Group Theory

1 Group theory Version 5: 22. January 2010 The purpose of this chapter is to present a number of important topics in the theory of groups. We have primarily chosen topics which are relevant to get a better understanding of finite groups, eg. to determine up to isomorphism all groups with certain given properties. This may be useful in later chapters to study Galois groups of finite normal field extensions. Most readers of this text may have already learned some basic facts about groups. Therefore some parts of this chapter is written in a rather compact form. A number of fairly elementary Exercises are included in the text. The text contains some references on the form (AT) or (AT; p. 17). They are references to the Lecture Notes by Anders Thorup: Algebra written in Danish. This is for the convenience of some readers. However most of the present text may be read independently of AT. The notes of Anders Thorup may be found here: http://www.math.ku.dk/noter/filer/alg12.pdf Changes from version 4: A new section on transfer has been added and also a number of other new results of various types. The section on permutation groups has been rewritten and extended. 1 Contents 1 Group theory 1.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . 1.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Powers and orders of elements . . . . . . . . . . . . . . . . . . . 1.4 Some subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Cosets and Lagrange’s theorem . . . . . . . . . . . . . . . . . . 1.6 Product of subgroups . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Normal subgroups and factor groups . . . . . . . . . . . . . . . 1.8 Homomorphisms, isomorphisms, automorphisms . . . . . . . . . 1.9 The homomorphism theorem. Noether’s isomorphism theorems . 1.10 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Conjugation. The class equation . . . . . . . . . . . . . . . . . . 1.12 Characteristic subgroups . . . . . . . . . . . . . . . . . . . . . . 1.13 The automorphism group of a cyclic group . . . . . . . . . . . . 1.14 Sylow’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Direct products. Abelian groups . . . . . . . . . . . . . . . . . . 1.16 Group actions and permutation groups . . . . . . . . . . . . . . 1.17 The transfer map . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Chains of subgroups, composition series . . . . . . . . . . . . . . 1.19 Higher commutator subgroups. Solvable groups . . . . . . . . . 1.20 Semidirect products of groups . . . . . . . . . . . . . . . . . . . 1.21 Groups of a given order . . . . . . . . . . . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 5 6 8 10 12 12 15 17 19 22 26 27 28 31 34 40 44 45 47 49 1.1 Definitions and examples Definition 1.1 A group is a nonempty set G on which there is defined a law of composition (sometimes also referred to as a binary operation.) This is a map ? : G × G → G, (a, b) 7→ a ? b, satisfying the following properties: (1) For all a, b, c ∈ G : (a ? b) ? c = a ? (b ? c). (Associativity of ?) (2) There exists e ∈ G, such that for all a ∈ G : a ? e = e ? a = a. (Existence of a unit/neutral/trivial element in G.) (3) For all a ∈ G there exists an element a0 ∈ G such that a ? a0 = a0 ? a = e. (Existence of inverse elements in G.) Remark 1.2 Clearly there is only one element e ∈ G satisfying condition (2). Indeed if e0 ∈ G satisfies e0 ? a = a for all a ∈ G, then in particular e = e0 ? e = e0 . The unique element e occuring in condition (2) is called the unit element of G. It is also sometimes referred to as the trivial or the neutral element. Also clearly the element a0 in conditi