program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Exercises 13–19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Groups Acting on Sets 34 General definitions and results . . . . . . . . . . . . . . . . . . . . . . . . . 34 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Th e Todd-Coxeter algorith m. . . . . . . . . . . . . . . . . . . . . . . . . . 46 Primitive actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Exercises 20–33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 The Sylow Theorems; Applications 51 Th e Sylow th eorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Normal Series; Solvable and Nilpotent Groups 59 Normal Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Groups with operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Krull-Sch midt th eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Furth er reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 A Solutions to Exercises 72 B Review Problems 77 C Two-Hour Examination 82 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
ii Notations. We use the standard (Bourbaki) notations: N = {0, 1, 2,...}, Z = ring of integers, R = field of real numbers, C = field of complex numbers, F p = Z/pZ =fieldof p-elements, p aprimenumber. Given an equivalence relation, [∗] denotes the equivalence class containing ∗. Throughout the notes, p is a prime number, i.e., p =2, 3, 5, 7, 11,.... Let I and A be sets. A family of elements of A indexed by I, denoted (a i ) i∈I ,is a function i ↦→ a i : I → A. Rings are required to have an identity element 1, and homomorphisms of rings are required to take 1 to 1. X ⊂ Y X is a subset of Y (not necessarily proper). X = df Y X is defined to be Y ,orequalsY by definition. X ≈ Y X is isomorphic to Y . X ∼ = Y X and Y are canonically isomorphic (or there is a given or unique isomorphism). References. Artin, M., Algebra, Prentice Hall, 1991. Dummit, D., and Foote, R.M., Abstract Algebra, Prentice Hall, 1991. Rotman, J.J., An Introduction to the Theory of Groups, Third Edition, Springer, 1995. Also, FT: Milne, J.S., Fields and Galois Theory, available at www.jmilne.org.
Page 1: GROUP THEORY J.S. Milne Abstract Th
Page 5 and 6: 1 1 Basic Definitions Definitions D
Page 7 and 8: Subgroups 3 group. For example GL n
Page 9 and 10: Groups of small order 5 and G can b
Page 11 and 12: Homomorphisms 7 Homomorphisms Defin
Page 13 and 14: Normal subgroups 9 Example 1.17. If
Page 15 and 16: Quotients 11 Quotients The kernel o
Page 17 and 18: 13 2 Free Groups and Presentations
Page 19 and 20: Free groups 15 We say two words w,
Page 21 and 22: Generators and relations 17 Lemma 2
Page 23 and 24: Finitely presented groups 19 group
Page 25 and 26: 21 3 Isomorphism Theorems. Extensio
Page 27 and 28: Direct products 23 (b) H 1 ∩ H 2
Page 29 and 30: Automorphisms of groups 25 and so w
Page 31 and 32: Semidirect products 27 Remark 3.12.
Page 33 and 34: Semidirect products 29 Write G = N
Page 35 and 36: Extensions of groups 31 Extensions
Page 37 and 38: Exercises 13-19 33 Part A has been
Page 39 and 40: General definitions and results 35
Page 45 and 46: Permutation groups 41 Permutation g
Page 47 and 48: Permutation groups 43 Note that the
Page 49 and 50: Permutation groups 45 Note that A 4
Page 51 and 52: The Todd-Coxeter algorithm. 47 The
Page 53 and 54:
Exercises 20-33 49 Proof. =⇒: The
Page 55 and 56:
51 5 The Sylow Theorems; Applicatio
Page 57 and 58:
The Sylow theorems 53 the highest p
Page 59 and 60:
Applications 55 (a) α(V i ) ⊂ V
Page 61 and 62:
Applications 57 Now assume that P i
Page 63 and 64:
59 6 Normal Series; Solvable and Ni
Page 65 and 66:
Solvable groups 61 Similarly G/H 1
Page 67 and 68:
Solvable groups 63 (b) Let N be a n
Page 69 and 70:
Nilpotent groups 65 Nilpotent group
Page 71 and 72:
Nilpotent groups 67 Corollary 6.16.
Page 73 and 74:
Groups with operators 69 Example 6.
Page 75 and 76:
Further reading 71 Theorem 6.29 (Kr
Page 77 and 78:
73 10. The hint gives ab 3 a −1 =
Page 79 and 80:
75 Deduce that #C G (α) = 3, and s
Page 81 and 82:
77 B Review Problems 34. Prove that
Page 83 and 84:
79 (e) Describe all subgroups of G
Page 85 and 86:
81 73. (a) Prove that subgroups A a
Page 87 and 88:
Solutions 83 Solutions 1. (a) False
Page 89 and 90:
Index action doubly transitive, 36
show all

Milne - Group Theory.. - Free

Create successful ePaper yourself

Delete template?

Save as template?