aarjaneiro/planetmath huggingface dataset:
Archimedean semigroup

Archimedean semigroup

Let

S

be a

commutative semigroup

. We say an

element

x

divides

an element

y

, written

x

∣

y

, if there is an element

z

such that

x

⁢

z

=

y

.

An

Archimedean semigroup

S

is a commutative semigroup with the property that for all

x

,

y

∈

S

there is a

natural number

n

such that

x

∣

y

n

.

This is related to the Archimedean property of positive real numbers

ℝ

+

: if

x

,

y

>

0

then there is a natural number

n

such that

x

<

n

⁢

y

. Except that the notation is

additive

rather than multiplicative, this is the same as saying that

(

ℝ

+

,

+

)

is an Archimedean semigroup.

Title

Archimedean semigroup

Canonical name

ArchimedeanSemigroup

Date of creation

2013-03-22 13:08:06

Last modified on

2013-03-22 13:08:06

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20M14

Related topic

ArchimedeanProperty

Defines

divides

Defines

Archimedean

Generated on Fri Feb 9 19:34:05 2018 by

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commutative semigroup

commutative semigroup

A

semigroup

S

is

commutative

if the defining

binary operation

is commutative (

http://planetmath.org/Commutative

). That is, for all

x

,

y

∈

S

, the

identity

x

⁢

y

=

y

⁢

x

holds.

Although the term

Abelian semigroup

is sometimes used, it is more common simply to refer to such semigroups as

commutative semigroups

.

A monoid which is also a commutative semigroup is called a

commutative monoid

.

Title

commutative semigroup

Canonical

name

CommutativeSemigroup

Date of creation

2013-03-22 13:08:09

Last modified on

2013-03-22 13:08:09

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20M14

Synonym

Abelian semigroup

Related topic

AbelianGroup

Related topic

AbelianGroup2

Defines

commutative

Defines

commutative monoid

Generated on Fri Feb 9 19:34:06 2018 by

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regular semigroup

regular semigroup

\PMlinkescapephrase

completely regular

\PMlinkescapephrase

generated by

Let

S

be a

semigroup

.

x

∈

S

is

regular

if there is a

y

∈

S

such that

x

=

x

⁢

y

⁢

x

.

y

∈

S

is an

inverse

(or a

relative inverse

) for

x

if

x

=

x

⁢

y

⁢

x

and

y

=

y

⁢

x

⁢

y

.

1

Regular semigroups

S

is a

regular semigroup

if all its elements are regular.
The phrase ’

von Neumann regular

’ is sometimes used, after the definition for rings.

In a regular semigroup, every

principal ideal

is generated by an

idempotent

.

Every regular element has at least one inverse.
To show this, suppose

a

∈

S

is regular,
so that

a

=

a

⁢

b

⁢

a

for some

b

∈

S

.
Put

c

=

b

⁢

a

⁢

b

.
Then

a

=

a

⁢

b

⁢

a

=

(

a

⁢

b

⁢

a

)

⁢

b

⁢

a

=

a

⁢

(

b

⁢

a

⁢

b

)

⁢

a

=

a

⁢

c

⁢

a

and

c

=

b

⁢

a

⁢

b

=

b

⁢

(

a

⁢

b

⁢

a

)

⁢

b

=

(

b

⁢

a

⁢

b

)

⁢

a

⁢

b

=

c

⁢

a

⁢

b

=

c

⁢

(

a

⁢

b

⁢

a

)

⁢

b

=

c

⁢

a

⁢

(

b

⁢

a

⁢

b

)

=

c

⁢

a

⁢

c

,

so

c

is an inverse of

a

.

2

Inverse semigroups

S

is an

inverse semigroup

if for all

x

∈

S

there is a

unique

y

∈

S

such that

x

=

x

⁢

y

⁢

x

and

y

=

y

⁢

x

⁢

y

.

In an inverse semigroup every principal ideal is generated by a

unique

idempotent.

In an inverse semigroup the set of idempotents is a

subsemigroup

, in particular a

commutative

band (

http://planetmath.org/ASemilatticeIsACommutativeBand

).

The

bicyclic semigroup

is an example of an inverse semigroup.
The

symmetric inverse semigroup

(on some set

X

) is another example.
Of course, every group is also an inverse semigroup.

3

Motivation

Both of these notions generalise the definition of a group. In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents. Apart from inverse semigroups, there are

orthodox semigroups

where the set of idempotents is a subsemigroup, and

Clifford semigroups

where the idempotents are central.

4

Additional

S

is called

eventually regular

(or

π

-regular

) if a power of every element is regular.

S

is called

group-bound

(or

strongly

π

-regular

, or an

epigroup

) if a power of every element is in a

subgroup

of

S

.

S

is called

completely regular

if every element is in a subgroup of

S

.

Title

regular semigroup

Canonical name

RegularSemigroup

Date of creation

2013-03-22 14:23:17

Last modified on

2013-03-22 14:23:17

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Definition

Classification

msc 20M17

Classification

msc 20M18

Related topic

ACharacterizationOfGroups

Defines

regular

Defines

π

-regular

Defines

eventually regular

Defines

strongly

π

-regular

Defines

group-bound

Defines

inverse semigroup

Defines

Clifford semigroup

Defines

orthodox semigroup

Defines

completely regular

Defines

epigroup

Defines

regular element

Defines

inverse

Defines

relative inverse

Generated on Fri Feb 9 19:34:08 2018 by

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McAlister covering theorem

McAlister covering theorem

A subset

X

in an

inverse semigroup

S

is called

unitary

if for any

elements

x

∈

X

and

s

∈

S

,

x

⁢

s

∈

X

or

s

⁢

x

∈

X

implies

s

∈

X

.

An inverse semigroup is

E-unitary

if its

semigroup

of

idempotents

is unitary.

Theorem

.

Let

S

be an inverse semigroup; then, there exists an E-unitary inverse semigroup

P

and a

surjective

, idempotent-separating

homomorphism

θ

:

P

→

S

.

Also, if

S

is finite, then

P

may be chosen to be finite as well.

Note that a homomorphism is

idempotent-separating

if it is

injective

on idempotents.

References

1

M. Lawson,

Inverse Semigroups: The Theory of Partial

Symmetries

, World Scientific, 1998

Title

McAlister covering theorem

Canonical name

McAlisterCoveringTheorem

Date of creation

2013-03-22 14:37:19

Last modified on

2013-03-22 14:37:19

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 20M18

Defines

unitary

Defines

E-unitary

Defines

idempotent-separating

Generated on Fri Feb 9 19:34:10 2018 by

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symmetric inverse semigroup

symmetric inverse semigroup

Let

X

be a set. A

partial map

on

X

is an application defined from a subset of

X

into

X

. We denote by

𝔉

⁢

(

X

)

the set of partial map on

X

. Given

α

∈

𝔉

⁢

(

X

)

, we denote by

dom

⁢

(

α

)

and

ran

⁢

(

α

)

respectively the domain and the range of

α

, i.e.

dom

(

α

)

,

ran

α

⊆

X

,

α

:

dom

(

α

)

→

X

,

α

(

dom

(

α

)

)

=

ran

(

α

)

.

We define the

composition

of two partial map

α

,

β

∈

𝔉

⁢

(

X

)

as the partial map

α

∘

β

∈

𝔉

⁢

(

X

)

with domain

dom

⁢

(

α

∘

β

)

=

β

-

1

⁢

(

ran

⁢

(

β

)

∩

dom

⁢

(

α

)

)

=

{

x

∈

dom

⁢

(

β

)

|

α

⁢

(

x

)

∈

dom

⁢

(

β

)

}

defined by the common rule

α

∘

β

⁢

(

x

)

=

α

⁢

(

β

⁢

(

x

)

)

,

∀

x

∈

dom

⁢

(

α

∘

β

)

.

It is easily verified that the

𝔉

⁢

(

X

)

with the composition

∘

is a

semigroup

.

A partial map

α

∈

𝔉

⁢

(

X

)

is said

bijective

when it is bijective as a map

α

:

ran

⁢

(

α

)

→

dom

⁢

(

α

)

. It can be proved that the subset

ℑ

⁢

(

X

)

⊆

𝔉

⁢

(

X

)

of the partial bijective maps on

X

is an

inverse semigroup

(with the composition

∘

), that is called

symmetric inverse semigroup

on

X

. Note that the

symmetric group

on

X

is a

subgroup

of

ℑ

⁢

(

X

)

.

Title

symmetric inverse semigroup

Canonical

name

SymmetricInverseSemigroup

Date of creation

2013-03-22 16:11:14

Last modified on

2013-03-22 16:11:14

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

6

Author

Mazzu (14365)

Entry type

Definition

Classification

msc 20M18

Defines

partial map

Defines

composition of partial maps

Defines

symmetric inverse semigroup

Generated on Fri Feb 9 19:34:12 2018 by

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Wagner-Preston representation theorem

Wagner-Preston representation theorem

Let

S

be an

inverse semigroup

and

X

a set. An inverse semigroup

homomorphism

ϕ

:

S

→

ℑ

⁢

(

X

)

, where

ℑ

⁢

(

X

)

denotes the

symmetric inverse semigroup

, is called a

representation

of

S

by

bijective

partial maps on

X

. The representation is said to be

faithful

if

ϕ

is a

monomorphism

, i.e. it is

injective

.

Given

s

∈

S

, we define

ρ

s

∈

ℑ

⁢

(

S

)

as the bijective partial map with domain

dom

⁢

(

ρ

s

)

=

S

⁢

s

-

1

=

{

t

⁢

s

-

1

|

t

∈

S

}

and defined by

ρ

s

⁢

(

t

)

=

t

⁢

s

,

∀

t

∈

dom

⁢

(

ρ

s

)

.

Then the map

s

↦

ρ

s

is a representation called the

Wagner-Preston representation

of

S

. The following result, due to Wagner and Preston, is analogous to the Cayley representation theorem for groups.

Theorem 1

(Wagner-Preston representation theorem)

The Wagner-Preston representation of an inverse semigroup is faithful.

References

1

N. Petrich,

Inverse Semigroups

, Wiley, New York, 1984.

2

G.B. Preston,

Representation of

inverse

semi-groups

, J. London Math. Soc. 29 (1954), 411-419.

Title

Wagner-Preston representation theorem

Canonical name

WagnerPrestonRepresentationTheorem

Date of creation

2013-03-22 16:11:16

Last modified on

2013-03-22 16:11:16

Owner

Mazzu (14365)

Last modified by

Mazzu (14365)

Numerical id

10

Author

Mazzu (14365)

Entry type

Theorem

Classification

msc 20M18

Defines

representation by bijective partial maps

Defines

faithful representation

Defines

Wagner-Preston representation

Generated on Fri Feb 9 19:34:13 2018 by

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word problem

word problem

Let

(

X

;

R

)

be a

presentation

for the group

G

=

Gp

⁢

⟨

X

∣

R

⟩

. It is well known that

G

is a

quotient group

of the

free monoid with involution

on

X

, i.e.

G

=

(

X

∐

X

-

1

)

∗

/

θ

for some

congruence

θ

⊆

(

X

∐

X

-

1

)

∗

×

(

X

∐

X

-

1

)

∗

. We recall that

R

⊂

(

X

∐

X

-

1

)

∗

is a set of words all representing the

identity

1

G

of the group, i.e.

[

r

]

θ

=

1

G

for all

r

∈

R

. The

word problem

in the category of groups consists in establish whether or not two given words

v

,

w

∈

(

X

∐

X

-

1

)

∗

represent

the same element of

G

, i.e. whether or not

[

v

]

θ

=

[

w

]

θ

.

Let

(

X

;

T

)

be a presentation for the

inverse

monoid

M

=

Inv

1

⁢

⟨

X

∣

T

⟩

=

(

X

∐

X

-

1

)

∗

/

τ

, where

τ

=

(

ρ

X

∪

T

)

c

. The

concept

of presentation for inverse monoid is analogous to the group’s one, but now

T

is a

binary relation

on

(

X

∐

X

-

1

)

∗

, i.e.

T

⊆

(

X

∐

X

-

1

)

∗

×

(

X

∐

X

-

1

)

∗

.
The

word problem

in the

category

of inverse monoids consists in establish whether or not two given words

v

,

w

∈

(

X

∐

X

-

1

)

∗

represent the same element of

M

, i.e. whether or not

[

v

]

τ

=

[

w

]

τ

.

We can modify the last paragraph to introduce the

word problem

in the category of inverse semigroups as well.

A classical results in combinatorial group theory says that the word problem in the category of groups is

undecidable

, so it is undecidable also for the larger categories of inverse semigroups and inverse monoids.

References

1

W. W. Boone,

Certain simple unsolvable problems in

group theory

, I, II, III,
IV, V, VI, Nederl. Akad.Wetensch Proc. Ser. A57, 231-237,492- 497 (1954), 58,
252-256,571-577 (1955), 60, 22-27,227-232 (1957).

2

R. Lyndon and P. Schupp,

Combinatorial Group Theory

, Springer-Verlag, 1977.

3

P.S. Novikov,

On the algorithmic unsolvability of the word problem in group theory

,
Trudy Mat. Inst. Steklov 44, 1-143 (1955).

4

J.B. Stephen,

Presentation of inverse monoids

, J. Pure Appl.

Algebra

63 (1990) 81-
112.

Title

word problem

Canonical name

WordProblem

Date of creation

2013-05-17 17:00:19

Last modified on

2013-05-17 17:00:19

Owner

Mazzu (14365)

Last modified by

unlord (1)

Numerical id

11

Author

Mazzu (1)

Entry type

Definition

Classification

msc 20M18

Classification

msc 20M05

Defines

word problem

Generated on Fri Feb 9 19:34:16 2018 by

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Cayley’s theorem for semigroups

Cayley’s theorem for semigroups

Let

X

be a set.
We can define on

X

X

,
the set of

functions

from

X

to itself,
a

structure

of

semigroup

by putting

f

⊗

g

=

g

∘

f

.
Such semigroup is actually a monoid,
whose

identity element

is the

identity function

of

X

.

Theorem 1

(Cayley’s theorem for semigroups)

For every semigroup

(

S

,

⋅

)

there exist a set

X

and an injective map

ϕ

:

S

→

X

X

which is a

morphism

of semigroups from

(

S

,

⋅

)

to

(

X

X

,

⊗

)

.

In other words,
every semigroup is

isomorphic

to
a

semigroup of transformations

of some set.
This is an

extension

of Cayley’s

theorem

on groups,
which states that every group is isomorphic to
a group of

invertible

transformations

of some set.

Proof of Theorem

1

.

The

argument

is

similar

to the one for Cayley’s theorem on groups.
Let

X

=

S

, the set of elements of the semigroup.

First, suppose

(

S

,

⋅

)

is a monoid with unit

e

.
For

s

∈

S

define

f

s

:

S

→

S

as

f

s

⁢

(

x

)

=

x

⋅

s

⁢

∀

x

∈

S

.

(1)

Then for every

s

,

t

,

x

∈

S

we have

f

s

⋅

t

⁢

(

x

)

=

x

⋅

(

s

⋅

t

)

=

(

x

⋅

s

)

⋅

t

=

f

t

⁢

(

x

⋅

s

)

=

f

t

⁢

(

f

s

⁢

(

x

)

)

=

(

f

t

∘

f

s

)

⁢

(

x

)

=

(

f

s

⊗

f

t

)

⁢

(

x

)

,

so

ϕ

⁢

(

s

)

=

f

s

is a

homomorphism

of monoids,
with

f

e

=

id

S

.
This homomorphism is

injective

,
because if

f

s

=

f

t

,
then

s

=

f

s

⁢

(

e

)

=

f

t

⁢

(

e

)

=

t

.

Next, suppose

(

S

,

⋅

)

is a semigroup but not a monoid.
Let

e

∉

S

.

Construct

a monoid

(

M

,

∗

)

by putting

M

=

S

∪

{

e

}

and defining

s

∗

t

=

{

s

⋅

t

if

⁢

s

,

t

∈

S

,

s

if

⁢

s

∈

S

,

t

=

e

,

t

if

⁢

s

=

e

,

t

∈

S

,

e

if

⁢

s

=

t

=

e

.

Then

(

M

,

∗

)

is isomorphic to a

submonoid

of

(

M

M

,

⊗

)

as by (

1

).
For

s

∈

S

put

g

s

=

f

s

|

S

:
then

g

s

∈

S

S

for every

s

,

g

s

⋅

t

=

f

s

∗

t

|

S

,
and

(

S

,

⋅

)

is isomorphic to

(

Σ

,

⊗

)

with

Σ

=

{

g

s

∣

s

∈

S

}

.

□

Observe that the theorem remains valid
if

f

⊗

g

is defined as

f

∘

g

.
In this case, the morphism

ϕ

is defined by

f

s

⁢

(

x

)

=

s

⋅

x

⁢

∀

x

∈

S

.

Title

Cayley’s theorem for semigroups

Canonical name

CayleysTheoremForSemigroups

Date of creation

2013-03-22 19:04:37

Last modified on

2013-03-22 19:04:37

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

8

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 20M20

Classification

msc 20M15

Related topic

CayleysTheorem

Generated on Fri Feb 9 19:34:18 2018 by

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semigroup of transformations

semigroup of transformations

Let

X

be a set. A

transformation

of

X

is a

function

from

X

to

X

.

If

α

and

β

are transformations on

X

, then their

product

α

⁢

β

is defined (writing functions on the right) by

(

x

)

⁢

(

α

⁢

β

)

=

(

(

x

)

⁢

α

)

⁢

β

.

With this definition, the set of all transformations on

X

becomes a

semigroup

, the

full semigroupf of transformations

on

X

, denoted

𝒯

X

.

More generally, a

semigroup of transformations

is any

subsemigroup

of a full set of transformations.

When

X

is finite, say

X

=

{

x

1

,

x

2

,

…

,

x

n

}

, then the transformation

α

which maps

x

i

to

y

i

(with

y

i

∈

X

, of course) is often written:

α

=

(

x

1

x

2

…

x

n

y

1

y

2

…

y

n

)

With this notation it is quite easy to products. For example, if

X

=

{

1

,

2

,

3

,

4

}

, then

(

1

2

3

4

3

2

1

2

)

⁢

(

1

2

3

4

2

3

3

4

)

=

(

1

2

3

4

3

3

2

3

)

When

X

is

infinite

, say

X

=

{

1

,

2

,

3

,

…

}

, then this notation is still useful for illustration in cases where the transformation

pattern

is apparent. For example, if

α

∈

𝒯

X

is given by

α

:

n

↦

n

+

1

, we can write

α

=

(

1

2

3

4

…

2

3

4

5

…

)

Title

semigroup of transformations

Canonical name

SemigroupOfTransformations

Date of creation

2013-03-22 13:07:36

Last modified on

2013-03-22 13:07:36

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20M20

Synonym

transformation semigroup

Defines

full transformation semigroup

Generated on Fri Feb 9 19:34:20 2018 by

LaTeXML
subcommutative

subcommutative

A

semigroup

(

S

,

⋅

)

is said to be

left subcommutative

if for any two of its elements

a

and

b

, there exists its element

c

such that

a

⁢

b

=

c

⁢

a

.

(1)

A semigroup

(

S

,

⋅

)

is said to be

right subcommutative

if for any two of its elements

a

and

b

, there exists its element

d

such that

a

⁢

b

=

b

⁢

d

.

(2)

If

S

is both left subcommutative and right subcommutative, it is

subcommutative

.

The

commutativity

is a special case of all the three kinds of subcommutativity.

Example 1.

The following operation table defines a right subcommutative semigroup

{

0

,

1

,

2

,

3

}

which is not left subcommutative (e.g.

0

⋅

3

=

2

≠

c

⋅

0

):

⋅

0

1

2

3

0

0

0

2

2

1

0

1

2

3

2

0

0

2

2

3

0

1

2

3

Example 2.

The group of the

square matrices

over a field is both left and right subcommutative (but not

commutative

), since the

equations

(1) and (2) are satisfied by

c

=

a

⁢

b

⁢

a

-

1

and

d

=

b

-

1

⁢

a

⁢

b

.

Remark.

One uses the above also for a ring

(

S

,

+

,

⋅

)

if its

multiplicative

semigroup

(

S

,

⋅

)

satisfies the corresponding requirements.

References

1

S. Lajos

: “On

(

m

,

n

)

-ideals in subcommutative semigroups”.  –

Elemente der Mathematik

24

(1969).

2

V. P. Elizarov

: “Subcommutative Q-rings”.  –

Mathematical notes

2

(1967).

Title

subcommutative

Canonical name

Subcommutative

Date of creation

2013-03-22 19:13:45

Last modified on

2013-03-22 19:13:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 20M25

Classification

msc 20M99

Related topic

Commutative

Related topic

Klein4Ring

Related topic

Anticommutative

Defines

left subcommutative

Defines

right subcommutative

Generated on Fri Feb 9 19:34:22 2018 by

LaTeXML
centralizer of a k-cycle

centralizer of a k-cycle

Theorem 1

.

Let

σ

be a

k

-cycle in

S

n

. Then the

centralizer

of

σ

is

C

S

n

⁢

(

σ

)

=

{

σ

i

⁢

τ

∣

0

≤

i

≤

k

-

1

,

τ

∈

S

n

-

k

}

where

S

n

-

k

is the

subgroup

of

S

n

consisting of those

permutations

that fix all elements appearing in

σ

.

Proof.

This is fundamentally a counting argument. It is clear that

σ

commutes with each element in the set given, since

σ

commutes with powers of itself and also commutes with disjoint permutations. The size of the given set is

k

⋅

(

n

-

k

)

!

. However, the number of

conjugates

of

σ

is the index of

C

S

n

⁢

(

σ

)

in

S

n

by the

orbit-stabilizer theorem

, so to determine

|

C

S

n

⁢

(

σ

)

|

we need only count the number of conjugates of

σ

, i.e. the number of

k

-cycles.

In a

k

-cycle

(

a

1

⁢

…

⁢

a

k

)

, there are

n

choices for

a

1

,

n

-

1

choices for

a

2

, and so on. So there are

n

⁢

(

n

-

1

)

⁢

⋯

⁢

(

n

-

k

+

1

)

choices for the elements of the cycle. But this counts each cycle

k

times, depending on which element appears as

a

1

. So the number of

k

-cycles is

n

⁢

(

n

-

1

)

⁢

⋯

⁢

(

n

-

k

+

1

)

k

Finally,

n

!

=

|

S

n

|

=

n

⁢

(

n

-

1

)

⁢

⋯

⁢

(

n

-

k

+

1

)

k

⁢

|

C

S

n

⁢

(

σ

)

|

so that

|

C

S

n

⁢

(

σ

)

|

=

k

⋅

n

!

n

⁢

(

n

-

1

)

⁢

⋯

⁢

(

n

-

k

+

1

)

=

k

⋅

(

n

-

k

)

!

and we see that the elements in the list above must exhaust

C

S

n

⁢

(

σ

)

.
∎

Title

centralizer of a k-cycle

Canonical name

CentralizerOfAKcycle

Date of creation

2013-03-22 17:18:00

Last modified on

2013-03-22 17:18:00

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20M30

Generated on Fri Feb 9 19:34:24 2018 by

LaTeXML
counting theorem

counting theorem

Given a

group action

of a

finite group

G

on a

finite set

X

, the following expression gives the number of distinct orbits

1

|

G

|

⁢

∑

g

∈

G

stab

g

⁡

(

X

)

Where

stab

g

⁡

(

X

)

is the number of

elements

fixed by the action of

g

.

Title

counting theorem

Canonical name

CountingTheorem

Date of creation

2013-03-22 12:22:23

Last modified on

2013-03-22 12:22:23

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 20M30

Synonym

Cauchy-Frobenius-Burnside formula

Generated on Fri Feb 9 19:34:29 2018 by

LaTeXML
example of counting theorem

example of counting theorem

We shall find the number of distinct ways to colour the faces of a cube with 3 colours. We say that two

colourings

are distinct if one cannot be rotated so that it looks the same as the other. We find this by considering the action of the group

G

of

rotations

of a cube on the set

X

of colourings ignoring rotations. So

|

X

|

=

3

6

and

|

G

|

=

24

. The number of distinct colourings is the number of orbits of

G

in

X

.

First we must consider what types of rotation there are in

G

. We have the

identity

rotation, rotations about an axis through the centre of two opposite faces, rotations about an axis through two opposite vertices, and rotations about an axis through the centre of two opposite edges. We need to find

stab

g

⁡

(

X

)

for each

g

∈

G

.

The identity rotation will fix all

3

6

members of

X

. There are 6 rotations of order 4 through the centre of opposite faces, and these fix

3

3

members of

X

as the faces through which the axis passes can be any colour, but the other faces must all be the same colour to be fixed by this type of rotation. There are 3 rotations of order 2 through the centre of opposite faces, and these fix

3

4

members of

X

. There are 8 rotations of order 3 through opposite vertices, and each of these fixes

3

2

members of

X

as the faces

adjacent

to each corner through which the axis passes must be the same colour. There are 6 rotations through the centre of opposite edges, and these have order 2. They each fix

3

3

members of

X

.

So we have

∑

g

∈

G

stab

g

⁡

(

X

)

=

1

⋅

3

6

+

6

⋅

3

3

+

3

⋅

3

4

+

8

⋅

3

2

+

6

⋅

3

3

=

1368

hence

1

|

G

|

⁢

∑

g

∈

G

stab

g

⁡

(

X

)

=

1368

24

=

57

and so there are 57 distinct ways to colour a cube with three colours.

Title

example of

counting theorem

Canonical name

ExampleOfCountingTheorem

Date of creation

2013-03-22 14:26:35

Last modified on

2013-03-22 14:26:35

Owner

aoh45 (5079)

Last modified by

aoh45 (5079)

Numerical id

4

Author

aoh45 (5079)

Entry type

Example

Classification

msc 20M30

Generated on Fri Feb 9 19:34:31 2018 by

LaTeXML
orbit

orbit

Let

G

be a group,

X

a set, and

⋅

:

G

×

X

⟶

X

a

group action

. For any

x

∈

X

, the

orbit

of

x

under the group action is the set

{

g

⋅

x

∣

g

∈

G

}

⊂

X

.

Title

orbit

Canonical name

Orbit

Date of creation

2013-03-22 12:12:20

Last modified on

2013-03-22 12:12:20

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 20M30

Generated on Fri Feb 9 19:34:33 2018 by

LaTeXML