Table Of ContentIntroduction to Group Theory and
Applications
Gruppentheorie mit Anwendungen in der
Kombinatorik
Ju(cid:127)rgen Bierbrauer
September 2, 2000
2
Contents
1 Introduction to Group Theory 5
1.1 De(cid:12)nition of groups . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Groups of symmetry . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Group tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Permutations and the symmetric groups . . . . . . . . . . . . 11
1.5 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Cosets and Lagrange's theorem . . . . . . . . . . . . . . . . . 15
1.7 Divisors and the Euclidean algorithm . . . . . . . . . . . . . . 16
1.8 Congruences and the cyclic groups . . . . . . . . . . . . . . . 19
1.9 Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.11 Calculating in cyclic groups . . . . . . . . . . . . . . . . . . . 21
1.12 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.13 Normal subgroups and factor groups . . . . . . . . . . . . . . 23
1.14 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . 25
1.15 The signum and alternating groups . . . . . . . . . . . . . . . 26
1.16 Permutation representations . . . . . . . . . . . . . . . . . . . 29
1.17 Orbits and the orbit lemma . . . . . . . . . . . . . . . . . . . 30
1.18 The dihedral groups . . . . . . . . . . . . . . . . . . . . . . . 32
1.19 The cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.20 Prime (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.21 Finite (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.22 Linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.23 Group automorphisms and conjugation . . . . . . . . . . . . . 46
1.24 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.25 Characteristic subgroups . . . . . . . . . . . . . . . . . . . . . 48
1.26 The semidirect product . . . . . . . . . . . . . . . . . . . . . . 48
3
4 CONTENTS
1.27 Permutation representations inside G . . . . . . . . . . . . . . 49
1.28 Conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.29 Products of subgroups . . . . . . . . . . . . . . . . . . . . . . 52
1.30 p-groups and Sylow's theorems . . . . . . . . . . . . . . . . . . 53
1.31 Proof of the Sylow theorems . . . . . . . . . . . . . . . . . . . 55
1.32 Simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.33 Composition series . . . . . . . . . . . . . . . . . . . . . . . . 57
1.34 Solvable and nilpotent groups . . . . . . . . . . . . . . . . . . 58
1.35 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2 Permutation groups 63
2.1 Normal subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 The classical simple groups 69
3.1 Scalar products . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 The symplectic groups . . . . . . . . . . . . . . . . . . . . . . 70
3.2.1 Hyperbolic planes . . . . . . . . . . . . . . . . . . . . . 70
3.2.2 Case n > 2 . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 The unitary groups . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4 The orthogonal groups in odd characteristic . . . . . . . . . . 74
3.5 Witt's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.1 Proof of Witt's theorem . . . . . . . . . . . . . . . . . 81
3.6 Symmetric scalar products in characteristic 2 . . . . . . . . . . 83
3.7 Quadratic forms in characteristic 2 . . . . . . . . . . . . . . . 84
3.7.1 The 3-dimensional case in characteristic 2 . . . . . . . 88
3.8 A table of groups, orders and isomorphisms . . . . . . . . . . 88
4 Polya's enumeration theory 91
4.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5 Block designs 97
5.1 Some easy direct constructions . . . . . . . . . . . . . . . . . . 99
5.2 Steiner triple systems . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Some easy recursive constructions . . . . . . . . . . . . . . . . 104
5.4 A link to permutation groups . . . . . . . . . . . . . . . . . . 106
5.5 The groups PGL2(q) . . . . . . . . . . . . . . . . . . . . . . . 106
5.5.1 Circle geometries . . . . . . . . . . . . . . . . . . . . . 108
CONTENTS 5
5.5.2 A class of 4-designs . . . . . . . . . . . . . . . . . . . . 109
5.6 The projective linear groups . . . . . . . . . . . . . . . . . . . 111
5.7 The simplicity of PSLn(q): . . . . . . . . . . . . . . . . . . . . 112
5.8 Quadrics and ovals . . . . . . . . . . . . . . . . . . . . . . . . 115
5.9 The large Witt designs . . . . . . . . . . . . . . . . . . . . . . 116
5.10 Golay code and Witt designs . . . . . . . . . . . . . . . . . . . 121
5.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A Transfer and fusion 125
B Transitive extensions, Mathieu groups 129
C The small Witt design 133
6 CONTENTS
Chapter 1
Introduction to Group Theory
1.1 De(cid:12)nition of groups
We start with a formal de(cid:12)nition.
1.1 De(cid:12)nition. Let G be a set and let a product operation : G(cid:2)G (cid:0)! G
be de(cid:12)ned. Then (G;(cid:1)) is a group if the following hold:
(cid:15) g1(g2g3) = (g1g2)g3 for all gi 2 G (associativity).
(cid:15) There is an element e 2 G (the neutral element) such that
eg = ge = g for all g 2 G:
(cid:0)1
(cid:15) For every g 2 G there is an element g 2 G ( the inverse element)
(cid:0)1 (cid:0)1
such that gg = g g = e:
0
Observe that the neutral element is uniquely determined (ife is a neutral
0 0
element, then by de(cid:12)nition ee = e = e). We also write the neutral element
as 1: A (cid:12)rst general result are the cancellation laws, which hold in any
group.
1.2 Theorem (cancellation laws). Let G be a group. Then the following
hold:
(cid:15) If ax = ay; then x = y
(cid:15) If xa = ya; then x = y:
7
8 CHAPTER 1. INTRODUCTION TO GROUP THEORY
A B
C D
Figure 1.1: The rectangular playing card
(cid:0)1
Proof. Assume ax = ay: Multiply by the inverse a from the left and use
associativity:
(cid:0)1 (cid:0)1
x = a ax = a ay = y:
The same procedure works in the other case.
As a consequence we see that every element a has a uniquely deter-
(cid:0)1
mined inverse a and that equations ax = b and xa = b have unique
(cid:0)1 (cid:0)1
solutions x (x = a b in the (cid:12)rst case, x = ba in the second case).
A group G is (cid:12)nite if the set G is (cid:12)nite. 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 2 G: If ab = ba we also say that a and b commute.
It is one objective of this course to show that allsorts 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 (cid:12)rst 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 de(cid:12)ned as the set of all permutations of the corners
1.2. GROUPS OF SYMMETRY 9
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 fA;B;C;Dg is a symmetry if and
only if
(cid:15) the image of any long edge is a long edge,
(cid:15) the image of any short edge is a short edge, and
(cid:15) the image of any non-edge is a non-edge
(cid:18) (cid:19)
A B C D
Letusconsiderthepermutation :Thisnotationmeans
A C D B
that the permutation maps A 7! A;B 7! C;C 7! D;D 7! B: Is this a sym-
metry of the rectangular playing card or not? No, because the pair fA;Bg of
corners (a long edge) is mapped to fA;Cg (a short edge). Let us determine
all symmetries of our card: one symmetry is always there, the permutation
(cid:18) (cid:19)
A B C D
e = ; which does nothing(the neutral element). Anything
A B C D
else? Geometric intuition will help: if we imagine a horizontal axis through
the centers of edges AC and BD and we rotate our card about that axis
(cid:18) (cid:19)
A B C D
(in 3-space), we obtain the symmetry a = : An analo-
C D A B
(cid:18) (cid:19)
A B C D
gous rotation about a vertical axis yields b = : Another
B A D C
idea is to re(cid:13)ect at the center of the card. This gives us the symmetry
(cid:18) (cid:19)
A B C D
c = : Is this all? Yes. It is not hard to convince ourselves
D C B A
that any symmetry is uniquely determined as soon as we know the image of
A: If for example A 7! D; then B; the unique partner forming a long edge
with A; must be mapped to C; the unique partner to form a long edge with
D; and so forth.
1.3 Theorem. The symmetry group of the rectangular card is the group
V = fe;a;b;cg of order 4.
10 CHAPTER 1. INTRODUCTION TO GROUP THEORY
Thisisthe(cid:12)rstgroupweactuallysaw, anditisaninterestinggroup. Why
is it a group and what is the group operation? Each symmetry is a mapping,
more speci(cid:12)cally a permutation. When two permutations are given, we can
form the composite function. If each of f and g is a symmetry, then also the
compositions f (cid:14)g and g(cid:14)f are symmetries. We can check the group axioms
from De(cid:12)nition 1.1. Associativity is obvious as composition of functions is
(cid:0)1
associative. The neutral element is the lazy permutation e; the inverse f
of permutation (symmetry) f is the inverse mapping.
1.4 De(cid:12)nition. We de(cid:12)ne the product of two permutations (on the same
set) as the composition of functions. Here we adopt the convention that fg
is the composite function, which is obtained by applying f (cid:12)rst and then g:
With this terminology we have for example ab = c (in words: if we apply
(cid:12)rst permutation a; then b; the result is permutation c:) More in general we
see, that the product of any two di(cid:11)erent non-neutral elements of V is the
third.
Here is another example, the quadratic card.
This time there are only edges and non-edges. A symmetry is a permu-
tation of the corners, which satis(cid:12)es
(cid:15) the image of any edge is an edge, and
(cid:15) the image of any non-edge is a non-edge
In fact, it su(cid:14)ces that the (cid:12)rst condition be satis(cid:12)ed. If the image of
every edge is an edge, then the non-edges will automatically be mapped to
non-edges. It is also clear, that the group V from Theorem 1.3 is contained
in the symmetry group of the quadratic card. Is that all? No, here is a
symmetry of the quadratic card, which is not a symmetry of the rectangular
(cid:18) (cid:19)
A B C D
card: u = : By now we know that the symmetries form
A C B D
a group, with the group product from De(cid:12)nition 1.4. As we already have
the symmetries from V; each new symmetry like u gives us at least four
new symmetries: u;v = ua;w = ub;x = uc: These must be di(cid:11)erent of
each other and of the elements of V; because of the cancellation laws (see
(cid:18) (cid:19) (cid:18) (cid:19)
A B C D A B C D
Theorem 1.2). We obtain v = ; w = ;
C A D B B D A C
(cid:18) (cid:19)
A B C D
x = : We have eight symmetries so far. This is all. In
D B C A
