theory from orders of classical groups

theory from orders of classical groups

One

formula

for the order of

G

⁢

L

⁢

(

d

,

q

)

can be given as:

|

G

⁢

L

⁢

(

d

,

q

)

|

=

q

(

d

2

)

⁢

(

q

-

1

)

d

⁢

∏

i

=

2

d

(

q

i

-

1

+

⋯

+

q

+

1

)

.

In Orders of

Classical Groups

(

http://planetmath.org/OrdersAndStructureOfClassicalGroups

) we see an accounting for this orders from an

elementary

linear algebra

perspective and from there derive order of related classical groups. However, many of these formulas can be explicitly observed with the group theoretic

structure

of the various classical groups. We explore this presently.

We describe three families of

subgroups

U

,

T

,

S

i

which

intersect

trivially and whose orders are:

|

U

|

=

q

(

d

2

)

,

|

T

|

=

(

q

-

1

)

d

,

and

⁢

|

S

i

|

=

q

i

-

1

+

⋯

+

q

+

1

,

2

≤

i

≤

d

.

1

The Unipotent matrices

U

Proposition 1

.

Given

q

=

p

s

a Sylow

p

-subgroup of

G

⁢

L

⁢

(

d

,

q

)

has order

p

s

⁢

(

d

2

)

and is

isomorphic

to the group of lower-unitriangular matrices.

Proof.

Observe for any

i

>

0

,

q

i

-

1

≡

-

1

(

mod

q

)

. Therefore

q

i

-

1

≡

-

1

(

mod

p

)

as

q

=

p

s

. Therefore

p

is relatively prime to the

∏

i

=

1

d

(

q

i

-

1

)

factor of the order of

G

⁢

L

⁢

(

d

,

q

)

. Therefore the order of a Sylow

p

-subgroup
of

G

⁢

L

⁢

(

d

,

q

)

is

q

(

d

2

)

)

=

p

s

⁢

(

d

d

)

.

From

matrix multiplication

we observe the that

product

of two

lower triangular
matrices

is lower triangular. And the same for lower unitriangular. Therefore
the lower unitriangular matrices form a subgroup of

G

⁢

L

⁢

(

d

,

k

)

. Moreover, they
have

(

d

2

)

entries which can range over

G

⁢

F

⁢

(

q

)

giving a total order of

q

(

d

2

)

.

[

1

0

⋯

*

1

*

*

1

⋮

*

⋮

*

1

]

∎

Remark 2

.

Unitriangular subgroups are often called

unipotent

because every

element

in the group has all

eigenvalues

equal to 1.

Definition 3

.

A unipotent subgroup of

G

⁢

L

⁢

(

V

)

is a subgroup in which every element has
all eigenvalues equal to 1.

It can be shown that every unipotent group, under some choice of basis, is a set
of lower unitriangular matrices. Of course

upper triangular matrices

may also be used.

2

The Diagonal (Toral) matrices

T

We will now account for the

(

q

-

1

)

d

term in the order.

Proposition 4

.

The

diagonal matrices

of

G

⁢

L

⁢

(

d

,

k

)

form an

abelian group

isomorphic to

(

k

×

)

d

. So in

G

⁢

L

⁢

(

d

,

q

)

, the diagonal matrices have order

(

q

-

1

)

d

.

Proof.

The

isomorphism

is easily explained with matrix multiplication:

[

a

1

a

2

⋱

a

d

]

⁢

[

b

1

b

2

⋱

b

d

]

=

[

a

1

⁢

b

1

a

2

⁢

b

2

⋱

a

d

⁢

b

d

]

.

Note that in order to be

invertible

each

a

i

≠

0

.
∎

Remark 5

.

In general theory, diagonal matrices are replaced with the term

toral

subgroup. This is on account of the fact that when

k

=

C

,

C

×

is

homotopic

to the circle,

S

1

. Therefore, as a
topology,

(

C

×

)

d

is

S

1

×

S

1

×

⋯

×

S

1

,
that is, the

d

-dimensional torus.

Definition 6

.

A toral subgroup of

G

⁢

L

⁢

(

V

)

is a subgroup in which can be simultaneously
diagonalized, possibly requiring a

field extension

. A maximal torus is on which is not properly

contained

in
any other.

3

The Singer Cycles

S

The last point illustrate where the remaining pieces of the order of

G

⁢

L

⁢

(

V

)

come from. When we accounted for

(

q

-

1

)

d

we used diagonal matrices. However we could have used any subgroup of matrices which can be diagonalized, and then choose a

maximal

such subgroup (proof that this is

sufficient

is beyond the scope at the momenet.) So we should ask, when is a matrix

diagonalizable

.
The answer is when all the eigenvalues of the matrix lie in the field and the matrix is

symmetric

, meaning

X

=

X

t

.

As our fields

G

⁢

F

⁢

(

q

)

are finite, they are not

algebraically closed

, and so some matrices may have eigenvalues that are not in the field. For example:

A

=

[

0

1

-

1

0

]

has

characteristic polynomial

x

2

+

1

which requires a

-

1

. This is not an element of

ℤ

p

when

p

is odd. Therefore we must create a

quadratic extension

to

G

⁢

F

⁢

(

p

2

)

to

diagonalize

this matrix. In particular we
would then be abel to write:

[

-

1

0

0

-

-

1

]

as the diagonalization of

A

. If

p

=

2

the example

B

=

[

1

1

1

0

]

has

irreducible

characteristic polynomial

x

2

+

x

+

1

and requires a

quadratic field

extension

as well. Notice that

|

B

|

=

3

just as a

generator

of

G

⁢

F

⁢

(

4

)

×

requires.

When one extends to a field of order

p

2

then the

multiplicative

subgroup of

G

⁢

F

⁢

(

p

2

)

×

has order

p

2

-

1

=

(

p

-

1

)

⁢

(

p

+

1

)

. Here we finally see the source of the remaining parts of the order of

G

⁢

L

⁢

(

V

)

.
We now collect this idea into a

theorem

.

Proposition 7

.

G

⁢

L

⁢

(

d

/

e

,

q

e

)

embeds in

G

⁢

L

⁢

(

d

,

q

)

for every

e

|

d

. In particular,

G

⁢

F

⁢

(

q

e

)

×

embeds in

G

⁢

L

⁢

(

d

,

q

)

for every

e

|

d

.

Proof.

Given

W

a

d

/

e

-dimensional

vector space

over

G

⁢

F

⁢

(

q

e

)

where

e

|

d

,
then

W

is a

d

-dimensional vector space over

G

⁢

F

⁢

(

q

)

as

G

⁢

F

⁢

(

q

)

≤

G

⁢

F

⁢

(

q

e

)

.
Furthermore, if

f

:

W

→

W

is a

G

⁢

F

⁢

(

q

e

)

-

linear map

, then it is
also a

G

⁢

F

⁢

(

q

)

-linear map. Thus we can interpret

G

⁢

L

⁢

(

d

/

e

,

q

e

)

as a
subgroup of

G

⁢

L

⁢

(

d

,

q

)

.

To see that

G

⁢

F

⁢

(

q

e

)

×

embeds in

G

⁢

L

⁢

(

d

,

q

)

simply observe that the

scalar matrices

of

G

⁢

L

⁢

(

d

/

e

,

q

e

)

are isomorphic to

G

⁢

F

⁢

(

q

e

)

.
∎

Caution: although the scalar matrices of

G

⁢

L

⁢

(

e

,

q

d

/

e

)

are

central

in

G

⁢

L

⁢

(

e

,

q

d

/

e

)

, then need not be central in

G

⁢

L

⁢

(

d

,

q

)

.

Definition 8

.

A

Singer cycle

is an element

f

∈

G

⁢

L

⁢

(

V

)

which is the image of
a scalar transform of some field extension.

Remark 9

.

It is not uncommon for Singer cycles to refer only to

maximal order

scalar
transforms.

Finally, as

G

⁢

L

⁢

(

i

,

q

)

embeds in

G

⁢

L

⁢

(

d

,

q

)

for every

1

≤

i

≤

d

we now
conclude:

Corollary 10

.

G

⁢

F

⁢

(

q

i

)

embeds in

G

⁢

L

⁢

(

d

,

q

)

for every

1

≤

i

≤

d

. Moreover, the
order

∏

i

=

2

d

(

q

i

-

1

+

⋯

+

q

+

1

)

are represented by the Singer cycles of

G

⁢

L

⁢

(

d

,

q

)

.

Proof.

G

⁢

F

⁢

(

q

i

)

×

embeds in

G

⁢

L

⁢

(

d

,

q

)

and has order

q

i

-

1

=

(

q

-

1

)

⁢

(

q

i

1

+

⋯

+

q

+

1

)

. As it is cyclic, it has only one

q

-

1

order subgroup and this must therefore correspond to the

G

⁢

F

⁢

(

q

)

×

subgroup. These are accounted for with toral subgroups we already mentioned. The remaining

q

i

-

1

+

⋯

+

q

+

1

cyclic subgroup

of

G

⁢

F

⁢

(

q

i

)

×

are
Singer cycles of

G

⁢

L

⁢

(

d

,

q

)

. In particular, they account for the remaining
portion of the order formula of

G

⁢

L

⁢

(

d

,

q

)

.
∎

Unfortunately, encoding a Singer cycle as a matrix is typically difficult. One can expect this by considering that these matrices must all be matrices which are not triangular. Thus a geometric pattern is out of the question. Yet one will also observe that for this very reason, there are many Singer cycles so at random there is a high probablity of finding such an

invertible matrix

. This is a very useful tool in practical algorithms for matrices which rely on non-deterministic approaches to remain efficient.

If the matrices of

G

⁢

L

⁢

(

d

,

q

)

are interpreted as matrices of

G

⁢

L

⁢

(

d

,

q

¯

)

, where

q

¯

denotes the

algebraic closure

of the field

G

⁢

F

⁢

(

q

)

, then every Singer cycle is diagonalizable as all the eigenvalues now lie in the field. In this case every matrix of

G

⁢

L

⁢

(

d

,

q

)

can be triangularized and so we only need to talk of unipotent and toral subgroups.

The study of Singer cycles has largely been replaced by the more generally applicable study of toral subgroups. Singer cycles generally do not have

determinant

1 and so the study of these elements in projective and special groups is difficult. The study of

semi-simple

elements from Lie theory and algebraic groups replaces the use of Singer cycles.

Title

theory from orders of classical groups

Canonical name

TheoryFromOrdersOfClassicalGroups

Date of creation

2013-03-22 15:56:48

Last modified on

2013-03-22 15:56:48

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

10

Author

Algeboy (12884)

Entry type

Derivation

Classification

msc 11E57

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