Notes on Algebra
Donu Arapura
December 5, 2017
,Contents
1 The idea of a group 3
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The group of permutations 11
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Rotations and reflections in the plane 15
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Cyclic groups and dihedral groups 19
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Finite sets, counting and group theory 24
5.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 More counting problems with groups 29
6.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7 Kernels and quotients 36
7.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Rings and modular arithmetic 40
8.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9 Z∗p is cyclic 45
9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10 Matrices over Zp 49
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11 The sign of a permutation 52
11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
12 Determinants 56
12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1
,13 The 3 dimensional rotation group 60
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
14 Finite subgroups of the rotation group 64
14.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
15 Quaternions 69
15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
16 The Spin group 73
16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2
, Chapter 1
The idea of a group
One of our goals in this class is to make precise the idea of symmetry, which is
important in math, other parts of science, and art. Something like a square has
a lot of symmetry, but circle has even more. But what does this mean? One
way of expressing this is to a view a symmetry of a given shape as a motion
which takes the shape to itself. Let us start with the example of an equilateral
triangle with vertices labelled by 1, 2, 3.
3
1 2
We want to describe all the symmetries, which are the motions (both rota-
tions and flips) which takes the triangle to itself. First of all, we can do nothing.
We call this I, which stands for identity. In terms of the vertices, I sends 1 → 1,
2 → 2 and 3 → 3. We can rotate once counterclockwise.
R+ : 1 → 2 → 3 → 1.
We can rotate once clockwise
R− : 1 → 3 → 2 → 1.
We can also flip it in various ways
F12 : 1 → 2, 2 → 1, 3 fixed
F13 : 1 → 3, 3 → 1, 2 fixed
3
Donu Arapura
December 5, 2017
,Contents
1 The idea of a group 3
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The group of permutations 11
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Rotations and reflections in the plane 15
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Cyclic groups and dihedral groups 19
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Finite sets, counting and group theory 24
5.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 More counting problems with groups 29
6.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7 Kernels and quotients 36
7.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8 Rings and modular arithmetic 40
8.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
9 Z∗p is cyclic 45
9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10 Matrices over Zp 49
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11 The sign of a permutation 52
11.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
12 Determinants 56
12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1
,13 The 3 dimensional rotation group 60
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
14 Finite subgroups of the rotation group 64
14.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
15 Quaternions 69
15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
16 The Spin group 73
16.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2
, Chapter 1
The idea of a group
One of our goals in this class is to make precise the idea of symmetry, which is
important in math, other parts of science, and art. Something like a square has
a lot of symmetry, but circle has even more. But what does this mean? One
way of expressing this is to a view a symmetry of a given shape as a motion
which takes the shape to itself. Let us start with the example of an equilateral
triangle with vertices labelled by 1, 2, 3.
3
1 2
We want to describe all the symmetries, which are the motions (both rota-
tions and flips) which takes the triangle to itself. First of all, we can do nothing.
We call this I, which stands for identity. In terms of the vertices, I sends 1 → 1,
2 → 2 and 3 → 3. We can rotate once counterclockwise.
R+ : 1 → 2 → 3 → 1.
We can rotate once clockwise
R− : 1 → 3 → 2 → 1.
We can also flip it in various ways
F12 : 1 → 2, 2 → 1, 3 fixed
F13 : 1 → 3, 3 → 1, 2 fixed
3
