Part IA — Groups
Based on lectures by J. Goedecke
Notes taken by Dexter Chua
Michaelmas 2014
These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.
Examples of groups
Axioms for groups. Examples from geometry: symmetry groups of regular polygons,
cub e, tetrahedron. Permutations on a set; the symmetric group. Subgroups and
homomorphisms. Symmetry groups as subgroups of general permutation groups. The
M¨obius group; crossratios, preservation of circles, the p oint at infinity. Conjugation.
Fixed p oints of M¨obius maps and iteration. [4]
Lagrange’s theorem
Cosets. Lagrange’s theorem. Groups of small order (up to order 8). Quaternions.
FermatEuler theorem from the grouptheoretic point of view. [5]
Group actions
Group actions; orbits and stabilizers. Orbitstabilizer theorem. Cayley’s theorem
(every group is isomorphic to a subgroup of a permutation group). Conjugacy classes.
Cauchy’s theorem. [4]
Quotient groups
Normal subgroups, quotient groups and the isomorphism theorem. [4]
Matrix groups
The general and special linear groups; relation with the M¨obius group. The orthogonal
and special orthogonal groups. Proof (in
R
3
) that every element of the orthogonal
group is the product of reflections and every rotation in
R
3
has an axis. Basis change
as an example of conjugation. [3]
Permutations
Permutations, cycles and transpositions. The sign of a permutation. Conjugacy in
S
n
and in A
n
. Simple groups; simplicity of A
5
. [4]
Contents
0 Introduction
1 Groups and homomorphisms
1.1 Groups
1.2 Homomorphisms
1.3 Cyclic groups
1.4 Dihedral groups
1.5 Direct products of groups
2 Symmetric group I
2.1 Symmetric groups
2.2 Sign of permutations
3 Lagrange’s Theorem
3.1 Small groups
3.2 Left and right cosets
4 Quotient groups
4.1 Normal subgroups
4.2 Quotient groups
4.3 The Isomorphism Theorem
5 Group actions
5.1 Group acting on sets
5.2 Orbits and Stabilizers
5.3 Important actions
5.4 Applications
6 Symmetric groups II
6.1 Conjugacy classes in S
n
6.2 Conjugacy classes in A
n
7 Quaternions
8 Matrix groups
8.1 General and special linear groups
8.2 Actions of GL
n
(C)
8.3 Orthogonal groups
8.4 Rotations and reflections in R
2
and R
3
8.5 Unitary groups
9 More on regular polyhedra
9.1 Symmetries of the cube
9.2 Symmetries of the tetrahedron
10 M¨obius group
10.1 M¨obius maps
10.2 Fixed points of M¨obius maps
10.3 Permutation properties of M¨obius maps
10.4 Crossratios
11 Projective line (nonexaminable)
0 Introduction
Group theory is an example of algebra. In pure mathematics, algebra (usually)
does not refer to the boring mindless manipulation of symbols. Instead, in
algebra, we have some set of objects with some operations on them. For example,
we can take the integers with addition as the operation. However, in algebra, we
allow any set and any operations, not just numbers.
Of course, such a definition is too broad to be helpful. We categorize algebraic
structures into different types. In this course, we will study a particular kind of
structures, groups. In the IB Groups, Rings and Modules course, we will study
rings and modules as well.
These different kinds of structures are defined by certain axioms. The group
axioms will say that the operation must follow certain rules, and any set and
operation that satisfies these rules will be considered to form a group. We will
then have a different set of axioms for rings, modules etc.
As mentioned above, the most familiar kinds of algebraic structures are
number systems such as integers and rational numbers. The focus of group
theory, however, is not on things that resemble “numbers”. Instead, it is the
study of symmetries.
First of all, what is a symmetry? We are all familiar with, say, the symmetries
of an (equilateral) triangle (we will always assume the triangle is equilateral). We
rotate a triangle by 120
◦
, and we get the original triangle. We say that rotating
by 120
◦
is a symmetry of a triangle. In general, a symmetry is something we do
to an object that leaves the object intact.
Of course, we don’t require that the symmetry leaves everything intact.
Otherwise, we would only be allowed to do nothing. Instead, we require certain
important things to be intact. For example, when considering the symmetries
of a triangle, we only care about how the resultant object looks, but don’t care
about where the individual vertices went.
In the case of the triangle, we have six symmetries: three rotations (rotation
by 0
◦
, 120
◦
and 240
◦
), and three reflections along the axes below:
These six together form the underlying set of the group of symmetries. A more
sophisticated example is the symmetries of
R
3
. We define these as operations on
R
3
that leave distances between points unchanged. These include translations,
rotations, reflections, and combinations of these.
So what is the operation? This operation combines two symmetries to give a
new symmetry. The natural thing to do is to do the symmetry one after another.
For example, if we combine the two 120
◦
rotations, we get a 240
◦
rotation.
Now we are studying algebra, not geometry. So to define the group, we
abstract away the triangle. Instead, we define the group to be six objects, say
{e, r, r
2
, s, rs, r
2
s}
, with rules defining how we combine two elements to get a
third. Officially, we do not mention the triangle at all when defining the group.
We can now come up with the group axioms. What rules should the set of
symmetries obey? First of all, we must have a “do nothing” symmetry. We call
this the identity element. When we compose the identity with another symmetry,
the other symmetry is unchanged.
Secondly, given a symmetry, we can do the reverse symmetry. So for any
element, there is an inverse element that, when combined with the original, gives
the identity.
Finally, given three symmetries, we can combine them, one after another. If
we denote the operation of the group as
∗
, then if we have three symmetries,
x, y, z
, we should be able to form
x ∗ y ∗ z
. If we want to define it in terms of
the binary operation
∗
, we can define it as (
x ∗ y
)
∗ z
, where we first combine
the first two symmetries, then combine the result with the third. Alternatively,
we can also define it as
x ∗
(
y ∗ z
). Intuitively, these two should give the same
result, since both are applying
x
after
y
after
z
. Hence we have the third rule
x ∗ (y ∗ z) = (x ∗ y) ∗ z.
Now a group is any set with an operation that satisfies the three rules above.
In group theory, the objective is to study the properties of groups just assuming
these three axioms. It turns out that there is a lot we can talk about.
1 Groups and homomorphisms
1.1 Groups
Definition
(Binary operation)
.
A (binary) operation is a way of combining two
elements to get a new element. Formally, it is a map ∗ : A × A → A.
Definition
(Group)
.
A group is a set
G
with a binary operation
∗
satisfying
the following axioms:
1. There is some e ∈ G such that for all a, we have
a ∗ e = e ∗ a = a. (identity)
2. For all a ∈ G, there is some a
−1
∈ G such that
a ∗ a
−1
= a
−1
∗ a = e. (inverse)
3. For all a, b, c ∈ G, we have
(a ∗ b) ∗ c = a ∗ (b ∗ c). (associativity)
Definition
(Order of group)
.
The order of the group, denoted by
G
, is the
number of elements in G. A group is a finite group if the order is finite.
Note that technically, the inverse axiom makes no sense, since we have not
specified what
e
is. Even if we take it to be the
e
given by the identity axiom,
the identity axiom only states there is some
e
that satisfies that property, but
there could be many! We don’t know which one
a ∗ a
−1
is supposed to be equal
to! So we should technically take that to mean there is some
a
−1
such that
a ∗ a
−1
and
a
−1
∗ a
satisfy the identity axiom. Of course, we will soon show that
identities are indeed unique, and we will happily talk about “the” identity.
Some people put a zeroth axiom called “closure”:
0. For all a, b ∈ G, we have a ∗ b ∈ G. (closure)
Technically speaking, this axiom also makes no sense — when we say
∗
is a
binary operation, by definition,
a ∗ b
must be a member of
G
. However, in
practice, we often have to check that this axiom actually holds. For example, if
we let G be the set of all matrices of the form
1 x y
0 1 z
0 0 1
under matrix multiplication, we will have to check that the product of two such
matrices is indeed a matrix of this form. Officially, we are checking that the
binary operation is a welldefined operation on G.
It is important to know that it is generally not true that
a∗b
=
b∗a
. There is
no a priori reason why this should be true. For example, if we are considering the
symmetries of a triangle, rotating and then reflecting is different from reflecting
and then rotating.
However, for some groups, this happens to be true. We call such groups
abelian groups.
Definition (Abelian group). A group is abelian if it satisfies
4. (∀a, b ∈ G) a ∗ b = b ∗ a. (commutativity)
If it is clear from context, we are lazy and leave out the operation
∗
, and
write
a ∗ b
as
ab
. We also write
a
2
=
aa
,
a
n
=
aaa · · · a
 {z }
n copies
,
a
0
=
e
,
a
−n
= (
a
−1
)
n
etc.
Example. The following are abelian groups:
(i) Z with +
(ii) Q with +
(iii) Z
n
(integers mod n) with +
n
(iv) Q
∗
with ×
(v) {−1, 1} with ×
The following are nonabelian groups:
(vi)
Symmetries of an equilateral triangle (or any
n
gon) with composition.
(D
2n
)
(vii) 2 × 2 invertible matrices with matrix multiplication (GL
2
(R))
(viii) Symmetry groups of 3D objects
Recall that the first group axiom requires that there exists an identity element,
which we shall call
e
. Then the second requires that for each
a
, there is an inverse
a
−1
such that
a
−1
a
=
e
. This only makes sense if there is only one identity
e
, or
else which identity should a
−1
a be equal to?
We shall now show that there can only be one identity. It turns out that the
inverses are also unique. So we will talk about the identity and the inverse.
Proposition. Let (G, ∗) be a group. Then
(i) The identity is unique.
(ii) Inverses are unique.
Proof.
(i)
Suppose
e
and
e
0
are identities. Then we have
ee
0
=
e
0
, treating
e
as an
inverse, and ee
0
= e, treating e
0
as an inverse. Thus e = e
0
.
(ii)
Suppose
a
−1
and
b
both satisfy the inverse axiom for some
a ∈ G
. Then
b = be = b(aa
−1
) = (ba)a
−1
= ea
−1
= a
−1
. Thus b = a
−1
.
Proposition. Let (G, ∗) be a group and a, b ∈ G. Then
(i) (a
−1
)
−1
= a
(ii) (ab)
−1
= b
−1
a
−1
Proof.
(i) Given a
−1
, both a and (a
−1
)
−1
satisfy
xa
−1
= a
−1
x = e.
By uniqueness of inverses, (a
−1
)
−1
= a.
(ii) We have
(ab)(b
−1
a
−1
) = a(bb
−1
)a
−1
= aea
−1
= aa
−1
= e
Similarly, (
b
−1
a
−1
)
ab
=
e
. So
b
−1
a
−1
is an inverse of
ab
. By the uniqueness
of inverses, (ab)
−1
= b
−1
a
−1
.
Sometimes if we have a group
G
, we might want to discard some of the
elements. For example if
G
is the group of all symmetries of a triangle, we might
one day decide that we hate reflections because they reverse orientation. So
we only pick the rotations in
G
and form a new, smaller group. We call this a
subgroup of G.
Definition
(Subgroup)
.
A
H
is a subgroup of
G
, written
H ≤ G
, if
H ⊆ G
and
H with the restricted operation ∗ from G is also a group.
Example.
– (Z, +) ≤ (Q, +) ≤ (R, +) ≤ (C, +)
– (e, ∗) ≤ (G, ∗) (trivial subgroup)
– G ≤ G
– ({±1}, ×) ≤ (Q
∗
, ×)
According to the definition, to prove that
H
is a subgroup of
G
, we need to
make sure
H
satisfies all group axioms. However, this is often tedious. Instead,
there are some simplified criteria to decide whether H is a subgroup.
Lemma (Subgroup criteria I). Let (G, ∗) be a group and H ⊆ G. H ≤ G iff
(i) e ∈ H
(ii) (∀a, b ∈ H) ab ∈ H
(iii) (∀a ∈ H) a
−1
∈ H
Proof. The group axioms are satisfied as follows:
0. Closure: (ii)
1.
Identity: (i). Note that
H
and
G
must have the same identity. Suppose that
e
H
and
e
G
are the identities of
H
and
G
respectively. Then
e
H
e
H
=
e
H
.
Now
e
H
has an inverse in
G
. Thus we have
e
H
e
H
e
−1
H
=
e
H
e
−1
H
. So
e
H
e
G
= e
G
. Thus e
H
= e
G
.
2. Inverse: (iii)
3. Associativity: inherited from G.
Humans are lazy, and the test above is still too complicated. We thus come
up with an even simpler test:
Lemma (Subgroup criteria II). A subset H ⊆ G is a subgroup of G iff:
(I) H is nonempty
(II) (∀a, b ∈ H) ab
−1
∈ H
Proof. (I) and (II) follow trivially from (i), (ii) and (iii).
To prove that (I) and (II) imply (i), (ii) and (iii), we have
(i) H must contain at least one element a. Then aa
−1
= e ∈ H.
(iii) ea
−1
= a
−1
∈ H.
(ii) a(b
−1
)
−1
= ab ∈ H.
Proposition.
The subgroups of (
Z,
+) are exactly
nZ
, for
n ∈ N
(
nZ
is the
integer multiples of n).
Proof.
Firstly, it is trivial to show that for any
n ∈ N
,
nZ
is a subgroup. Now
show that any subgroup must be in the form nZ.
Let
H ≤ Z
. We know 0
∈ H
. If there are no other elements in
H
, then
H = 0Z. Otherwise, pick the smallest positive integer n in H. Then H = nZ.
Otherwise, suppose (
∃a ∈ H
)
n  a
. Let
a
=
pn
+
q
, where 0
< q < n
. Since
a − pn ∈ H
,
q ∈ H
. Yet
q < n
but
n
is the smallest member of
H
. Contradiction.
So every
a ∈ H
is divisible by
n
. Also, by closure, all multiples of
n
must be in
H. So H = nZ.
1.2 Homomorphisms
It is often helpful to study functions between different groups. First, we need to
define what a function is. These definitions should be familiar from IA Numbers
and Sets.
Definition
(Function)
.
Given two sets
X
,
Y
, a function
f
:
X → Y
sends each
x ∈ X
to a particular
f
(
x
)
∈ Y
.
X
is called the domain and
Y
is the codomain.
Example.
–
Identity function: for any set
X
, 1
X
:
X → X
with 1
X
(
x
) =
x
is a function.
This is also written as id
X
.
–
Inclusion map:
ι
:
Z → Q
:
ι
(
n
) =
n
. Note that this differs from the
identity function as the domain and codomain are different in the inclusion
map.
– f
1
: Z → Z: f
1
(x) = x + 1.
– f
2
: Z → Z: f
2
(x) = 2x.
– f
3
: Z → Z: f
3
(x) = x
2
.
– For g : {0, 1, 2, 3, 4} → {0, 1, 2, 3, 4}, we have:
◦ g
1
(x) = x + 1 if x < 4; g
1
(4) = 4.
◦ g
2
(x) = x + 1 if x < 4; g
1
(4) = 0.
Definition
(Composition of functions)
.
The composition of two functions is a
function you get by applying one after another. In particular, if
f
:
X → Y
and
G : Y → Z, then g ◦ f : X → Z with g ◦ f(x) = g(f(x)).
Example. f
2
◦ f
1
(
x
) = 2
x
+ 2.
f
1
◦ f
2
(
x
) = 2
x
+ 1. Note that function
composition is not commutative.
Definition
(Injective functions)
.
A function
f
is injective if it hits everything
at most once, i.e.
(∀x, y ∈ X) f(x) = f (y) ⇒ x = y.
Definition
(Surjective functions)
.
A function is surjective if it hits everything
at least once, i.e.
(∀y ∈ Y )(∃x ∈ X) f(x) = y.
Definition
(Bijective functions)
.
A function is bijective if it is both injective
and surjective. i.e. it hits everything exactly once. Note that a function has an
inverse iff it is bijective.
Example. ι
and
f
2
are injective but not subjective.
f
3
and
g
1
are neither. 1
X
,
f
1
and g
2
are bijective.
Lemma. The composition of two bijective functions is bijective
When considering sets, functions are allowed to do all sorts of crazy things,
and can send any element to any element without any restrictions. However, we
are currently studying groups, and groups have additional structure on top of
the set of elements. Hence we are not interested in arbitrary functions. Instead,
we are interested in functions that “respect” the group structure. We call these
homomorphisms.
Definition
(Group homomorphism)
.
Let (
G, ∗
) and (
H, ×
) be groups. A
function f : G → H is a group homomorphism iff
(∀g
1
, g
2
∈ G) f(g
1
) × f(g
2
) = f(g
1
∗ g
2
),
Definition
(Group isomorphism)
.
Isomorphisms are bijective homomorphisms.
Two groups are isomorphic if there exists an isomorphism between them. We
write G
∼
=
H.
We will consider two isomorphic groups to be “the same”. For example, when
we say that there is only one group of order 2, it means that any two groups of
order 2 must be isomorphic.
Example.
– f
:
G → H
defined by
f
(
g
) =
e
, where
e
is the identity of
H
, is a
homomorphism.
–
1
G
:
G → G
and
f
2
:
Z →
2
Z
are isomorphisms.
ι
:
Z → Q
and
f
2
:
Z → Z
are homomorphisms.
– exp : (R, +) → (R
+
, ×) with exp(x) = e
x
is an isomorphism.
–
Take (
Z
4
,
+) and
H
: (
{e
ikπ/2
:
k
= 0
,
1
,
2
,
3
}, ×
). Then
f
:
Z
4
→ H
by
f(a) = e
iπa/2
is an isomorphism.
– f
:
GL
2
(
R
)
→ R
∗
with
f
(
A
) =
det
(
A
) is a homomorphism, where
GL
2
(
R
)
is the set of 2 × 2 invertible matrices.
Proposition. Suppose that f : G → H is a homomorphism. Then
(i) Homomorphisms send the identity to the identity, i.e.
f(e
G
) = e
H
(ii) Homomorphisms send inverses to inverses, i.e.
f(a
−1
) = f(a)
−1
(iii) The composite of 2 group homomorphisms is a group homomorphism.
(iv) The inverse of an isomorphism is an isomorphism.
Proof.
(i)
f(e
G
) = f(e
2
G
) = f(e
G
)
2
f(e
G
)
−1
f(e
G
) = f(e
G
)
−1
f(e
G
)
2
f(e
G
) = e
H
(ii)
e
H
= f(e
G
)
= f(aa
−1
)
= f(a)f(a
−1
)
Since inverses are unique, f(a
−1
) = f(a)
−1
.
(iii)
Let
f
:
G
1
→ G
2
and
g
:
G
2
→ G
3
. Then
g
(
f
(
ab
)) =
g
(
f
(
a
)
f
(
b
)) =
g(f(a))g(f(b)).
(iv) Let f : G → H be an isomorphism. Then
f
−1
(ab) = f
−1
n
f
f
−1
(a)
f
f
−1
(b)
o
= f
−1
n
f
f
−1
(a)f
−1
(b)
o
= f
−1
(a)f
−1
(b)
So
f
−1
is a homomorphism. Since it is bijective,
f
−1
is an isomorphism.
Definition
(Image of homomorphism)
.
If
f
:
G → H
is a homomorphism, then
the image of f is
im f = f(G) = {f (g) : g ∈ G}.
Definition (Kernel of homomorphism). The kernel of f, written as
ker f = f
−1
({e
H
}) = {g ∈ G : f(g) = e
H
}.
Proposition.
Both the image and the kernel are subgroups of the respective
groups, i.e. im f ≤ H and ker f ≤ G.
Proof.
Since
e
H
∈ im f
and
e
G
∈ ker f
,
im f
and
ker f
are nonempty. Moreover,
suppose
b
1
, b
2
∈ im f
. Now
∃a
1
, a
2
∈ G
such that
f
(
a
i
) =
b
i
. Then
b
1
b
−1
2
=
f(a
1
)f(a
−1
2
) = f(a
1
a
−1
2
) ∈ im f.
Then consider
b
1
, b
2
∈ ker f
. We have
f
(
b
1
b
−1
2
) =
f
(
b
1
)
f
(
b
2
)
−1
=
e
2
=
e
. So
b
1
b
−1
2
∈ ker f.
Proposition.
Given any homomorphism
f
:
G → H
and any
a ∈ G
, for all
k ∈ ker f, aka
−1
∈ ker f.
This proposition seems rather pointless. However, it is not. All subgroups
that satisfy this property are known as normal subgroups, and normal subgroups
have very important properties. We will postpone the discussion of normal
subgroups to later lectures.
Proof. f(aka
−1
) = f(a)f(k)f(a)
−1
= f(a)ef(a)
−1
= e. So aka
−1
∈ ker f.
Example. Images and kernels for previously defined functions:
(i) For the function that sends everything to e, im f = {e} and ker f = G.
(ii) For the identity function, im 1
G
= G and ker 1
G
= {e}.
(iii) For the inclusion map ι : Z → Q, we have im ι = Z and ker ι = {0}
(iv) For f
2
: Z → Z and f
2
(x) = 2x, we have im f
2
= 2Z and ker f
2
= {0}.
(v)
For
det
:
GL
2
(
R
)
→ R
∗
, we have
im det
=
R
∗
and
ker det
=
{A
:
det A
=
1} = SL
2
(R)
Proposition. For all homomorphisms f : G → H, f is
(i) surjective iff im f = H
(ii) injective iff ker f = {e}
Proof.
(i) By definition.
(ii)
We know that
f
(
e
) =
e
. So if
f
is injective, then by definition
ker f
=
{e}
. If
ker f
=
{e}
, then given
a, b
such that
f
(
a
) =
f
(
b
),
f
(
ab
−1
) =
f(a)f(b)
−1
= e. Thus ab
−1
∈ ker f = {e}. Then ab
−1
= e and a = b.
So far, the definitions of images and kernels seem to be just convenient
terminology to refer to things. However, we will later prove an important
theorem, the first isomorphism theorem, that relates these two objects and
provides deep insights (hopefully).
Before we get to that, we will first study some interesting classes of groups
and develop some necessary theory.
1.3 Cyclic groups
The simplest class of groups is cyclic groups. A cyclic group is a group of the
form
{e, a, a
2
, a
2
, · · · , a
n−1
}
, where
a
n
=
e
. For example, if we consider the
group of all rotations of a triangle, and write
r
= rotation by 120
◦
, the elements
will be {e, r, r
2
} with r
3
= e.
Officially, we define a cyclic group as follows:
Definition (Cyclic group C
n
). A group G is cyclic if
(∃a)(∀b)(∃n ∈ Z) b = a
n
,
i.e. every element is some power of a. Such an a is called a generator of G.
We write C
n
for the cyclic group of order n.
Example.
(i) Z is cyclic with generator 1 or −1. It is the infinite cyclic group.
(ii) ({+1, −1}, ×) is cyclic with generator −1.
(iii) (Z
n
, +) is cyclic with all numbers coprime with n as generators.
Notation.
Given a group
G
and
a ∈ G
, we write
hai
for the cyclic group
generated by
a
, i.e. the subgroup of all powers of
a
. It is the smallest subgroup
containing a.
Definition
(Order of element)
.
The order of an element
a
is the smallest integer
n
such that
a
n
=
e
. If
n
doesn’t exist,
a
has infinite order. Write
ord
(
a
) for the
order of a.
We have given two different meanings to the word “order”. One is the order
of a group and the other is the order of an element. Since mathematicians
are usually (but not always) sensible, the name wouldn’t be used twice if they
weren’t related. In fact, we have
Lemma. For a in g, ord(a) = hai.
Proof.
If
ord
(
a
) =
∞
,
a
n
6
=
a
m
for all
n 6
=
m
. Otherwise
a
m−n
=
e
. Thus
hai = ∞ = ord(a).
Otherwise, suppose
ord
(
a
) =
k
. Thus
a
k
=
e
. We now claim that
hai
=
{e, a
1
, a
2
, · · · a
k−1
}
. Note that
hai
does not contain higher powers of
a
as
a
k
=
e
and higher powers will loop back to existing elements. There are also no repeating
elements in the list provided since a
m
= a
n
⇒ a
m−n
= e. So done.
It is trivial to show that
Proposition. Cyclic groups are abelian.
Definition
(Exponent of group)
.
The exponent of a group
G
is the smallest
integer n such that a
n
= e for all a ∈ G.
1.4 Dihedral groups
Definition
(Dihedral groups
D
2n
)
.
Dihedral groups are the symmetries of a
regular
n
gon. It contains
n
rotations (including the identity symmetry, i.e.
rotation by 0
◦
) and n reflections.
We write the group as
D
2n
. Note that the subscript refers to the order of
the group, not the number of sides of the polygon.
The dihedral group is not hard to define. However, we need to come up with
a presentation of D
2n
that is easy to work with.
We first look at the rotations. The set of all rotations is generated by
r
=
360
◦
n
.
This r has order n.
How about the reflections? We know that each reflection has order 2. Let
s
be our favorite reflection. Then using some geometric arguments, we can show
that any reflection can be written as a product of
r
m
and
s
for some
m
. We also
have srs = r
−1
.
Hence we can define
D
2n
as follows:
D
2n
is a group generated by
r
and
s
,
and every element can be written as a product of
r
’s and
s
’s. Whenever we see
r
n
and s
2
, we replace it by e. When we see srs, we replace it by r
−1
.
It then follows that every element can be written in the form r
m
s.
Formally, we can write D
2n
as follows:
D
2n
= hr, s  r
n
= s
2
= e, srs
−1
= r
−1
i
= {e, r, r
2
, · · · r
n−1
, s, rs, r
2
s, · · · r
n−1
s}
This is a notation we will commonly use to represent groups. For example, a
cyclic group of order n can be written as
C
n
= ha  a
n
= ei.
1.5 Direct products of groups
Recall that if we have to sets
X, Y
, then we can obtain the product
X × Y
=
{(x, y) : x ∈ X, y ∈ Y }. We can do the same if X and Y are groups.
Definition
(Direct product of groups)
.
Given two groups (
G, ◦
) and (
H, •
),
we can define a set
G × H
=
{
(
g, h
) :
g ∈ G, h ∈ H}
and an operation
(a
1
, a
2
) ∗ (b
1
, b
2
) = (a
1
◦ b
1
, a
2
• b
2
). This forms a group.
Why would we want to take the product of two groups? Suppose we have
two independent triangles. Then the symmetries of this system include, say
rotating the first triangle, rotating the second, or rotating both. The symmetry
group of this combined system would then be D
6
× D
6
.
Example.
C
2
× C
2
= {(0, 0), (0, 1), (1, 0), (1, 1)}
= {e, x, y, xy} with everything order 2
= hx, y  x
2
= y
2
= e, xy = yxi
Proposition. C
n
× C
m
∼
=
C
nm
iff hcf(m, n) = 1.
Proof.
Suppose that
hcf
(
m, n
) = 1. Let
C
n
=
hai
and
C
m
=
hbi
. Let
k
be the
order of (
a, b
). Then (
a, b
)
k
= (
a
k
, b
k
) =
e
. This is possible only if
n  k
and
m  k
, i.e.
k
is a common multiple
n
and
m
. Since the order is the minimum
value of k that satisfies the above equation, k = lcm(n, m) =
nm
hcf(n,m)
= nm.
Now consider
h
(
a, b
)
i ≤ C
n
× C
m
. Since (
a, b
) has order
nm
,
h
(
a, b
)
i
has
nm
elements. Since
C
n
× C
m
also has
nm
elements,
h
(
a, b
)
i
must be the whole of
C
n
× C
m
. And we know that h(a, b)i
∼
=
C
nm
. So C
n
× C
m
∼
=
C
nm
.
On the other hand, suppose
hcf
(
m, n
)
6
= 1. Then
k
=
lcm
(
m, n
)
6
=
mn
. Then
for any (
a, b
)
∈ C
n
× C
m
,we have (
a, b
)
k
= (
a
k
, b
k
) =
e
. So the order of any (
a, b
)
is at most
k < mn
. So there is no element of order
mn
. So
C
n
× C
m
is not a
cyclic group of order nm.
Given a complicated group
G
, it is sometimes helpful to write it as a product
H × K
, which could make things a bit simpler. We can do so by the following
theorem:
Proposition
(Direct product theorem)
.
Let
H
1
, H
2
≤ G
. Suppose the following
are true:
(i) H
1
∩ H
2
= {e}.
(ii) (∀a
i
∈ H
i
) a
1
a
2
= a
2
a
1
.
(iii) (∀a ∈ G)(∃a
i
∈ H
i
) a = a
1
a
2
. We also write this as G = H
1
H
2
.
Then G
∼
=
H
1
× H
2
.
Proof.
Define
f
:
H
1
×H
2
→ G
by
f
(
a
1
, a
2
) =
a
1
a
2
. Then it is a homomorphism
since
f((a
1
, a
2
) ∗ (b
1
, b
2
)) = f(a
1
b
1
, a
2
b
2
)
= a
1
b
1
a
2
b
2
= a
1
a
2
b
1
b
2
= f(a
1
, a
2
)f(b
1
, b
2
).
Surjectivity follows from (iii). We’ll show injectivity by showing that the kernel
is
{e}
. If
f
(
a
1
, a
2
) =
e
, then we know that
a
1
a
2
=
e
. Then
a
1
=
a
−1
2
. Since
a
1
∈ H
1
and
a
−1
2
∈ H
2
, we have
a
1
=
a
−1
2
∈ H
1
∩ H
2
=
{e}
. Thus
a
1
=
a
2
=
e
and ker f = {e}.
2 Symmetric group I
We will devote two full chapters to the study of symmetric groups, because
it is really important. Recall that we defined a symmetry to be an operation
that leaves some important property of the object intact. We can treat each
such operation as a bijection. For example, a symmetry of
R
2
is a bijection
f
:
R
2
→ R
2
that preserves distances. Note that we must require it to be a
bijection, instead of a mere function, since we require each symmetry to be an
inverse.
We can consider the case where we don’t care about anything at all. So a
“symmetry” would be any arbitrary bijection
X → X
, and the set of all bijections
will form a group, known as the symmetric group. Of course, we will no longer
think of these as “symmetries” anymore, but just bijections.
In some sense, the symmetric group is the most general case of a symmetry
group. In fact, we will later (in Chapter 5) show that every group can be written
as a subgroup of some symmetric group.
2.1 Symmetric groups
Definition
(Permutation)
.
A permutation of
X
is a bijection from a set
X
to
X itself. The set of all permutations on X is Sym X.
When composing permutations, we treat them as functions. So if
σ
and
ρ
are permutations, σ ◦ ρ is given by first applying ρ, then applying σ.
Theorem. Sym X with composition forms a group.
Proof. The groups axioms are satisfied as follows:
0.
If
σ
:
X → X
and
τ
:
X → X
, then
σ ◦ τ
:
X → X
. If they are both
bijections, then the composite is also bijective. So if
σ, τ ∈ Sym X
, then
σ ◦ τ ∈ Sym X.
1.
The identity 1
X
:
X → X
is clearly a permutation, and gives the identity
of the group.
2.
Every bijective function has a bijective inverse. So if
σ ∈ Sym X
, then
σ
−1
∈ Sym X.
3. Composition of functions is associative.
Definition
(Symmetric group
S
n
)
.
If
X
is finite, say
X
=
n
(usually use
X
=
{
1
,
2
, · · · , n}
), we write
Sym X
=
S
n
. The is the symmetric group of degree
n.
It is important to note that the degree of the symmetric group is different
from the order of the symmetric group. For example,
S
3
has degree 3 but order
6. In general, the order of S
n
is n!.
There are two ways to write out an element of the symmetric group. The
first is the two row notation.
Notation.
(Two row notation) We write 1
,
2
,
3
, · · · n
on the top line and their
images below, e.g.
1 2 3
2 3 1
∈ S
3
and
1 2 3 4 5
2 1 3 4 5
∈ S
5
In general, if σ : X → X, we write
1 2 3 · · · n
σ(1) σ(2) σ(3) · · · σ(n)
Example. For small n, we have
(i) When n = 1, S
n
=
1
1
= {e}
∼
=
C
1
.
(ii) When n = 2, S
n
=
1 2
1 2
,
1 2
2 1
∼
=
C
2
(iii) When n = 3,
S
n
=
1 2 3
1 2 3
,
1 2 3
2 3 1
,
1 2 3
3 1 2
1 2 3
2 1 3
,
1 2 3
3 2 1
,
1 2 3
1 3 2
∼
=
D
6
.
Note that
S
3
is not abelian. Thus
S
n
is not abelian for
n ≥
3 since we can
always view S
3
as a subgroup of S
n
by fixing 4, 5, 6, · · · n.
In general, we can view
D
2n
as a subgroup of
S
n
because each symmetry is
a permutation of the corners.
While the two row notation is fully general and can represent any (finite)
permutation, it is clumsy to write and wastes a lot of space. It is also very
annoying to type using L
A
T
E
X. Hence, most of the time, we actually use the
cycle notation.
Notation
(Cycle notation)
.
If a map sends 1
7→
2, 2
7→
3, 3
7→
1, then we
write it as a cycle (1 2 3). Alternatively, we can write (2 3 1) or (3 1 2), but by
convention, we usually write the smallest number first. We leave out numbers
that don’t move. So we write (1 2) instead of (1 2)(3).
For more complicated maps, we can write them as products of cycles. For
example, in S
4
, we can have things like (1 2)(3 4).
The order of each cycle is the length of the cycle, and the inverse is the cycle
written the other way round, e.g. (1 2 3)
−1
= (3 2 1) = (1 3 2).
Example.
(i)
Suppose we want to simplify (1 2 3)(1 2). Recall that composition is from
right to left. So 1 gets mapped to 3 ((1 2) maps 1 to 2, and (1 2 3) further
maps it to 3). Then 3 gets mapped to 1. 2 is mapped to 2 itself. So
(1 2 3)(1 2) = (1 3)(2)
(ii) (1 2 3 4)(1 4) = (1)(2 3 4) = (2 3 4).
Definition
(
k
cycles and transpositions)
.
We call (
a
1
a
2
a
3
· · · a
k
) a
k
cycle.
2cycles are called transpositions. Two cycles are disjoint if no number appears
in both cycles.
Example. (1 2) and (3 4) are disjoint but (1 2 3) and (1 2) are not.
Lemma. Disjoint cycles commute.
Proof.
If
σ, τ ∈ S
n
are disjoint cycles. Consider any
n
. Show that:
σ
(
τ
(
a
)) =
τ
(
σ
(
a
)). If
a
is in neither of
σ
and
τ
, then
σ
(
τ
(
a
)) =
τ
(
σ
(
a
)) =
a
. Otherwise,
wlog assume that
a
is in
τ
but not in
σ
. Then
τ
(
a
)
∈ τ
and thus
τ
(
a
)
6∈ σ
. Thus
σ
(
a
) =
a
and
σ
(
τ
(
a
)) =
τ
(
a
). Therefore we have
σ
(
τ
(
a
)) =
τ
(
σ
(
a
)) =
τ
(
a
).
Therefore τ and σ commute.
In general, nondisjoint cycles may not commute. For example, (1 3)(2 3) =
(1 3 2) while (2 3)(1 3) = (1 2 3).
Theorem.
Any permutation in
S
n
can be written (essentially) uniquely as a
product of disjoint cycles. (Essentially unique means unique up to reordering of
cycles and rotation within cycles, e.g. (1 2) and (2 1))
Proof.
Let
σ ∈ S
n
. Start with (1
σ
(1)
σ
2
(1)
σ
3
(1)
· · ·
). As the set
{
1
,
2
,
3
· · · n}
is finite, for some
k
, we must have
σ
k
(1) already in the list. If
σ
k
(1) =
σ
l
(1),
with
l < k
, then
σ
k−l
(1) = 1. So all
σ
i
(1) are distinct until we get back to 1.
Thus we have the first cycle (1 σ(1) σ
2
(1) σ
3
(1) · · · σ
k−1
(1)).
Now choose the smallest number that is not yet in a cycle, say
j
. Repeat to
obtain a cycle (
j σ
(
j
)
σ
2
(
j
)
· · · σ
l−1
(
j
)). Since
σ
is a bijection, nothing in this
cycle can be in previous cycles as well.
Repeat until all
{
1
,
2
,
3
· · · n}
are exhausted. This is essentially unique
because every number
j
completely determines the whole cycle it belongs to,
and whichever number we start with, we’ll end up with the same cycle.
Definition
(Cycle type)
.
Write a permutation
σ ∈ S
n
in disjoint cycle notation.
The cycle type is the list of cycle lengths. This is unique up to reordering. We
often (but not always) leave out singleton cycles.
Example. (1 2) has cycle type 2 (transposition). (1 2)(3 4) has cycle type 2, 2
(double transposition). (1 2 3)(4 5) has cycle type 3, 2.
Lemma.
For
σ ∈ S
n
, the order of
σ
is the least common multiple of cycle
lengths in the disjoint cycle notation. In particular, a kcycle has order k.
Proof.
As disjoint cycles commute, we can group together each cycle when we take
powers. i.e. if σ = τ
1
τ
2
· · · τ
l
with τ
i
all disjoint cycles, then σ
m
= τ
m
1
τ
m
2
· · · τ
m
l
.
Now if cycle
τ
i
has length
k
i
, then
τ
k
i
i
=
e
, and
τ
m
i
=
e
iff
k
i
 m
. To get an
m
such that
σ
m
=
e
, we need all
k
i
to divide
m
. i.e.
m
is a common multiple of
k
i
. Since the order is the least possible
m
such that
σ
m
=
e
, the order is the
least common multiple of k
i
.
Example. Any transpositions and double transpositions have order 2.
(1 2 3)(4 5) has order 6.
2.2 Sign of permutations
To classify different permutations, we can group different permutations according
to their cycle type. While this is a very useful thing to do, it is a rather fine
division. In this section, we will assign a “sign” to each permutation, and each
permutation can either be odd or even. This highlevel classification allows us
to separate permutations into two sets, which is also a useful notion.
To define the sign, we first need to write permutations as products of
transpositions.
Proposition. Every permutation is a product of transpositions.
This is not a deep or mysterious fact. All it says is that you can rearrange
things however you want just by swapping two objects at a time.
Proof.
As each permutation is a product of disjoint cycles, it suffices to prove
that each cycle is a product of transpositions. Consider a cycle (
a
1
a
2
a
3
· · · a
k
).
This is in fact equal to (
a
1
a
2
)(
a
2
a
3
)
· · ·
(
a
k−1
a
k
). Thus a
k
cycle can be written
as a product of k − 1 transpositions.
Note that the product is not unique. For example,
(1 2 3 4 5) = (1 2)(2 3)(3 4)(4 5) = (1 2)(2 3)(1 2)(3 4)(1 2)(4 5).
However, the number of terms in the product, mod 2, is always the same.
Theorem.
Writing
σ ∈ S
n
as a product of transpositions in different ways,
σ
is
either always composed of an even number of transpositions, or always an odd
number of transpositions.
The proof is rather magical.
Proof.
Write #(
σ
) for the number of cycles in disjoint cycle notation, including
singleton cycles. So #(
e
) =
n
and #((1 2)) =
n −
1. When we multiply
σ
by a
transposition τ = (c d) (wlog assume c < d),
–
If
c, d
are in the same
σ
cycle, say, (
c a
2
· · · a
k−1
d a
k+1
· · · a
k+l
)(
c d
) =
(c a
k+1
a
k+2
· · · a
k+l
)(d a
2
a
3
· · · a
k−1
). So #(στ) = #(σ) + 1 .
– If c, d are in different σcycles, say
(d a
2
a
3
· · · a
k−1
)(c a
k+1
a
k+2
· · · a
k+l
)(c d)
= (c a
2
· · · a
k−1
d a
k+1
· · · a
k+l
)(c d)(c d)
= (c a
2
· · · a
k−1
d a
k+1
· · · a
k+l
) and #(στ) = #(σ) − 1.
Therefore for any transposition τ, #(στ ) ≡ #(σ) + 1 (mod 2).
Now suppose
σ
=
τ
1
· · · τ
l
=
τ
0
1
· · · τ
0
k
. Since disjoint cycle notation is unique,
#(σ) is uniquely determined by σ.
Now we can construct
σ
by starting with
e
and multiplying the transpositions
one by one. Each time we add a transposition, we increase #(
σ
) by 1 (
mod
2).
So #(
σ
)
≡
#(
e
) +
l
(
mod
2). Similarly, #(
σ
)
≡
#(
e
) +
k
(
mod
2). So
l ≡ k
(mod 2).
Definition
(Sign of permutation)
.
Viewing
σ ∈ S
n
as a product of transpo
sitions,
σ
=
τ
1
· · · τ
l
, we call
sgn
(
σ
) = (
−
1)
l
. If
sgn
(
σ
) = 1, we call
σ
an even
permutation. If sgn(σ) = −1, we call σ an odd permutation.
While
l
itself is not welldefined, it is either always odd or always even, and
(−1)
l
is welldefined.
Theorem. For n ≥ 2, sgn : S
n
→ {±1} is a surjective group homomorphism.
Proof.
Suppose
σ
1
=
τ
1
· · · τ
l
1
and
σ
2
=
τ
0
1
· · · τ
l
2
. Then
sgn
(
σ
1
σ
2
) = (
−
1)
l
1
+l
2
=
(−1)
l
1
(−1)
l
2
= sgn(σ
1
) sgn(σ
2
). So it is a homomorphism.
It is surjective since sgn(e) = 1 and sgn((1 2)) = −1.
It is this was rather trivial to prove. The hard bit is showing that
sgn
is
well defined. If a question asks you to show that
sgn
is a welldefined group
homomorphism, you have to show that it is welldefined.
Lemma. σ
is an even permutation iff the number of cycles of even length is
even.
Proof.
A
k
cycle can be written as
k −
1 transpositions. Thus an evenlength
cycle is odd, vice versa.
Since
sgn
is a group homomorphism, writing
σ
in disjoint cycle notation,
σ
=
σ
1
σ
2
· · · σ
l
, we get
sgn
(
σ
) =
sgn
(
σ
1
)
· · · sgn
(
σ
l
). Suppose there are
m
even
length cycles and
n
oddlength cycles, then
sgn
(
σ
) = (
−
1)
m
1
n
. This is equal to
1 iff (−1)
m
= 1, i.e. m is even.
Rather confusingly, odd length cycles are even, and even length cycles are
odd.
Definition
(Alternating group
A
n
)
.
The alternating group
A
n
is the kernel of
sgn
, i.e. the even permutations. Since
A
n
is a kernel of a group homomorphism,
A
n
≤ S
n
.
Among the many uses of the
sgn
homomorphism, it is used in the definition
of the determinant of a matrix: if A
n×n
is a square matrix, then
det A =
X
σ∈S
n
sgn(σ)a
1σ(1)
· · · a
nσ(n)
.
Proposition.
Any subgroup of
S
n
contains either no odd permutations or
exactly half.
Proof.
If
S
n
has at least one odd permutation
τ
, then there exists a bijection
between the odd and even permutations by
σ 7→ στ
(bijection since
σ 7→ στ
−1
is a welldefined inverse). So there are as many odd permutations as even
permutations.
After we prove the isomorphism theorem later, we can provide an even shorter
proof of this.
3 Lagrange’s Theorem
One can model a Rubik’s cube with a group, with each possible move correspond
ing to a group element. Of course, Rubik’s cubes of different sizes correspond to
different groups.
Suppose I have a 4
×
4
×
4 Rubik’s cube, but I want to practice solving a
2
×
2
×
2 Rubik’s cube. It is easy. I just have to make sure every time I make a
move, I move two layers together. Then I can pretend I am solving a 2
×
2
×
2
cube. This corresponds to picking a particular subgroup of the 4
×
4
×
4 group.
Now what if I have a 3
×
3
×
3 cube? I can still practice solving a 2
×
2
×
2
one. This time, I just look at the corners and pretend that the edges and centers
do not exist. Then I am satisfied when the corners are in the right positions,
while the centers and edges can be completely scrambled. In this case, we are
not taking a subgroup. Instead, we are identifying certain moves together. In
particular, we are treating two moves as the same as long as their difference is
confined to the centers and edges.
Let
G
be the 3
×
3
×
3 cube group, and
H
be the subgroup of
G
that only
permutes the edges and centers. Then for any
a, b ∈ G
, we think
a
and
b
are “the
same” if
a
−1
b ∈ H
. Then the set of things equivalent to
a
is
aH
=
{ah
:
h ∈ H}
.
We call this a coset, and the set of cosets form a group.
An immediate question one can ask is: why not
Ha
=
{ha
:
h ∈ H}
? In
this particular case, the two happen to be the same for all possible
a
. However,
for a general subgroup
H
, they need not be. We can still define the coset
aH
=
{ah
:
h ∈ H}
, but these are less interesting. For example, the set of all
{aH}
will no longer form a group. We will look into these more indepth in the
next chapter. In this chapter, we will first look at results for general cosets. In
particular, we will, step by step, prove the things we casually claimed above.
Definition
(Cosets)
.
Let
H ≤ G
and
a ∈ G
. Then the set
aH
=
{ah
:
h ∈ H}
is a left coset of H and Ha = {ha : h ∈ H} is a right coset of H.
Example.
(i)
Take 2
Z ≤ Z
. Then 6 + 2
Z
=
{all even numbers}
= 0 + 2
Z
. 1 + 2
Z
=
{all odd numbers} = 17 + 2Z.
(ii) Take G = S
3
, let H = h(1 2)i = {e, (1 2)}. The left cosets are
eH = (1 2)H = {e, (1 2)}
(1 3)H = (1 2 3)H = {(1 3), (1 2 3)}
(2 3)H = (1 3 2)H = {(2 3), (1 3 2)}
(iii)
Take
G
=
D
6
(which is isomorphic to
S
3
). Recall
D
6
=
hr, s  r
3
e
=
s
2
, rs
=
sr
−1
i
.Take
H
=
hsi
=
{e, s}
. We have left coset
rH
=
{r, rs
=
sr
−1
}
and
the right coset Hr = {r, sr}. Thus rH 6= Hr.
Proposition. aH = bH ⇔ b
−1
a ∈ H.
Proof.
(
⇒
) Since
a ∈ aH
,
a ∈ bH
. Then
a
=
bh
for some
h ∈ H
. So
b
−1
a
=
h ∈ H.
(
⇐
). Let
b
−1
a
=
h
0
. Then
a
=
bh
0
. Then
∀ah ∈ aH
, we have
ah
=
b
(
h
0
h
)
∈
bH. So aH ⊆ bH. Similarly, bH ⊆ aH. So aH = bH.
Definition
(Partition)
.
Let
X
be a set, and
X
1
, · · · X
n
be subsets of
X
. The
X
i
are called a partition of
X
if
S
X
i
=
X
and
X
i
∩ X
j
=
∅
for
i 6
=
j
. i.e. every
element is in exactly one of X
i
.
Lemma.
The left cosets of a subgroup
H ≤ G
partition
G
, and every coset has
the same size.
Proof.
For each
a ∈ G
,
a ∈ aH
. Thus the union of all cosets gives all of
G
. Now
we have to show that for all
a, b ∈ G
, the cosets
aH
and
bH
are either the same
or disjoint.
Suppose that
aH
and
bH
are not disjoint. Let
ah
1
=
bh
2
∈ aH ∩ bH
. Then
b
−1
a = h
2
h
−1
1
∈ H. So aH = bH.
To show that they each coset has the same size, note that
f
:
H → aH
with
f
(
h
) =
ah
is invertible with inverse
f
−1
(
h
) =
a
−1
h
. Thus there exists a bijection
between them and they have the same size.
Definition
(Index of a subgroup)
.
The index of
H
in
G
, written
G
:
H
, is the
number of left cosets of H in G.
Theorem
(Lagrange’s theorem)
.
If
G
is a finite group and
H
is a subgroup of
G, then H divides G. In particular,
HG : H = G.
Note that the converse is not true. If
k
divides
G
, there is not necessarily a
subgroup of order
k
, e.g.
A
4

= 12 but there is no subgroup of order 6. However,
we will later see that this is true if k is a prime (cf. Cauchy’s theorem).
Proof.
Suppose that there are
G
:
H
left cosets in total. Since the left cosets
partition G, and each coset has size H, we have
HG : H = G.
Again, the hard part of this proof is to prove that the left cosets partition
G
and have the same size. If you are asked to prove Lagrange’s theorem in exams,
that is what you actually have to prove.
Corollary.
The order of an element divides the order of the group, i.e. for any
finite group G and a ∈ G, ord(a) divides G.
Proof.
Consider the subgroup generated by
a
, which has order
ord
(
a
). Then by
Lagrange’s theorem, ord(a) divides G.
Corollary.
The exponent of a group divides the order of the group, i.e. for any
finite group G and a ∈ G, a
G
= e.
Proof.
We know that
G
=
k ord
(
a
) for some
k ∈ N
. Then
a
G
= (
a
ord(a)
)
k
=
e
k
= e.
Corollary.
Groups of prime order are cyclic and are generated by every non
identity element.
Proof.
Say
G
=
p
. If
a ∈ G
is not the identity, the subgroup generated by
a
must have order
p
since it has to divide
p
. Thus the subgroup generated by
a
has the same size as
G
and they must be equal. Then
G
must be cyclic since it
is equal to the subgroup generated by a.
A useful way to think about cosets is to view them as equivalence classes.
To do so, we need to first define what an equivalence class is.
Definition
(Equivalence relation)
.
An equivalence relation
∼
is a relation that
is reflexive, symmetric and transitive. i.e.
(i) (∀x) x ∼ x (reflexivity)
(ii) (∀x, y) x ∼ y ⇒ y ∼ x (symmetry)
(iii) (∀x, y, z) [(x ∼ y) ∧ (y ∼ z) ⇒ x ∼ z] (transitivity)
Example. The following relations are equivalence relations:
(i) Consider Z. The relation ≡
n
defined as a ≡
n
b ⇔ n  (a − b).
(ii)
Consider the set (formally: class) of all finite groups. Then “is isomorphic
to” is an equivalence relation.
Definition
(Equivalence class)
.
Given an equivalence relation
∼
on
A
, the
equivalence class of a is
[a]
∼
= [a] = {b ∈ A : a ∼ b}
Proposition. The equivalence classes form a partition of A.
Proof.
By reflexivity, we have
a ∈
[
a
]. Thus the equivalence classes cover the
whole set. We must now show that for all
a, b ∈ A
, either [
a
] = [
b
] or [
a
]
∩
[
b
] =
∅
.
Suppose [
a
]
∩
[
b
]
6
=
∅
. Then
∃c ∈
[
a
]
∩
[
b
]. So
a ∼ c, b ∼ c
. By symmetry,
c ∼ b
.
By transitivity, we have
a ∼ b
. Now for all
b
0
∈
[
b
], we have
b ∼ b
0
. Thus by
transitivity, we have
a ∼ b
0
. Thus [
b
]
⊆
[
a
]. Similarly, [
a
]
⊆
[
b
] and [
a
] = [
b
].
Lemma.
Given a group
G
and a subgroup
H
, define the equivalence relation
on G with a ∼ b iff b
−1
a ∈ H. The equivalence classes are the left cosets of H.
Proof. First show that it is an equivalence relation.
(i) Reflexivity: Since aa
−1
= e ∈ H, a ∼ a.
(ii) Symmetry: a ∼ b ⇒ b
−1
a ∈ H ⇒ (b
−1
a)
−1
= a
−1
b ∈ H ⇒ b ∼ a.
(iii)
Transitivity: If
a ∼ b
and
b ∼ c
, we have
b
−1
a, c
−1
b ∈ H
. So
c
−1
bb
−1
a
=
c
−1
a ∈ H. So a ∼ c.
To show that the equivalence classes are the cosets, we have
a ∼ b ⇔ b
−1
a ∈
H ⇔ aH = bH.
Example.
Consider (
Z,
+), and for fixed
n
, take the subgroup
nZ
. The cosets
are 0 +
H,
1 +
H, · · ·
(
n −
1) +
H
. We can write these as [0]
,
[1]
,
[2]
· · ·
[
n
]. To
perform arithmetic “mod
n
”, define [
a
] + [
b
] = [
a
+
b
], and [
a
][
b
] = [
ab
]. We
need to check that it is welldefined, i.e. it doesn’t depend on the choice of the
representative of [a].
If [
a
1
] = [
a
2
] and [
b
1
] = [
b
2
], then
a
1
=
a
2
+
kn
and
b
1
=
b
2
+
kn
, then
a
1
+
b
1
=
a
2
+
b
2
+
n
(
k
+
l
) and
a
1
b
1
=
a
2
b
2
+
n
(
kb
2
+
la
2
+
kln
). So [
a
1
+
b
1
] =
[a
2
+ b
2
] and [a
1
b
1
] = [a
2
b
2
].
We have seen that (
Z
n
,
+
n
) is a group. What happens with multiplication?
We can only take elements which have inverses (these are called units, cf. IB
Groups, Rings and Modules). Call the set of them
U
n
=
{
[
a
] : (
a, n
) = 1
}
. We’ll
see these are the units.
Definition (Euler totient function). (Euler totient function) φ(n) = U
n
.
Example. If p is a prime, φ(n) = p − 1. φ(4) = 2.
Proposition. U
n
is a group under multiplication mod n.
Proof. The operation is welldefined as shown above. To check the axioms:
0.
Closure: if
a, b
are coprime to
n
, then
a · b
is also coprime to
n
. So
[a], [b] ∈ U
n
⇒ [a] · [b] = [a · b] ∈ U
n
1. Identity: [1]
2.
Let [
a
]
∈ U
n
. Consider the map
U
n
→ U
n
with [
c
]
7→
[
ac
]. This is injective:
if [
ac
1
] = [
ac
2
], then
n
divides
a
(
c
1
− c
2
). Since
a
is coprime to
n
,
n
divides
c
1
− c
2
, so [
c
1
] = [
c
2
]. Since
U
n
is finite, any injection (
U
n
→ U
n
) is also a
surjection. So there exists a c such that [ac] = [a][c] = 1. So [c] = [a]
−1
.
3. Associativity (and also commutativity): inherited from Z.
Theorem (FermatEuler theorem). Let n ∈ N and a ∈ Z coprime to n. Then
a
φ(n)
≡ 1 (mod n).
In particular, (Fermat’s Little Theorem) if
n
=
p
is a prime, then for any
a
not
a multiple of p.
a
p−1
≡ 1 (mod p).
Proof.
As
a
is coprime with
n
, [
a
]
∈ U
n
. Then [
a
]
U
n

= [1], i.e.
a
φ(n)
≡
1
(mod n).
3.1 Small groups
We will study the structures of certain small groups.
Example
(Using Lagrange theorem to find subgroups)
.
To find subgroups of
D
10
, we know that the subgroups must have size 1, 2, 5 or 10:
1: {e}
2: The groups generated by the 5 reflections of order 2
5:
The group must be cyclic since it has prime order 5. It is then generated
by an element of order 5, i.e.
r, r
2
, r
3
and
r
4
. They generate the same
group hri.
10: D
10
As for D
8
, subgroups must have order 1, 2, 4 or 8.
1: {e}
2: 5 elements of order 2, namely 4 reflections and r
2
.
4:
First consider the subgroup isomorphic to
C
4
, which is
hri
. There are two
other noncyclic group.
8: D
8
Proposition. Any group of order 4 is either isomorphic to C
4
or C
2
× C
2
.
Proof.
Let
G
= 4. By Lagrange theorem, possible element orders are 1 (
e
only),
2 and 4. If there is an element a ∈ G of order 4, then G = hai
∼
=
C
4
.
Otherwise all nonidentity elements have order 2. Then
G
must be abelian
(For any
a, b
, (
ab
)
2
= 1
⇒ ab
= (
ab
)
−1
⇒ ab
=
b
−1
a
−1
⇒ ab
=
ba
). Pick
2 elements of order 2, say
b, c ∈ G
, then
hbi
=
{e, b}
and
hci
=
{e, c}
. So
hbi ∩ hci
=
{e}
. As
G
is abelian,
hbi
and
hci
commute. We know that
bc
=
cb
has
order 2 as well, and is the only element of
G
left. So
G
∼
=
hbi × hci
∼
=
C
2
× C
2
by the direct product theorem.
Proposition.
A group of order 6 is either cyclic or dihedral (i.e. is isomorphic
to C
6
or D
6
). (See proof in next section)
3.2 Left and right cosets
As
aH
=
H
and similarly
H
=
Ha
, left and right cosets have the same size.
Are they necessarily the same? We’ve previously shown that they might not be
the same. In some other cases, they are.
Example.
(i)
Take
G
= (
Z,
+) and
H
= 2
Z
. We have 0 + 2
Z
= 2
Z
+ 0 = even numbers
and 1 + 2
Z
= 2
Z
+ 1 = odd numbers. Since
G
is abelian,
aH
=
Ha
for all
a, ∈ G, H ≤ G.
(ii)
Let
G
=
D
6
=
hr, s  r
3
=
e
=
s
2
, rs
=
sr
−1
i
. Let
U
=
hri
. Since
the cosets partition
G
, so one must be
U
and the other
sU
=
{s, sr
=
r
2
s, sr
2
= rs} = Us. So for all a ∈ G, aU = Ua.
(iii)
Let
G
=
D
6
and take
H
=
hsi
. We have
H
=
{e, s}
,
rH
=
{r, rs
=
sr
−1
}
and
r
2
H
=
{r
2
, r
s
}
; while
H
=
{e, s}, Hr
=
{r, sr}
and
Hr
2
=
{r
2
, sr
2
}
.
So the left and right subgroups do not coincide.
This distinction will become useful in the next chapter.
4 Quotient groups
In the previous section, when attempting to pretend that a 3
×
3
×
3 Rubik’s
cube is a 2
×
2
×
2 one, we came up with the cosets
aH
, and claimed that these
form a group. We also said that this is not the case for arbitrary subgroup
H
,
but only for subgroups that satisfy
aH
=
Ha
. Before we prove these, we first
study these subgroups a bit.
4.1 Normal subgroups
Definition (Normal subgroup). A subgroup K of G is a normal subgroup if
(∀a ∈ G)(∀k ∈ K) aka
−1
∈ K.
We write K C G. This is equivalent to:
(i) (∀a ∈ G) aK = Ka, i.e. left coset = right coset
(ii) (∀a ∈ G) aKa
−1
= K (cf. conjugacy classes)
From the example last time,
H
=
hsi ≤ D
6
is not a normal subgroup, but
K
=
hri C D
6
. We know that every group
G
has at least two normal subgroups
{e} and G.
Lemma.
(i) Every subgroup of index 2 is normal.
(ii) Any subgroup of an abelian group is normal.
Proof.
(i)
If
K ≤ G
has index 2, then there are only two possible cosets
K
and
G \ K
.
As
eK
=
Ke
and cosets partition
G
, the other left coset and right coset
must be G \ K. So all left cosets and right cosets are the same.
(ii) For all a ∈ G and k ∈ K, we have aka
−1
= aa
−1
k = k ∈ K.
Proposition. Every kernel is a normal subgroup.
Proof.
Given homomorphism
f
:
G → H
and some
a ∈ G
, for all
k ∈ ker f
, we
have
f
(
aka
−1
) =
f
(
a
)
f
(
k
)
f
(
a
)
−1
=
f
(
a
)
ef
(
a
)
−1
=
e
. Therefore
aka
−1
∈ ker f
by definition of the kernel.
In fact, we will see in the next section that all normal subgroups are kernels
of some homomorphism.
Example.
Consider
G
=
D
8
. Let
K
=
hr
2
i
is normal. Check: Any element of
G
is either
sr
`
or
r
`
for some
`
. Clearly
e
satisfies
aka
−1
∈ K
. Now check
r
2
:
For the case of
sr
`
, we have
sr
`
r
2
(
sr
`
)
−1
=
sr
`
r
2
r
−`
s
−1
=
sr
2
s
=
ssr
−2
=
r
2
.
For the case of r
`
, r
`
r
2
r
−`
= r
2
.
Proposition. A group of order 6 is either cyclic or dihedral (i.e.
∼
=
C
6
or D
6
).
Proof.
Let
G
= 6. By Lagrange theorem, possible element orders are 1
,
2
,
3 and
6. If there is an
a ∈ G
of order 6, then
G
=
hai
∼
=
C
6
. Otherwise, we can only
have elements of orders 2 and 3 other than the identity. If
G
only has elements
of order 2, the order must be a power of 2 by Sheet 1 Q. 8, which is not the case.
So there must be an element
r
of order 3. So
hri C G
as it has index 2. Now
G
must also have an element s of order 2 by Sheet 1 Q. 9.
Since
hri
is normal, we know that
srs
−1
∈ hri
. If
srs
−1
=
e
, then
r
=
e
,
which is not true. If
srs
−1
=
r
, then
sr
=
rs
and
sr
has order 6 (lcm of the
orders of
s
and
r
), which was ruled out above. Otherwise if
srs
−1
=
r
2
=
r
−1
,
then G is dihedral by definition of the dihedral group.
4.2 Quotient groups
Proposition.
Let
K C G
. Then the set of (left) cosets of
K
in
G
is a group
under the operation aK ∗ bK = (ab)K.
Proof.
First show that the operation is welldefined. If
aK
=
a
0
K
and
bK
=
b
0
K
,
we want to show that
aK ∗bK
=
a
0
K ∗b
0
K
. We know that
a
0
=
ak
1
and
b
0
=
bk
2
for some
k
1
, k
2
∈ K
. Then
a
0
b
0
=
ak
1
bk
2
. We know that
b
−1
k
1
b ∈ K
. Let
b
−1
k
1
b
=
k
3
. Then
k
1
b
=
bk
3
. So
a
0
b
0
=
abk
3
k
2
∈
(
ab
)
K
. So picking a different
representative of the coset gives the same product.
1. Closure: If aK, bK are cosets, then (ab)K is also a coset
2. Identity: The identity is eK = K (clear from definition)
3. Inverse: The inverse of aK is a
−1
K (clear from definition)
4. Associativity: Follows from the associativity of G.
Definition
(Quotient group)
.
Given a group
G
and a normal subgroup
K
, the
quotient group or factor group of
G
by
K
, written as
G/K
, is the set of (left)
cosets of K in G under the operation aK ∗ bK = (ab)K.
Note that the set of left cosets also exists for nonnormal subgroups (abnormal
subgroups?), but the group operation above is not well defined.
Example.
(i)
Take
G
=
Z
and
nZ
(which must be normal since
G
is abelian), the cosets
are
k
+
nZ
for 0
≤ k < n
. The quotient group is
Z
n
. So we can write
Z/
(
nZ
) =
Z
n
. In fact these are the only quotient groups of
Z
since
nZ
are
the only subgroups.
Note that if G is abelian, G/K is also abelian.
(ii)
Take
K
=
hri C D
6
. We have two cosets
K
and
sK
. So
D
6
/K
has order 2
and is isomorphic to C
2
.
(iii)
Take
K
=
hr
2
i C D
8
. We know that
G/K
should have
8
2
= 4 elements. We
have
G/K
=
{K, rK
=
r
3
K, sK
=
sr
2
K, srK
=
sr
3
K}
. We see that all
elements (except K) has order 2, so G/K
∼
=
C
2
× C
2
.
Note that quotient groups are not subgroups of
G
. They contain different
kinds of elements. For example,
Z/nZ
∼
=
C
n
are finite, but all subgroups of
Z
infinite.
Example.
(Nonexample) Consider
D
6
with
H
=
hsi
.
H
is not a normal
subgroup. We have
rH ∗ r
2
H
=
r
3
H
=
H
, but
rH
=
rsH
and
r
2
H
=
srH
(by
considering the individual elements). So we have
rsH ∗ srH
=
r
2
H 6
=
H
, and
the operation is not welldefined.
Lemma.
Given
K C G
, the quotient map
q
:
G → G/K
with
g 7→ gK
is a
surjective group homomorphism.
Proof. q
(
ab
) = (
ab
)
K
=
aKbK
=
q
(
a
)
q
(
b
). So
q
is a group homomorphism.
Also for all aK ∈ G/K, q(a) = aK. So it is surjective.
Note that the kernel of the quotient map is
K
itself. So any normal subgroup
is a kernel of some homomorphism.
Proposition. The quotient of a cyclic group is cyclic.
Proof.
Let
G
=
C
n
with
H ≤ C
n
. We know that
H
is also cyclic. Say
C
n
=
hci
and
H
=
hc
k
i
∼
=
C
`
, where
k`
=
n
. We have
C
n
/H
=
{H, cH, c
2
H, · · · c
k−1
H}
=
hcHi
∼
=
C
k
.
4.3 The Isomorphism Theorem
Now we come to the Really Important Theorem
TM
.
Theorem
(The Isomorphism Theorem)
.
Let
f
:
G → H
be a group homomor
phism with kernel K. Then K C G and G/K
∼
=
im f.
Proof.
We have proved that
K C G
before. We define a group homomorphism
θ : G/K → im f by θ(aK) = f(a).
First check that this is welldefined: If a
1
K = a
2
K, then a
−1
2
a
1
∈ K. So
f(a
2
)
−1
f(a
1
) = f(a
−1
2
a
1
) = e.
So f (a
1
) = f(a
2
) and θ(a
1
K) = θ(a
2
K).
Now we check that it is a group homomorphism:
θ(aKbK) = θ(abK) = f(ab) = f(a)f(b) = θ(aK)θ(bK).
To show that it is injective, suppose
θ
(
aK
) =
θ
(
bK
). Then
f
(
a
) =
f
(
b
). Hence
f(b)
−1
f(a) = e. Hence b
−1
a ∈ K. So aK = bK.
By definition,
θ
is surjective since
im θ
=
im f
. So
θ
gives an isomorphism
G/K
∼
=
im f ≤ H.
If
f
is injective, then the kernel is
{e}
, so
G/K
∼
=
G
and
G
is isomorphic to
a subgroup of
H
. We can think of
f
as an inclusion map. If
f
is surjective, then
im f = H. In this case, G/K
∼
=
H.
Example.
(i)
Take
f
:
GL
n
(
R
)
→ R
∗
with
A 7→ det A
,
ker f
=
SL
N
(
R
).
im f
=
R
∗
as for
all
λ ∈ R
∗
,
det
λ 0 · · · 0
0 1 · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 1
=
λ
. So we know that
GL
n
(
R
)
/SL
n
(
R
)
∼
=
R
∗
.
(ii)
Define
θ
: (
R,
+)
→
(
C∗, ×
) with
r 7→ exp
(2
πir
). This is a group homomor
phism since
θ
(
r
+
s
) =
exp
(2
πi
(
r
+
s
)) =
exp
(2
πir
)
exp
(2
πis
) =
θ
(
r
)
θ
(
s
).
We know that the kernel is
Z C R
. Clearly the image is the unit circle
(S
1
, ×). So R/Z
∼
=
(S
1
, ×).
(iii) G
= (
Z
∗
p
, ×
) for prime
p 6
= 2. We have
f
:
G → G
with
a 7→ a
2
. This
is a homomorphism since (
ab
)
2
=
a
2
b
2
(
Z
∗
p
is abelian). The kernel is
{±
1
}
=
{
1
, p −
1
}
. We know that
im f
∼
=
G/ ker f
with order
p−1
2
. These
are known as quadratic residues.
Lemma.
Any cyclic group is isomorphic to either
Z
or
Z/
(
nZ
) for some
n ∈ N
.
Proof.
Let
G
=
hci
. Define
f
:
Z → G
with
m 7→ c
m
. This is a group
homomorphism since
c
m
1
+m
2
=
c
m
1
c
m
2
.
f
is surjective since
G
is by definition
all c
m
for all m. We know that ker f C Z. We have three possibilities. Either
(i) ker f = {e}, so F is an isomorphism and G
∼
=
Z; or
(ii) ker f = Z, then G
∼
=
Z/Z = {e} = C
1
; or
(iii) ker f
=
nZ
(since these are the only proper subgroups of
Z
), then
G
∼
=
Z/(nZ).
Definition
(Simple group)
.
A group is simple if it has no nontrivial proper
normal subgroup, i.e. only {e} and G are normal subgroups.
Example. C
p
for prime
p
are simple groups since it has no proper subgroups
at all, let alone normal ones.
A
5
is simple, which we will prove after Chapter 6.
The finite simple groups are the building blocks of all finite groups. All finite
simple groups have been classified (The Atlas of Finite Groups). If we have
K C G
with
K 6
=
G
or
{e}
, then we can “quotient out”
G
into
G/K
. If
G/K
is not simple, repeat. Then we can write
G
as an “inverse quotient” of simple
groups.
5 Group actions
Recall that we came up with groups to model symmetries and permutations.
Intuitively, elements of groups are supposed to “do things”. However, as we
developed group theory, we abstracted these away and just looked at how
elements combine to form new elements. Group actions recapture this idea and
make each group element correspond to some function.
5.1 Group acting on sets
Definition
(Group action)
.
Let
X
be a set and
G
be a group. An action of
G
on X is a homomorphism ϕ : G → Sym X.
This means that the homomorphism
ϕ
turns each element
g ∈ G
into a
permutation of X, in a way that respects the group structure.
Instead of writing ϕ(g)(x), we usually directly write g(x) or gx.
Alternatively, we can define the group action as follows:
Proposition.
Let
X
be a set and
G
be a group. Then
ϕ
:
G → Sym X
is a
homomorphism (i.e. an action) iff
θ
:
G × X → X
defined by
θ
(
g, x
) =
ϕ
(
g
)(
x
)
satisfies
0. (∀g ∈ G)(x ∈ X) θ(g, x) ∈ X.
1. (∀x ∈ X) θ(e, x) = x.
2. (∀g, h ∈ G)(∀x ∈ X) θ(g, θ(h, x)) = θ(gh, x).
This criteria is almost the definition of a homomorphism. However, here we
do not explicitly require
θ
(
g, ·
) to be a bijection, but require
θ
(
e, ·
) to be the
identity function. This automatically ensures that
θ
(
g, ·
) is a bijection, since
when composed with
θ
(
g
−1
, ·
), it gives
θ
(
e, ·
), which is the identity. So
θ
(
g, ·
)
has an inverse. This is usually an easier thing to show.
Example.
(i)
Trivial action: for any group
G
acting on any set
X
, we can have
ϕ
(
g
) = 1
X
for all g, i.e. G does nothing.
(ii) S
n
acts on {1, · · · n} by permutation.
(iii) D
2n
acts on the vertices of a regular ngon (or the set {1, · · · , n}).
(iv)
The rotations of a cube act on the faces/vertices/diagonals/axes of the
cube.
Note that different groups can act on the same sets, and the same group can
act on different sets.
Definition
(Kernel of action)
.
The kernel of an action
G
on
X
is the kernel of
ϕ, i.e. all g such that ϕ(g) = 1
X
.
Note that by the isomorphism theorem,
ker ϕ C G
and
G/K
is isomorphic to
a subgroup of Sym X.
Example.
(i) D
2n
acting on {1, 2 · · · n} gives ϕ : D
2n
→ S
n
with kernel {e}.
(ii)
Let
G
be the rotations of a cube and let it act on the three axes
x, y, z
through the faces. We have
ϕ
:
G → S
3
. Then any rotation by 180
◦
doesn’t
change the axes, i.e. act as the identity. So the kernel of the action has
at least 4 elements:
e
and the three 180
◦