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GROUP THEORY

J.S. Milne

Abstract

These notes, which are a revision of those handed out during a course taughtto first-year graduate students, give a concise introduction to the theory ofgroups. They are intended to include exactly the material that every mathe-matician must know.

They are freely available at www.jmilne.org.Please send comments and corrections to me at math@jmilne.org.

v2.01 (August 21, 1996). First version on the web; 57 pages.

v2.1. (January 28, 2002). Fixed misprints; made many improvements tothe exposition; added an index, 80 exercises (30 with solutions), and an exam-ination; 86 pages.

Contents

Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiReferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

1 Basic Definitions 1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Groups of small order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Multiplication tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Normal subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Exercises 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Free Groups and Presentations 13

Free semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

0Copyright 1996, 2002. J.S. Milne. You may make one copy of these notes for your own personaluse.

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Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Finitely presented groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Exercises 512 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Isomorphism Theorems. Extensions. 21

Theorems concerning homomorphisms . . . . . . . . . . . . . . . . . . . . 21

Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Automorphisms of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Extensions of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

The Holder program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Exercises 1319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Groups Acting on Sets 34

General definitions and results . . . . . . . . . . . . . . . . . . . . . . . . . 34

Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

The Todd-Coxeter algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 46

Primitive actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Exercises 2033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 The Sylow Theorems; Applications 51

The Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Normal Series; Solvable and Nilpotent Groups 59

Normal Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Groups with operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Krull-Schmidt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Solutions to Exercises 72

B Review Problems 77

C Two-Hour Examination 82

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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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, Fp = Z/pZ = field of

p-elements, p a prime number.

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 (ai)iI, isa function i ai : I A.

Rings are required to have an identity element 1, and homomorphisms of ringsare required to take 1 to 1.

X Y X is a subset ofY (not necessarily proper).

Xdf= Y X is defined to be Y, or equals Y 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.