8Matrix groups

IA Groups

8.1 General and special linear groups

Consider

M

n×n

(

F

), i.e. the set of

n × n

matrices over the field

F

=

R

or

C

(or

F

p

). We know that matrix multiplication is associative (since they represent

functions) but are, in general, not commutative. To make this a group, we want

the identity matrix

I

to be the identity. To ensure everything has an inverse, we

can only include invertible matrices.

(We do not necessarily need to take

I

as the identity of the group. We can,

for example, take

e

=

0 0

0 1

and obtain a group in which every matrix is of

the form

0 0

0 a

for some non-zero

a

. This forms a group, albeit a boring one

(it is simply

∼

=

R

∗

))

Definition (General linear group GL

n

(F )).

GL

n

(F ) = {A ∈ M

n×n

(F ) : A is invertible}

is the general linear group.

Alternatively, we can define

GL

n

(

F

) as matrices with non-zero determinants.

Proposition. GL

n

(F ) is a group.

Proof.

Identity is

I

, which is in

GL

n

(

F

) by definition (

I

is its self-inverse). The

composition of invertible matrices is invertible, so is closed. Inverse exist by

definition. Multiplication is associative.

Proposition. det : GL

n

(F ) → F \ {0} is a surjective group homomorphism.

Proof. det AB

=

det A det B

. If

A

is invertible, it has non-zero determinant and

det A ∈ F \ {0}.

To show it is surjective, for any

x ∈ F \ {

0

}

, if we take the identity matrix

and replace I

11

with x, then the determinant is x. So it is surjective.

Definition

(Special linear group

SL

n

(

F

))

.

The special linear group

SL

n

(

F

) is

the kernel of the determinant, i.e.

SL

n

(F ) = {A ∈ GL

n

(F ) : det A = 1}.

So SL

n

(F ) C GL

n

(F ) as it is a kernel. Note that Q

8

≤ SL

2

(C)