elementary proof of orders

elementary proof of orders

When possible, our proofs avoid matrices so that the proofs retain some value to infinite dimensional settings. When we use

k

we mean any field, and

G

⁢

F

⁢

(

q

)

indicates the special case of a

finite field

of order

q

.

V

is always our

vector space

.

Remark 1

.

There are many alternative methods for computing orders of

classical groups

,
for

instance

observing special

subgroups

(

http://planetmath.org/TheoryFromOrdersOfClassicalGroups2

) or from Lie theory and
the study of

Chevalley groups

. The method explored here is intended to be

elementary

linear algebra

.

The basic starting point in computing orders of classical groups is an application of elementary linear algebra rephrased in

group theory

terms.

Proposition 2

.

G

⁢

L

⁢

(

V

)

acts regularly on the set of

ordered bases

of vector space

V

over a field

k

.

Proof.

Given any two bases

B

=

{

b

i

:

i

∈

I

}

and

C

=

{

c

i

:

i

∈

I

}

of
a vector space

V

, define the map

f

:

V

→

V

by

(

∑

i

∈

I

l

i

⁢

b

i

)

⁢

f

=

∑

i

∈

I

l

i

⁢

c

i

.

(1)

[Note the above sum has only finitely many non-zero

l

i

∈

k

.]
It follows

f

is an

invertible linear transformation

so

f

∈

G

⁢

L

⁢

(

V

)

.
Furthermore, any

linear transformation

g

∈

G

⁢

L

⁢

(

V

)

with

(

b

i

)

⁢

g

=

c

i

must

satisfy

(

1

) to be linear so indeed

g

=

f

. Therefore

G

⁢

L

⁢

(

V

)

acts regularly on ordered bases of

V

.
∎

In the world of group theory, a

regular

action is a typical substitute for knowing the order of a group. In particular, any two groups, even

infinite

, have the same order if they have a regular action on the same set. However, we are presently after specific order of

finite groups

so we return to the case of

V

a finite

dimension

vector space over a finite field

k

=

G

⁢

F

⁢

(

q

)

. We do however attempt to establish the orders through

bijections

with other sets and groups so that the results apply in more general

contexts

as well.

Theorem 3

.

|

S

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

∏

i

=

2

d

(

q

i

-

1

)

,

|

P

⁢

S

⁢

L

⁢

(

d

,

q

)

|

=

|

S

⁢

L

⁢

(

d

,

q

)

|

(

d

,

q

-

1

)

,

|

G

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

,

|

P

⁢

G

⁢

L

⁢

(

d

,

q

)

|

=

|

S

⁢

L

⁢

(

d

,

q

)

|

,

|

Γ

⁢

L

⁢

(

d

,

q

)

|

=

(

q

-

1

)

⁢

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

,

|

P

⁢

Γ

⁢

L

⁢

(

d

,

q

)

|

=

|

Γ

⁢

L

⁢

(

d

,

q

)

|

q

-

1

.

Proof.

When

V

=

G

⁢

F

⁢

(

q

)

d

, the number of ordered bases can be counted. A basis
is a set

B

=

{

b

1

,

…

,

b

d

}

of

linearly independent

vectors. So

b

1

may be chosen freely from

V

-

{

0

}

, providing

q

d

-

1

possible choices.
Next

b

2

must be chosen

independent

form

b

1

so

b

2

can be freely
chosen from

V

-

⟨

b

1

⟩

leaving

q

d

-

q

choices. In a

similar

fashion

b

3

has

q

d

-

q

2

possiblities and so continuing by

induction

we
find the total number of ordered bases to be:

(

q

d

-

1

)

⁢

(

q

d

-

q

)

⁢

⋯

⁢

(

q

d

-

q

d

-

1

)

.

Now we treat

q

as the

variable

of a

polynomial

and factor this number into:

q

(

d

2

)

⁢

∏

i

=

1

d

(

q

i

-

1

)

.

As

G

⁢

L

⁢

(

V

)

acts regularly on ordered bases of

V

, this is the order of

G

⁢

L

⁢

(

V

)

.

For the order of

S

⁢

L

⁢

(

V

)

recall the

S

⁢

L

⁢

(

V

)

is the kernel of the

determinant

homomorphism

det

:

G

⁢

L

⁢

(

V

)

→

k

×

. Furthermore,

det

is

surjective

as a

diagonal matrix

can be used to exhibit any determinant we seek. We conclude

[

G

L

(

d

,

q

)

:

S

L

(

d

,

q

)

]

=

|

G

F

(

q

)

×

|

=

q

-

1

so that

|

S

⁢

L

⁢

(

d

,

q

)

|

=

|

G

⁢

L

⁢

(

d

,

q

)

|

/

(

q

-

1

)

.

In a similar process,

P

⁢

G

⁢

L

⁢

(

V

)

=

G

⁢

L

⁢

(

V

)

/

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

so if we derive the order of the
center of

G

⁢

L

⁢

(

V

)

we derive the order of

P

⁢

G

⁢

L

⁢

(

V

)

. The

central

transforms
are scalar (they must

preserve

every

eigenspace

of every linear transform) so

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

is

isomorphic

to

k

×

. Thus the order of

P

⁢

G

⁢

L

⁢

(

V

)

is the same as the order of

S

⁢

L

⁢

(

V

)

.

For

Γ

⁢

L

⁢

(

V

)

and

P

⁢

Γ

⁢

L

⁢

(

V

)

simply notice

Γ

⁢

L

⁢

(

V

)

=

G

⁢

L

⁢

(

V

)

⋊

k

×

so when

k

=

G

⁢

F

⁢

(

q

)

we get an additional

q

-

1

term.

Finally, we consider

P

⁢

S

⁢

L

⁢

(

V

)

=

S

⁢

L

⁢

(

V

)

/

(

S

⁢

L

⁢

(

V

)

∩

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

)

. The order of

S

⁢

L

⁢

(

V

)

∩

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

must be computed. So we require scalar transforms with determinant 1. As the such, if

r

is the eigen value of the scalar transform we need

r

d

=

1

in

G

⁢

F

⁢

(

q

)

. From finite field theory we know

G

⁢

F

⁢

(

q

)

×

≅

ℤ

q

-

1

. As this group is cyclic we know that every

element

r

∈

G

⁢

F

⁢

(

q

)

×

satisfying

r

d

=

1

also satisfies

r

q

-

1

=

1

and

r

lies in the unique subgroup of order

(

d

,

q

-

1

)

of

G

⁢

F

⁢

(

q

)

×

.
Thus

|

S

⁢

L

⁢

(

V

)

∩

Z

⁢

(

G

⁢

L

⁢

(

V

)

)

|

=

(

d

,

q

-

1

)

.
∎

Title

elementary proof of orders

Canonical

name

ElementaryProofOfOrders

Date of creation

2013-03-22 15:56:52

Last modified on

2013-03-22 15:56:52

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Proof

Classification

msc 11E57

Classification

msc 05E15

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