Finite groups and fields [Lecture notes]

130 Pages·2008·0.639 MB·English

by  Jürgen Bierbrauer

Finite groups and fields
MA 5301
Fall 2008
Ju¨rgen Bierbrauer
December 1, 2008
2
Contents
I Introduction to Group Theory 7
1 Groups and symmetries 9
1.1 Definition of groups . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Groups of symmetry . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Group tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Permutations and the symmetric groups . . . . . . . . . . . . 16
1.5 Quasigroups and latin squares . . . . . . . . . . . . . . . . . . 18
2 Subgroups and cosets 23
2.1 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Cosets and Lagrange’s theorem . . . . . . . . . . . . . . . . . 25
3 Some basic number theory 27
3.1 Divisors and the Euclidean algorithm . . . . . . . . . . . . . . 27
3.2 Congruences and the cyclic groups . . . . . . . . . . . . . . . 29
4 Generators and isomorphisms 33
4.1 Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Direct products 35
5.1 Calculating in cyclic groups . . . . . . . . . . . . . . . . . . . 35
5.2 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Factor groups and homomorphisms 39
6.1 Normal subgroups and factor groups . . . . . . . . . . . . . . 39
6.2 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . 41
3
4 CONTENTS
7 permutation representations 43
7.1 The signum and alternating groups . . . . . . . . . . . . . . . 43
7.2 Permutation representations . . . . . . . . . . . . . . . . . . . 46
7.3 Orbits and the orbit lemma . . . . . . . . . . . . . . . . . . . 47
7.4 Equivalence and automorphisms . . . . . . . . . . . . . . . . . 49
8 Dihedral groups and graphs 51
8.1 The dihedral groups . . . . . . . . . . . . . . . . . . . . . . . 51
8.2 The cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9 Finite fields and linear groups 61
9.1 Prime fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.2 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
9.3 Linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.4 Projective geometries . . . . . . . . . . . . . . . . . . . . . . . 69
10 Automorphisms, conjugation 73
10.1 Group automorphisms and conjugation . . . . . . . . . . . . . 73
10.2 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.3 Characteristic subgroups . . . . . . . . . . . . . . . . . . . . . 75
10.4 Elementary-abelian groups . . . . . . . . . . . . . . . . . . . . 76
11 Permutation representations on subsets 77
11.1 The semidirect product . . . . . . . . . . . . . . . . . . . . . . 77
11.2 Permutation representations inside G . . . . . . . . . . . . . . 78
12 Conjugacy classes 81
12.1 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . 81
12.2 Products of subgroups . . . . . . . . . . . . . . . . . . . . . . 82
13 The Sylow theorems 85
13.1 p-groups and Sylow’s theorems . . . . . . . . . . . . . . . . . . 85
13.2 Proof of the Sylow theorems . . . . . . . . . . . . . . . . . . . 86
13.3 Existence of complements . . . . . . . . . . . . . . . . . . . . 87
13.4 The structure of abelian groups . . . . . . . . . . . . . . . . . 89
14 Simple groups, composition series 91
14.1 Simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
14.2 Composition series . . . . . . . . . . . . . . . . . . . . . . . . 93
CONTENTS 5
14.3 Solvable and nilpotent groups . . . . . . . . . . . . . . . . . . 94
15 Transfer and fusion 95
16 Permutation groups 101
II Classical groups 107
17 The linear groups 109
17.1 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
17.2 PGL (q) and the projective line . . . . . . . . . . . . . . . . . 111
2
17.3 The simplicity of PSL (q). . . . . . . . . . . . . . . . . . . . . 116
n
17.4 The uniqueness of GL (2). . . . . . . . . . . . . . . . . . . . . 119
3
18 Some classical groups 123
18.1 Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
18.2 The symplectic groups . . . . . . . . . . . . . . . . . . . . . . 124
18.3 The notion of a classical group . . . . . . . . . . . . . . . . . . 127
6 CONTENTS
Part I
Introduction to Group Theory
7

Chapter 1
Groups and symmetries
1.1 Definition of groups
We start with a formal definition.
1.1 Definition. Let G be a set and let a product operation : G×G −→ G
be defined. Then (G,·) is a group if the following hold:
• g (g g ) = (g g )g for all g ∈ G (associativity).
1 2 3 1 2 3 i
• There is an element e ∈ G (the neutral element), the letter e probably
stands for Einheit) such that
eg = ge = g for all g ∈ G.
• For every g ∈ G there is an element g−1 ∈ G (the inverse element)
such that gg−1 = g−1g = e.
´
The notion of a group goes back to Evariste Galois (1811-1832), a genius
and political revolutionary, who died before reaching his twenty-first birth-
day. For more on his life see Chapter 2 of Mark Ronan’s book Symmetry
and the Monster [10].
We start by drawing some elementary consequences from the axioms.
Observe that the neutral element is uniquely determined (if e′ is a neutral
element, thenby definition ee′ = e = e′). The letter eprobably stands for the
German word Einheit. We also write the neutral element as 1. The following
cancellation laws hold in any group.
9
10 CHAPTER 1. GROUPS AND SYMMETRIES
1.2 Theorem (cancellation laws). Let G be a group. Then the following
hold:
• If ax = ay, then x = y.
• If xa = ya, then x = y.
Proof. Assume ax = ay. Multiply by the inverse a−1 from the left and use
associativity:
x = a−1ax = a−1ay = y.
The same procedure works in the other case.
As a consequence we see that every element a has a uniquely deter-
mined inverse a−1 and that equations ax = b and xa = b have unique
solutions x (x = a−1b in the first case, x = ba−1 in the second case).
A group G is finite if the set G is finite. The cardinality of G is then called
the order of the group. A group G is commutative (or abelian) if ab = ba
for all a,b ∈ G. If ab = ba we also say that a and b commute. Here the
term abelian is in honour of the Norwegian mathematician Niels Henrik
Abel (1802-1829).
It is one objective of this course to show that all sorts of algebraic, combi-
natorial or geometric structures give rise to groups in a natural way. Group
theory helps understanding the situation in all these seemingly diverse cases.
Our first class of examples are groups of symmetry.
1.2 Groups of symmetry
As a toy example consider a rectangular playing card. The symmetry
group of the card is defined as the set of all permutations of the corners
A,B,C,D which have the property that the card looks alike before and after
the permutation is applied. Recall that a permutation of a set is a bijective
(onto and one-to-one) mapping of the set. There are three types of pairs of
corners in our card: those pairs connected by a long edge, those connected
by a short edge and those not connected by an edge. We can reformulate our
condition: a permutation of the corners {A,B,C,D} is a symmetry if and
only if
• the image of any long edge is a long edge,