1Groups
IB Groups, Rings and Modules
1.1 Basic concepts
We will begin by quickly recapping some definitions and results from IA Groups.
Definition (Group). A group is a triple (
G, ·, e
), where
G
is a set,
·
:
G×G → G
is a function and e ∈ G is an element such that
(i) For all a, b, c ∈ G, we have (a · b) · c = a · (b · c). (associativity)
(ii) For all a ∈ G, we have a · e = e · a = a. (identity)
(iii)
For all
a ∈ G
, there exists
a
−1
∈ G
such that
a ·a
−1
=
a
−1
·a
=
e
.(inverse)
Some people add a stupid axiom that says
g ·h ∈ G
for all
g, h ∈ G
, but this
is already implied by saying
·
is a function to
G
. You can write that down as
well, and no one will say you are stupid. But they might secretly think so.
Lemma. The inverse of an element is unique.
Proof. Let a
−1
, b be inverses of a. Then
b = b · e = b ·a · a
−1
= e · a
−1
= a
−1
.
Definition (Subgroup). If (
G, ·, e
) is a group and
H ⊆ G
is a subset, it is a
subgroup if
(i) e ∈ H,
(ii) a, b ∈ H implies a · b ∈ H,
(iii) · : H × H → H makes (H, ·, e) a group.
We write H ≤ G if H is a subgroup of G.
Note that the last condition in some sense encompasses the first two, but we
need the first two conditions to hold before the last statement makes sense at all.
Lemma.
H ⊆ G
is a subgroup if
H
is non-empty and for any
h
1
, h
2
∈ H
, we
have h
1
h
−1
2
∈ H.
Definition (Abelian group). A group
G
is abelian if
a · b
=
b · a
for all
a, b ∈ G
.
Example. We have the following familiar examples of groups
(i) (Z, +, 0), (Q, +, 0), (R, +, 0), (C, +, 0).
(ii) We also have groups of symmetries:
(a)
The symmetric group
S
n
is the collection of all permutations of
{1, 2, ··· , n}.
(b) The dihedral group D
2n
is the symmetries of a regular n-gon.
(c)
The group
GL
n
(
R
) is the group of invertible
n × n
real matrices,
which also is the group of invertible
R
-linear maps from the vector
space R
n
to itself.
(iii) The alternating group A
n
≤ S
n
.
(iv) The cyclic group C
n
≤ D
2n
.
(v)
The special linear group
SL
n
(
R
)
≤ GL
n
(
R
), the subgroup of matrices of
determinant 1.
(vi) The Klein-four group C
2
× C
2
.
(vii)
The quaternions
Q
8
=
{±
1
, ±i, ±j, ±k}
with
ij
=
k, ji
=
−k
,
i
2
=
j
2
=
k
2
= −1, (−1)
2
= 1.
With groups and subgroups, we can talk about cosets.
Definition (Coset). If H ≤ G, g ∈ G, the left coset gH is the set
gH = {x ∈ G : x = g ·h for some h ∈ H}.
For example, since
H
is a subgroup, we know
e ∈ H
. So for any
g ∈ G
, we
must have g ∈ gH.
The collection of
H
-cosets in
G
forms a partition of
G
, and furthermore,
all
H
-cosets
gH
are in bijection with
H
itself, via
h 7→ gh
. An immediate
consequence is
Theorem (Lagrange’s theorem). Let G be a finite group, and H ≤ G. Then
|G| = |H||G : H|,
where |G : H| is the number of H-cosets in G.
We can do exactly the same thing with right cosets and get the same
conclusion.
We have implicitly used the following notation:
Definition (Order of group). The order of a group is the number of elements
in G, written |G|.
Instead of order of the group, we can ask what the order of an element is.
Definition (Order of element). The order of an element
g ∈ G
is the smallest
positive n such that g
n
= e. If there is no such n, we say g has infinite order.
We write ord(g) = n.
A basic lemma is as follows:
Lemma. If G is a finite group and g ∈ G has order n, then n | |G|.
Proof. Consider the following subset:
H = {e, g, g
2
, ··· , g
n−1
}.
This is a subgroup of
G
, because it is non-empty and
g
r
g
−s
=
g
r−s
is on the list
(we might have to add
n
to the power of
g
to make it positive, but this is fine
since
g
n
=
e
). Moreover, there are no repeats in the list: if
g
i
=
g
j
, with wlog
i ≥ j
, then
g
i−j
=
e
. So
i − j < n
. By definition of
n
, we must have
i − j
= 0,
i.e. i = j.
Hence Lagrange’s theorem tells us n = |H| | |G|.