aarjaneiro/planetmath huggingface dataset:
virtually abelian group

virtually abelian group

A group

G

is

virtually abelian

(or

abelian-by-finite

)
if it has an

abelian

subgroup

(

http://planetmath.org/Subgroup

) of finite index (

http://planetmath.org/Coset

).

More generally, let

χ

be a property of groups.
A group

G

is

virtually

χ

if it has a subgroup of finite index with the property

χ

.
A group

G

is

χ

-by-finite

if it has a

normal subgroup

of finite index with the property

χ

.
Note that every

χ

-by-finite group is virtually

χ

,
and the converse also holds if the property

χ

is inherited by subgroups.

These notions are obviously only of relevance to

infinite groups

, as all

finite groups

are virtually trivial (and trivial-by-finite).

Title

virtually abelian group

Canonical name

VirtuallyAbelianGroup

Date of creation

2013-03-22 14:35:58

Last modified on

2013-03-22 14:35:58

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 20F99

Classification

msc 20E99

Synonym

abelian-by-finite group

Synonym

virtually-abelian group

Related topic

VirtuallyCyclicGroup

Defines

virtually abelian

Defines

abelian-by-finite

Defines

virtually nilpotent

Defines

virtually solvable

Defines

virtually polycyclic

Defines

virtually free

Defines

nilpotent-by-finite

Defines

polycyclic-by-finite

Defines

virtually nilpotent group

Defines

virtually solvable group

Defines

virtually polycyclic group

Defines

virtually free

Generated on Fri Feb 9 19:30:53 2018 by

LaTeXML
virtually abelian subgroup theorem

virtually abelian subgroup theorem

Let us suppose that

G

is

virtually abelian

and

H

is an

abelian

subgroup

of

G

with a the finite

right coset

partition

G

=

H

⁢

e

⊔

H

⁢

x

2

⊔

…

⊔

H

⁢

x

q

,

*

so if

K

is any other subgroup in

G

we are going to prove:

K

is also virtually abelian

Proof

: From

(

*

)

above we have

K

=

K

∩

G

=

K

∩

(

H

⁢

e

⊔

H

⁢

x

2

⊔

…

⊔

H

⁢

x

q

)

,

=

(

K

∩

H

)

⊔

(

K

∩

H

⁢

x

2

)

⊔

…

⊔

(

K

∩

H

⁢

x

q

)

.

**

Here we consider the two cases:

1)

x

i

∈

K

2)

x

j

∉

K

In the first case

K

=

K

⁢

x

i

, and then

K

∩

H

⁢

x

i

=

K

⁢

x

i

∩

H

⁢

x

i

=

(

K

∩

H

)

⁢

x

i

.
In the second, find

y

j

∈

K

∩

H

⁢

x

j

hence

K

∩

H

⁢

x

j

=

K

⁢

y

j

∩

H

⁢

y

j

=

(

K

∩

H

)

⁢

y

j

So, in the

equation

(

*

*

)

above we can replace (reordering subindexation perhaps) to get

K

=

(

K

∩

H

)

⊔

(

K

∩

H

)

⁢

x

2

⊔

…

⊔

(

K

∩

H

)

⁢

x

s

⏟

1

)

⊔

(

K

∩

H

)

⁢

y

s

+

1

⊔

…

⊔

(

K

∩

H

)

⁢

y

q

⏟

2

)

relation

which shows that the index

[

K

:

K

∩

H

]

≤

[

G

:

H

]

.

It could be

<

since it is posible that

K

∩

H

⁢

x

r

=

∅

for some indexes

r

□

Title

virtually abelian subgroup theorem

Canonical name

VirtuallyAbelianSubgroupTheorem

Date of creation

2013-03-22 18:58:42

Last modified on

2013-03-22 18:58:42

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

13

Author

juanman (12619)

Entry type

Theorem

Classification

msc 20F99

Classification

msc 20E99

Classification

msc 20E07

Synonym

subgroup theorem

Generated on Fri Feb 9 19:30:55 2018 by

LaTeXML
Liouville function

Liouville function

The

Liouville function

is defined by

λ

⁢

(

1

)

=

1

and

λ

⁢

(

n

)

=

(

-

1

)

k

1

+

k

2

+

⋯

+

k

r

, if the

prime factorization

of

n

>

1

is

n

=

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

r

k

r

(where each

p

i

is

positive

). This

function

is

completely multiplicative

and the

∑

d

|

n

λ

⁢

(

d

)

=

{

1

if

n

=

m

2

for some integer

m

0

otherwise,

where the sum runs over all positive

divisors

of

n

.

Title

Liouville function

Canonical name

LiouvilleFunction

Date of creation

2013-03-22 11:47:09

Last modified on

2013-03-22 11:47:09

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

12

Author

KimJ (5)

Entry type

Definition

Classification

msc 20G10

Classification

msc 11A25

Classification

msc 81-00

Generated on Fri Feb 9 19:30:57 2018 by

LaTeXML
computation of the order of GL⁡(n,𝔽_q)

computation of the order of

GL

⁡

(

n

,

𝔽

q

)

\PMlinkescapephrase

linearly independent

GL

⁡

(

n

,

𝔽

q

)

is the group of

invertible

n

×

n

matrices
over the

finite field

𝔽

q

.
Here is a proof that

|

GL

⁡

(

n

,

𝔽

q

)

|

=

(

q

n

-

1

)

⁢

(

q

n

-

q

)

⁢

⋯

⁢

(

q

n

-

q

n

-

1

)

.

Each

element

A

∈

GL

⁡

(

n

,

𝔽

q

)

is given by a

collection

of

n

𝔽

q

-linearly independent vectors (

http://planetmath.org/LinearIndependence

).
If one chooses the first

column vector

of

A

from

(

𝔽

q

)

n

there are

q

n

choices, but one can’t choose the

zero vector

since this would make the

determinant

of

A

zero.
So there are really only

q

n

-

1

choices.
To choose an

i

-th vector from

(

𝔽

q

)

n

which is linearly independent from

i

-

1

already chosen
linearly independent vectors

{

V

1

,

…

,

V

i

-

1

}

one must choose a vector not in
the span of

{

V

1

,

…

,

V

i

-

1

}

.
There are

q

i

-

1

vectors in this span,
so the number of choices is

q

n

-

q

i

-

1

.
Thus the number of linearly independent collections of

n

vectors in

𝔽

q

is

(

q

n

-

1

)

⁢

(

q

n

-

q

)

⁢

⋯

⁢

(

q

n

-

q

n

-

1

)

.

Title

computation of the order of

GL

⁡

(

n

,

𝔽

q

)

Canonical name

ComputationOfTheOrderOfoperatornameGLnmathbbFq

Date of creation

2013-03-22 13:06:50

Last modified on

2013-03-22 13:06:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Proof

Classification

msc 20G15

Related topic

OrderOfTheGeneralLinearGroupOverAFiniteField

Generated on Fri Feb 9 19:31:00 2018 by

LaTeXML
general linear group

general linear group

Given a

vector space

V

, the

general linear group

GL

⁡

(

V

)

is defined to be the group of

invertible linear transformations

from

V

to

V

. The

group operation

is defined by

composition

: given

T

:

V

⟶

V

and

T

′

:

V

⟶

V

in

GL

⁡

(

V

)

, the

product

T

⁢

T

′

is just the composition of the maps

T

and

T

′

.

If

V

=

𝔽

n

for some field

𝔽

, then the group

GL

⁡

(

V

)

is often denoted

GL

⁡

(

n

,

𝔽

)

or

GL

n

⁡

(

𝔽

)

. In this case, if one identifies each

linear transformation

T

:

V

⟶

V

with its matrix with respect to the

standard basis

, the group

GL

⁡

(

n

,

𝔽

)

becomes the group of

invertible

n

×

n

matrices with entries in

𝔽

, under the group operation of

matrix multiplication

.

One also discusses the general linear group on a module

M

over some ring

R

. There it is the set of

automorphisms

of

M

as an

R

-module. For example, one might take

GL

⁡

(

ℤ

⊕

ℤ

)

; this is

isomorphic

to the group of two-by-two matrices with integer entries having

determinant

±

1

. If

M

is a general

R

-module, there need not be a natural

interpretation

of

GL

⁡

(

M

)

as a

matrix group

.

The general linear group is an example of a

group scheme

; viewing it in this way ties together the properties of

GL

⁡

(

V

)

for different vector spaces

V

and different fields

F

. The general linear group is an algebraic group, and it is a Lie group if

V

is a real or

complex vector space

.

When

V

is a

finite-dimensional

Banach space

,

GL

⁡

(

V

)

has a natural topology coming from the

operator norm

; this is isomorphic to the

topology

coming from its

embedding

into the ring of matrices. When

V

is an infinite-dimensional vector space, some elements of

GL

⁡

(

V

)

may not be

continuous

and one generally looks instead at the set of

bounded operators

.

Title

general linear group

Canonical name

GeneralLinearGroup

Date of creation

2013-03-22 12:25:36

Last modified on

2013-03-22 12:25:36

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 20G15

Related topic

Group

Related topic

Representation

Related topic

SpecialLinearGroup

Generated on Fri Feb 9 19:31:07 2018 by

LaTeXML
irreducible representations of the special linear group over 𝔽_p

irreducible representations of the special linear group over

𝔽

p

Let

p

≥

5

be a prime and let

G

=

SL

⁡

(

2

,

𝔽

p

)

be the

special linear group

over the field with

p

elements.

Lemma

.

The group

G

=

SL

⁡

(

2

,

F

p

)

has no non-trivial

1

dimensional

irreducible representations

over

F

p

.

Proof.

See an attached entry (

http://planetmath.org/MathitSL2mathbbF_pHasNo1DimensionalIrreducibleRepresentationsOverMathbbF_p

) for the proof.
∎

Next, we construct several irreducible representations for

G

=

SL

⁡

(

2

,

𝔽

p

)

. For

0

≤

m

≤

p

-

1

, let

V

m

be the

𝔽

p

vector space

of

homogeneous polynomials

of degree

m

in the independent variables

x

and

y

(of course, for

m

=

0

, the

representation

is trivial). We give

V

m

a structure of

𝔽

p

⁢

[

G

]

-module as follows. Let

p

=

p

⁢

(

(

x

,

y

)

)

∈

V

m

and

A

∈

SL

⁡

(

2

,

𝔽

p

)

. We define:

A

⋅

p

:=

p

⁢

(

A

⋅

(

x

,

y

)

t

)

=

p

⁢

(

a

11

⁢

x

+

a

12

⁢

y

,

a

21

⁢

x

+

a

22

⁢

y

)

where

t

denotes transpose. The representations

V

m

are, in a sense, all the irreducible representations of

G

.

Theorem

.

For

0

≤

m

≤

p

-

1

, the representations

V

m

are irreducible representations of

dimension

m

+

1

over

F

p

. Furthermore, up to

isomorphism

, there are no other irreducible representations of

G

over

F

p

.

References

1

Charles B. Thomas,

Representations of Finite and Lie Groups

, Imperial College Press, London.

Title

irreducible representations of the special linear group over

𝔽

p

Canonical name

IrreducibleRepresentationsOfTheSpecialLinearGroupOvermathbbFp

Date of creation

2013-03-22 15:09:53

Last modified on

2013-03-22 15:09:53

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20G15

Related topic

GroupRepresentation

Related topic

SpinNetworksAndSpinFoams

Generated on Fri Feb 9 19:31:10 2018 by

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Nagao’s theorem

Nagao’s theorem

For any integral domain

k

, the group of

n

×

n

invertible matrices

with coefficients in

k

⁢

[

t

]

is the

amalgamated free product

of invertible matrices over

k

and

invertible

upper triangular matrices

over

k

⁢

[

t

]

, amalgamated over the upper triangular matrices of

k

. More compactly

GL

n

⁢

(

k

⁢

[

t

]

)

≅

GL

n

⁢

(

k

)

*

B

⁢

(

k

)

B

⁢

(

k

⁢

[

t

]

)

.

Title

Nagao’s theorem

Canonical name

NagaosTheorem

Date of creation

2013-03-22 13:58:31

Last modified on

2013-03-22 13:58:31

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20G15

Generated on Fri Feb 9 19:31:12 2018 by

LaTeXML
projective special linear group

projective special linear group

Definition

.

Let

V

be a

vector space

over a field

F

and let

SL

⁡

(

V

)

be the

special linear group

. Let

Z

be the center of

SL

⁡

(

V

)

. The

projective special linear group

associated to

V

is the

quotient group

SL

⁡

(

V

)

/

Z

and is usually denoted by

PSL

⁡

(

V

)

.

When

V

is a finite dimensional vector space over

F

(of

dimension

n

) then we write

PSL

⁡

(

n

,

F

)

or

PSL

n

⁡

(

F

)

. We also identify the

linear transformations

of

V

with

n

×

n

matrices, so

PSL

may be regarded as a quotient of the group of matrices

SL

⁡

(

n

,

F

)

by its center.

Note: see the entry on

projective space

for the origin of the terminology.

Theorem 1

.

The center

Z

of

SL

⁡

(

n

,

F

)

is the group of all

scalar matrices

λ

⋅

Id

where

λ

is an

n

th

root of unity

in

F

.

In particular, for

n

=

2

,

Z

=

{

±

Id

}

and:

PSL

⁡

(

2

,

F

)

=

SL

⁡

(

2

,

F

)

/

{

±

Id

}

.

As a consequence of the previous theorem, we obtain:

Theorem 2

.

For

n

≥

3

,

PSL

⁡

(

n

,

F

)

is a

simple group

.
Furthermore, if

F

is a

finite field

then the groups

PSL

⁡

(

n

,

𝔽

)

=

SL

⁡

(

n

,

𝔽

)

/

Z

,

n

≥

2

are all finite simple groups, except for

n

=

2

and

F

=

F

2

,

F

3

.

References

1

S. Lang,

Algebra

, Springer-Verlag, New York.

2

D. Dummit, R. Foote,

Abstract Algebra

,
Second Edition, Wiley.

Title

projective special linear group

Canonical name

ProjectiveSpecialLinearGroup

Date of creation

2013-03-22 15:09:46

Last modified on

2013-03-22 15:09:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 20G15

Synonym

PSL

Related topic

TheoremsOfSpecialLinearGroupOverAFiniteField

Generated on Fri Feb 9 19:31:14 2018 by

LaTeXML
SL(n;R) is connected

SL(n;R) is connected

The special feature is that although not every element of

S

⁢

L

⁢

(

n

,

ℝ

)

is in the image of the exponential map of

𝔰

⁢

𝔩

⁢

(

n

,

ℝ

)

,

S

⁢

L

⁢

(

n

,

ℝ

)

is still a

connected

Lie group. The proof below is a guideline and should be clarified a bit more at some points, but this was done intentionally.

To illustrate the point, first we show

Proposition 0.1

.

(

-

1

1

0

-

1

)

∉

exp

⁡

𝔰

⁢

𝔩

⁢

(

2

,

ℝ

)

, but it is in

S

⁢

L

⁢

(

2

,

R

)

.

Proof.

det

x

=

:

det

(

-

1

1

0

-

1

)

=

1

, so

x

∈

S

⁢

L

⁢

(

2

,

ℝ

)

. We see that

x

is not

diagonalizable

, it already is in Jordan normal form. Moreover, it has a double

eigenvalue

,

-

1

. Suppose that

x

=

exp

⁡

X

,

X

∈

𝔰

⁢

𝔩

⁢

(

2

,

ℝ

)

, then

tr

⁡

X

=

0

. Since

x

had a double eigenvalue, so does

X

, hence the eigenvalues of

X

both are

0

. But this implies the eigenvalues of

x

are

1

. This is a

contradiction

.
∎

Lemma 0.2

.

We have

∀

x

∈

S

⁢

L

⁢

(

n

,

R

)

:

x

=

exp

⁡

(

X

a

)

⁢

exp

⁡

(

X

s

)

with

X

a

t

=

-

X

a

,

X

s

t

=

X

s

∈

s

⁢

l

⁢

(

n

,

R

)

.

Proof.

The keyword here is

polar decomposition

.
We notice that

x

t

⁢

x

is

symmetric

and

positive definite

, since

∀

ψ

∈

ℝ

n

:

⟨

ψ

,

x

t

⁢

x

⁢

ψ

⟩

>

0

, with the standard

inner product

on

ℝ

n

. Hence, we can write

x

=

R

⁢

P

, with

P

=

(

x

t

⁢

x

)

1

2

and

R

=

x

⁢

P

-

1

.

P

is well defined, since any real symmetric,

positive definite matrix

is diagonalizable. It’s easy to check that

R

⁢

R

t

=

id

n

, hence

R

∈

O

⁢

(

n

)

. We had

det

⁡

P

>

0

and

det

⁡

x

=

1

, hence

det

⁡

(

R

)

>

0

⇒

det

⁡

R

=

1

⇒

R

∈

S

⁢

O

⁢

(

n

⁢

𝕟

)

and so

det

⁡

P

=

1

. Since the choice of positive root is unique,

R

and

P

are unique. Moreover,

S

⁢

O

⁢

(

n

)

is exactly generated by the set

{

X

∈

G

⁢

L

⁢

(

n

,

ℝ

)

|

X

t

=

-

X

}

and

Ω

, the set of real

symmetric matrices

of

determinant

1

, by

{

X

∈

G

⁢

L

⁢

(

n

,

ℝ

)

|

X

t

=

X

,

tr

⁡

X

=

0

}

, we have the wanted statement:

S

L

(

n

,

ℝ

⊂

S

O

(

n

)

×

exp

Ω

.
∎

The reverse

inclusion

is simply shown: any such

combination

is trivially in

S

⁢

L

⁢

(

n

,

ℝ

)

.

Corollary 0.3

.

S

⁢

L

⁢

(

n

,

ℝ

)

is connected.

Proof.

This is now clear from the fact that both

S

⁢

O

⁢

(

n

)

and

Ω

are connected and so

∀

s

,

t

∈

[

0

,

1

]

:

exp

⁡

s

⁢

X

⁢

exp

⁡

t

⁢

Y

∈

S

⁢

L

⁢

(

n

,

ℝ

)

, a fact easily checked by taking the determinant. So

S

⁢

L

⁢

(

n

,

ℝ

)

is

path-connected

, hence connected.
∎

Title

SL(n;R) is connected

Canonical name

SLnRIsConnected

Date of creation

2013-03-22 18:52:05

Last modified on

2013-03-22 18:52:05

Owner

Stephaninos (23208)

Last modified by

Stephaninos (23208)

Numerical id

5

Author

Stephaninos (23208)

Entry type

Result

Classification

msc 20G15

Generated on Fri Feb 9 19:31:17 2018 by

LaTeXML
special linear group

special linear group

Given a

vector space

V

, the

special linear group

SL

⁡

(

V

)

is defined to be the

subgroup

of the

general linear group

GL

⁡

(

V

)

consisting of all

invertible linear transformations

T

:

V

⟶

V

in

GL

⁡

(

V

)

that have

determinant

1.

If

V

=

𝔽

n

for some field

𝔽

, then the group

SL

⁡

(

V

)

is often denoted

SL

⁡

(

n

,

𝔽

)

or

SL

n

⁡

(

𝔽

)

, and if one identifies each

linear transformation

with its matrix with respect to the

standard basis

, then

SL

⁡

(

n

,

𝔽

)

consists of all

n

×

n

matrices with entries in

𝔽

that have determinant 1.

Title

special linear group

Canonical name

SpecialLinearGroup

Date of creation

2013-03-22 12:25:38

Last modified on

2013-03-22 12:25:38

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 20G15

Synonym

SL

Related topic

GeneralLinearGroup

Related topic

Group

Related topic

UnimodularMatrix

Generated on Fri Feb 9 19:31:20 2018 by

LaTeXML
theorems of special linear group over a finite field

theorems of special linear group over a finite field

Let

𝔽

q

be the finite field with

q

elements, and consider the

special linear group

SL

⁡

(

n

,

𝔽

q

)

over the field

𝔽

q

.

1.

SL

⁡

(

n

,

𝔽

q

)

is finite. Furthermore,

|

SL

⁡

(

n

,

𝔽

q

)

|

=

1

q

-

1

⁢

∏

i

=

0

n

-

1

(

q

n

-

q

i

)

.

2.

SL

⁡

(

n

,

𝔽

q

)

is a

perfect group

, meaning that

[

SL

⁡

(

n

,

𝔽

q

)

,

SL

⁡

(

n

,

𝔽

q

)

]

=

SL

⁡

(

n

,

𝔽

q

)

, where

[

,

]

is the

commutator bracket

with two exceptions:

SL

⁡

(

2

,

𝔽

2

)

and

SL

⁡

(

2

,

𝔽

3

)

.

Title

theorems of special linear group over a finite field

Canonical name

TheoremsOfSpecialLinearGroupOverAFiniteField

Date of creation

2013-03-22 14:55:54

Last modified on

2013-03-22 14:55:54

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

6

Author

Daume (40)

Entry type

Theorem

Classification

msc 20G15

Related topic

ProjectiveSpecialLinearGroup

Generated on Fri Feb 9 19:31:25 2018 by

LaTeXML
𝑆𝐿⁢(2,𝔽_p) has no 1 dimensional irreducible representations over 𝔽_p

𝑆𝐿

⁢

(

2

,

𝔽

p

)

has no

1

dimensional irreducible representations over

𝔽

p

Lemma

.

The group

G

=

SL

⁡

(

2

,

F

p

)

has no non-trivial

1

dimensional

irreducible representations

over

F

p

.

Proof.

Notice that a

1

dimensional

representations

over

𝔽

p

is just a

homomorphism

:

ρ

:

SL

⁡

(

2

,

𝔽

p

)

⟶

𝔽

p

×

.

Let

ρ

be as above. Then there exist an induced homomorphism for the

projective special linear group

:

ρ

¯

:

PSL

⁡

(

2

,

𝔽

p

)

⟶

𝔽

p

×

/

{

±

𝐼𝑑

}

defined by

ρ

¯

⁢

(

A

)

=

ρ

⁢

(

B

)

mod

{

±

𝐼𝑑

}

, where

B

is any lift of

A

to

SL

⁡

(

2

,

𝔽

p

)

(this is well defined because

ρ

⁢

(

-

𝐼𝑑

)

=

±

𝐼𝑑

). Since

PSL

⁡

(

2

,

𝔽

p

)

is simple, the image of

ρ

¯

is

trivial

, and therefore, the image of

ρ

is

contained

in

{

±

𝐼𝑑

}

.

However,

SL

⁡

(

2

,

𝔽

p

)

does not have

subgroups

of index

2

(a subgroup of index

2

is normal). For our purposes, it suffices to show that:

ρ

:

SL

⁡

(

2

,

𝔽

p

)

⟶

{

±

𝐼𝑑

}

satisfies

ρ

⁢

(

-

𝐼𝑑

)

=

𝐼𝑑

. Let

S

∈

SL

⁡

(

2

,

𝔽

p

)

be the matrix:

S

=

(

0

-

1

1

0

)

.

Notice that

S

2

=

-

𝐼𝑑

and so

ρ

⁢

(

S

2

)

=

(

ρ

⁢

(

S

)

)

2

=

𝐼𝑑

=

ρ

⁢

(

-

𝐼𝑑

)

, as desired.
∎

Title

𝑆𝐿

⁢

(

2

,

𝔽

p

)

has no

1

dimensional irreducible representations over

𝔽

p

Canonical name

mathitSL2mathbbFpHasNo1DimensionalIrreducibleRepresentationsOvermathbbFp

Date of creation

2013-03-22 15:09:56

Last modified on

2013-03-22 15:09:56

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20G15

Generated on Fri Feb 9 19:31:27 2018 by

LaTeXML
orthogonal group

orthogonal group

Let

Q

be a

non-degenerate

symmetric bilinear form

over the

real vector space

ℝ

n

. A

linear transformation

T

:

V

→

V

is said to

preserve

Q

if

Q

⁢

(

T

⁢

x

,

T

⁢

y

)

=

Q

⁢

(

x

,

y

)

for all vectors

x

,

y

∈

V

. The

subgroup

of the

general linear group

GL

⁡

(

V

)

consisting of all linear transformations that preserve

Q

is called the

orthogonal group

with respect to

Q

, and denoted

O

⁡

(

n

,

Q

)

.

If

Q

is also

positive definite

(i.e.,

Q

is an

inner product

), then

O

⁡

(

n

,

Q

)

is equivalent to the group of

invertible linear transformations

that preserve the standard inner product on

ℝ

n

, and in this case the group

O

⁡

(

n

,

Q

)

is usually denoted

O

⁡

(

n

)

.

Elements of

O

⁡

(

n

)

are called

orthogonal transformations

.
One can show that a linear transformation

T

is an orthogonal transformation if and only if

T

-

1

=

T

T

(i.e., the

inverse

of

T

equals the

transpose

of

T

).

Title

orthogonal group

Canonical

name

OrthogonalGroup

Date of creation

2013-03-22 12:25:54

Last modified on

2013-03-22 12:25:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 20G20

Defines

orthogonal transformation

Generated on Fri Feb 9 19:31:29 2018 by

LaTeXML
Ihara’s theorem

Ihara’s theorem

Let

Γ

be a discrete,

torsion-free

subgroup

of

SL

2

⁢

ℚ

p

(where

ℚ

p

is the field of

p

-adic numbers (

http://planetmath.org/PAdicIntegers

)). Then

Γ

is free.

Proof, or a sketch thereof.

There exists a

p

+

1

regular

tree

X

on which

SL

2

⁢

ℚ

p

acts, with

stabilizer

SL

2

⁢

ℤ

p

(here,

ℤ

p

denotes the ring of

p

-adic integers (

http://planetmath.org/PAdicIntegers

)). Since

ℤ

p

is

compact

in its

profinite topology

, so is

SL

2

⁢

ℤ

p

. Thus,

SL

2

⁢

ℤ

p

∩

Γ

must be compact, discrete and torsion-free. Since compact and discrete implies finite, the only such group is trivial. Thus,

Γ

acts freely on

X

. Since groups acting freely on trees are free,

Γ

is free.
∎

Title

Ihara’s theorem

Canonical name

IharasTheorem

Date of creation

2013-03-22 13:54:26

Last modified on

2013-03-22 13:54:26

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

9

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20G25

Generated on Fri Feb 9 19:31:31 2018 by

LaTeXML
𝑆𝐿_2⁢(F_3)

𝑆𝐿

2

⁢

(

F

3

)

The

special linear group

over the finite field

𝔽

3

is represented by

𝑆𝐿

2

⁢

(

𝔽

3

)

and consists of the

2

×

2

invertible matrices

with

determinant

equal to

1

and whose entries belong to

𝔽

3

.

Title

𝑆𝐿

2

⁢

(

F

3

)

Canonical

name

mathitSL2F3

Date of creation

2013-03-22 14:00:32

Last modified on

2013-03-22 14:00:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 20G40

Synonym

SL(F3)

Related topic

FiniteField

Related topic

Field

Related topic

Group

Generated on Fri Feb 9 19:31:33 2018 by

LaTeXML