aarjaneiro/planetmath huggingface dataset:
symmetric set

symmetric set

Definition

A subset

A

of a group

G

is said to be

symmetric

if

A

=

A

-

1

, where

A

-

1

=

{

a

-

1

:

a

∈

A

}

. In other ,

A

is symmetric if

a

-

1

∈

A

whenever

a

∈

A

.

If

A

is a subset of a

vector space

, then

A

is said to be

symmetric

if it is symmetric with respect to the

additive group

structure

of the vector space; that is, if

A

=

{

-

a

:

a

∈

A

}

[

1

]

.

0.0.1

Examples

1.

In

ℝ

, examples of

symmetric sets

are
intervals of the type

(

-

k

,

k

)

with

k

>

0

, and the sets

ℤ

and

{

-

1

,

1

}

.

2.

Any

vector subspace

in a vector space is a symmetric set.

3.

If

A

is any subset of a group, then

A

∩

A

-

1

and

A

∪

A

-

1

are symmetric sets.

References

1

R. Cristescu,

Topological vector spaces

,
Noordhoff International Publishing, 1977.

2

W. Rudin,

Functional Analysis

,
McGraw-Hill Book Company, 1973.

Title

symmetric set

Canonical

name

SymmetricSet

Date of creation

2013-03-22 13:48:26

Last modified on

2013-03-22 13:48:26

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 20A99

Classification

msc 22A05

Classification

msc 15-00

Classification

msc 46-00

Generated on Fri Feb 9 19:20:00 2018 by

LaTeXML
cycle notation

cycle notation

The

cycle notation

is a useful convention for writing down a

permutations

in terms of its constituent cycles. Let

S

be a

finite
set

, and

a

1

,

…

,

a

k

,

k

≥

2

distinct elements of

S

. The
expression

(

a

1

,

…

,

a

k

)

denotes the cycle whose action is

a

1

↦

a

2

↦

a

3

⁢

…

⁢

a

k

↦

a

1

.

Note there are

k

different expressions for the same cycle; the
following all

represent

the same cycle:

(

a

1

,

a

2

,

a

3

,

…

,

a

k

)

=

(

a

2

,

a

3

,

…

,

a

k

,

a

1

)

,

=

…

=

(

a

k

,

a

1

,

a

2

,

…

,

a

k

-

1

)

.

Also note that a 1-element cycle is
the same thing as the

identity

permutation, and thus there is not
much point in writing down such things. Rather, it is customary to
express the identity permutation simply as

(

)

or

(

1

)

.

Let

π

be a permutation of

S

, and let

S

1

,

…

,

S

k

⊂

S

,

k

∈

ℕ

be the orbits of

π

with more than 1

element

. For each

j

=

1

,

…

,

k

let

n

j

denote
the

cardinality

of

S

j

. Also, choose an

a

1

,

j

∈

S

j

, and
define

a

i

+

1

,

j

=

π

⁢

(

a

i

,

j

)

,

i

∈

ℕ

.

We can now express

π

as a

product

of

disjoint

cycles, namely

π

=

(

a

1

,

1

,

…

⁢

a

n

1

,

1

)

⁢

(

a

2

,

1

,

…

,

a

n

2

,

2

)

⁢

…

⁢

(

a

k

,

1

,

…

,

a

n

k

,

k

)

.

By way of illustration, here are the 24 elements of the

symmetric
group

on

{

1

,

2

,

3

,

4

}

expressed using the cycle notation, and grouped
according to their

conjugacy classes

:

(

)

,

(

12

)

,

(

13

)

,

(

14

)

,

(

23

)

,

(

24

)

,

(

34

)

(

123

)

,

(

213

)

,

(

124

)

,

(

214

)

,

(

134

)

,

(

143

)

,

(

234

)

,

(

243

)

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

(

1234

)

,

(

1243

)

,

(

1324

)

,

(

1342

)

,

(

1423

)

,

(

1432

)

Title

cycle notation

Canonical name

CycleNotation

Date of creation

2013-03-22 12:33:41

Last modified on

2013-03-22 12:33:41

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 20B05

Classification

msc 05A05

Related topic

Cycle2

Related topic

Permutation

Related topic

OneLineNotationForPermutations

Generated on Fri Feb 9 19:20:12 2018 by

LaTeXML
permutation group

permutation group

A

permutation group

is a pair

(

G

,

X

)

where

G

is an

abstract group

, and

X

is a set on
which

G

acts

faithfully

. Alternatively, this can be thought of as a group

G

equipped with a

homomorphism

in to

Sym

⁢

(

X

)

, the

symmetric group

on

X

.

Title

permutation group

Canonical

name

PermutationGroup

Date of creation

2013-03-22 13:16:34

Last modified on

2013-03-22 13:16:34

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 20B05

Generated on Fri Feb 9 19:20:14 2018 by

LaTeXML
Simple Groups

Simple Groups

Recall that a group

G

is simple if it has no

normal
subgroups

except itself and

{

e

}

. Let

G

be a finite
simple group and let

p

be a prime number.

(a) Suppose

G

has precisely

k

Sylow

p

-

subgroups

with

k

>

1

. Show that

G

is

isomorphic

to a subgroup of the

symmetric group

S

k

.

(b) With the same

hypothesis

, show that

G

is isomorphic
to a subgroup of the

alternating group

A

k

.

(c) Suppose

G

is a

simple group

that is a

proper
subgroup

of

A

k

and

k

≥

5

. Show that the index

[

A

k

:

G

]

≥

k

.

(d) Prove that if

G

is a group of order

120

then

G

is not a simple group. (Parts (b) and (c) may be helpful.)

…

Title

Simple Groups

Canonical name

SimpleGroups

Date of creation

2013-03-22 19:30:43

Last modified on

2013-03-22 19:30:43

Owner

jac (4316)

Last modified by

jac (4316)

Numerical id

6

Author

jac (4316)

Classification

msc 20B05

Generated on Fri Feb 9 19:20:16 2018 by

LaTeXML
doubly transitive groups are primitive

doubly transitive groups are primitive

Theorem

.

Every

doubly transitive

group is

primitive

(

http://planetmath.org/PrimativeTransitivePermutationGroupOnASet

).

Proof.

Let

G

acting on

X

be doubly transitive. To show the action is , we must show that all blocks are

trivial blocks

; to do this, it suffices to show that any block containing more than one element is all of

X

. So choose a block

Y

with two distinct elements

y

1

,

y

2

. Given an arbitrary

x

∈

X

, since

G

is doubly transitive, we can choose

σ

∈

G

such that

σ

⋅

(

y

1

,

y

2

)

=

(

y

1

,

x

)

But then

σ

⋅

Y

∩

Y

≠

∅

, since

y

1

is in both. Thus

σ

⋅

Y

=

Y

, so

x

∈

Y

as well. So

Y

=

X

and we are done.
∎

Title

doubly transitive groups are primitive

Canonical name

DoublyTransitiveGroupsArePrimitive

Date of creation

2013-03-22 17:21:50

Last modified on

2013-03-22 17:21:50

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20B15

Generated on Fri Feb 9 19:20:18 2018 by

LaTeXML
examples of primitive groups that are not doubly transitive

examples of primitive groups that are not doubly transitive

The group

𝒟

2

⁢

n

,

n

≥

3

, the

dihedral group

of order

2

⁢

n

, is the

symmetry group

of the

regular

n

-gon. (Note that we use the more common notation

𝒟

2

⁢

n

for this group rather than

𝒟

n

).

𝒟

2

⁢

n

is clearly not

doubly transitive

for

n

≥

4

, since it preserves “adjacency” in the vertices. Thus, for example, clearly no element of

𝒟

2

⁢

n

can take

(

1

,

2

)

to

(

1

,

3

)

. (

𝒟

2

⋅

3

=

𝒟

6

, the symmetry group of the

triangle

, is, however, doubly transitive).

We show that for

p

prime,

𝒟

2

⁢

p

is

primitive

. To prove this, we need only verify that any block containing two distinct

elements

is the entire set of vertices. Number the vertices consecutively

{

0

,

…

,

p

-

1

}

, and let

r

be the element of

𝒟

2

⁢

n

that takes each vertex into its

successor

(

mod

p

)

. Now, suppose a block contains two distinct elements

a

,

b

; assume wlog that

b

≠

0

. Iteratively apply

r

b

-

a

to these elements to get

a

b

b

2

⁢

b

-

a

2

⁢

b

-

a

3

⁢

b

-

a

…

…

Since blocks are either equal or

disjoint

, we see that the block in question contains

a

,

b

, and

n

⁢

b

-

a

for each

n

. But

a

≠

b

, so

n

⁢

b

-

a

runs through all residues (

http://planetmath.org/ResidueSystems

)

(

mod

p

)

and thus the block contains each vertex. Thus

D

2

⁢

p

is primitive.

For nonprime

n

,

𝒟

2

⁢

n

is not primitive. In this case, if

d

is a divisor of

n

, then the set of vertices that are

multiples

of

d

form a block.

Title

examples of primitive groups that are not doubly transitive

Canonical name

ExamplesOfPrimitiveGroupsThatAreNotDoublyTransitive

Date of creation

2013-03-22 17:22:37

Last modified on

2013-03-22 17:22:37

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Example

Classification

msc 20B15

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

LaTeXML
primitive permutation group

primitive permutation group

Let

X

be a set, and

G

a

transitive

permutation group

on

X

.
Then

G

is said to be a

primitive permutation group

if it has no nontrivial blocks (

http://planetmath.org/BlockSystem

).

For example, the

symmetric group

S

4

is a primitive permutation group on

{

1

,

2

,

3

,

4

}

.

Note that

D

8

is not a primitive permutation group on the vertices of a square, because the pairs of opposite points form a nontrivial block.

It can be shown that a transitive permutation group

G

on a set

X

is

primitive

if and only if the

stabilizer

Stab

G

⁡

(

x

)

is a

maximal subgroup

of

G

for all

x

∈

X

.

Title

primitive permutation group

Canonical name

PrimitivePermutationGroup

Date of creation

2013-03-22 14:00:49

Last modified on

2013-03-22 14:00:49

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

20

Author

Thomas Heye (1234)

Entry type

Definition

Classification

msc 20B15

Generated on Fri Feb 9 19:20:22 2018 by

LaTeXML
transitive actions are primitive if and only if stabilizers are maximal subgroups

transitive actions are primitive if and only if stabilizers are maximal subgroups

Theorem 1

.

If

G

is

transitive

on the set

A

, then

G

is

primitive

on

A

if and only if for each

a

∈

A

,

G

a

is a

maximal subgroup

of

G

. Here

G

a

=

Stab

G

⁡

(

a

)

is the

stabilizer

of

a

∈

A

.

Proof.

First claim that if

G

is transitive on

A

and

B

⊂

A

is a block (

http://planetmath.org/BlockSystem

) with

a

∈

B

, then

G

B

=

{

σ

∈

G

∣

σ

⁢

(

B

)

=

B

}

is a

subgroup

of

G

containing

G

a

. It is obvious that

G

B

is a subgroup, since

σ

∈

G

B

⇒

σ

⁢

(

B

)

=

B

⇒

σ

-

1

⁢

(

σ

⁢

(

B

)

)

=

σ

-

1

⁢

(

B

)

⇒

B

=

σ

-

1

⁢

(

B

)

⇒

σ

-

1

∈

G

B

σ

,

τ

∈

G

B

⇒

(

σ

⁢

τ

)

⁢

(

B

)

=

σ

⁢

(

τ

⁢

(

B

)

)

=

σ

⁢

(

B

)

=

B

⇒

σ

⁢

τ

∈

G

B

But also, if

σ

∈

G

a

for

a

∈

B

, then

σ

⁢

(

a

)

=

a

, so

σ

⁢

(

B

)

∩

B

≠

∅

and thus

σ

⁢

(

B

)

=

B

since

B

is a

block system

and thus

σ

∈

G

B

. This proves the claim.

To prove the

theorem

, note that for each

a

∈

A

, there is by the claim a

1

-

1

correspondence between containing

a

and subgroups of

G

containing

G

a

. Thus,

G

is primitive on

A

if and only if all blocks are either of size

1

or equal to

A

, if and only if any group containing

G

a

is either

G

a

itself or

G

, if and only if for all

a

∈

A

,

G

a

is maximal in

G

.
∎

Title

transitive actions are primitive if and only if stabilizers are maximal subgroups

Canonical name

TransitiveActionsArePrimitiveIfAndOnlyIfStabilizersAreMaximalSubgroups

Date of creation

2013-03-22 17:19:07

Last modified on

2013-03-22 17:19:07

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20B15

Generated on Fri Feb 9 19:20:25 2018 by

LaTeXML
alternative characterization of multiply transitive permutation groups

alternative characterization of multiply transitive permutation groups

This article derives an alternative

characterization

of

n

-

transitive groups

.

Theorem

.

For

n

>

1

,

G

is

n

-

transitive

on

X

if and only if for all

x

∈

X

,

G

x

is

(

n

-

1

)

-transitive on

X

-

{

x

}

.

Proof.

First assume

G

is

n

-transitive on

X

, and choose

x

∈

X

. To show

G

x

is

(

n

-

1

)

-transitive on

X

-

{

x

}

, choose

x

1

,

…

,

x

n

-

1

,

y

1

,

…

,

y

n

-

1

∈

X

. Since

G

is

n

-transitive on

X

, we can choose

σ

∈

G

such that

σ

⋅

(

x

1

,

…

,

x

n

-

1

,

x

)

=

(

y

1

,

…

,

y

n

-

1

,

x

)

But obviously

σ

∈

G

x

, and

σ

restricted to

X

-

{

x

}

is the desired

permutation

.

To prove the

converse

, choose

x

1

,

…

,

x

n

,

y

1

,

…

,

y

n

∈

X

. Choose

σ

1

∈

G

x

n

such that

σ

1

⋅

(

x

1

,

…

,

x

n

-

1

)

=

(

y

1

,

…

,

y

n

-

1

)

and choose

σ

2

∈

G

y

1

such that

σ

2

⋅

(

y

2

,

…

,

y

n

-

1

,

x

n

)

=

(

y

2

,

…

,

y

n

-

1

,

y

n

)

Then

σ

2

⁢

σ

1

is the desired permutation.
∎

Note that this definition of

n

-

transitivity

affords a straightforward proof of the statement that

A

n

is

(

n

-

2

)

-transitive: by inspection,

A

3

is

1

-transitive; the result follows by

induction

using the

theorem

. (The corresponding statement that

S

n

is

n

-transitive is obvious).

Finally, note that the most common cases of

n

-transitivity are for

n

=

1

(

transitive

), and

n

=

2

(

doubly transitive

).

Title

alternative characterization of multiply transitive permutation groups

Canonical name

AlternativeCharacterizationOfMultiplyTransitivePermutationGroups

Date of creation

2013-03-22 17:21:47

Last modified on

2013-03-22 17:21:47

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Derivation

Classification

msc 20B20

Defines

doubly transitive

Generated on Fri Feb 9 19:20:28 2018 by

LaTeXML
example of multiply transitive

example of multiply transitive

Theorem 1

.

1.

The

general linear group

G

⁢

L

⁢

(

V

)

acts transitively on the set of points (1-dimensional

subspaces

) in the

projective geometry

P

⁢

G

⁢

(

V

)

.

2.

P

⁢

G

⁢

L

⁢

(

V

)

is

doubly transitive

on the set of all of points in

P

⁢

G

⁢

(

V

)

.

3.

P

⁢

G

⁢

L

⁢

(

V

)

is not 3-transitive on the set of all points in

P

⁢

G

⁢

(

V

)

if

dim

⁡

V

≠

2

.

Proof.

Evidently 2 implies 1. So suppose we have pairs of distinct points

(

P

,

Q

)

and

(

R

,

S

)

. Then take

P

=

⟨

x

⟩

,

Q

=

⟨

y

⟩

,

R

=

⟨

z

⟩

and

S

=

⟨

w

⟩

.
As

P

≠

Q

,

x

and

y

are

linearly independent

, just as

z

and

w

are. Therefore extending

{

x

,

y

}

to a basis

B

and

{

z

,

w

}

to a basis

C

, we know there is a

linear transformation

f

∈

G

⁢

L

⁢

(

V

)

taking

B

to

C

– consider the

change of basis matrix

. Therefore

G

⁢

L

⁢

(

V

)

is
2-transitive.

Now suppose

dim

⁡

V

≥

2

. Then there exists a linearly indepedent set

{

x

,

y

,

z

}

which
gives three distinct non-collinear points

(

P

,

Q

,

R

)

,

P

=

⟨

x

⟩

,

Q

=

⟨

y

⟩

and

R

=

⟨

z

⟩

. But then we also have three

collinear points

(

P

,

Q

,

S

)

where

S

=

⟨

x

+

y

⟩

. As

G

⁢

L

⁢

(

V

)

prevserves the

geometry

of

P

⁢

G

⁢

(

V

)

,
we cannot have a map in

G

⁢

L

⁢

(

V

)

send

(

P

,

Q

,

R

)

to

(

P

,

Q

,

S

)

.
∎

Note that the action of

G

⁢

L

⁢

(

V

)

on

P

⁢

G

⁢

(

V

)

is not

faithful

so we use instead

P

⁢

G

⁢

L

⁢

(

V

)

.

Title

example of

multiply transitive

Canonical name

ExampleOfMultiplyTransitive

Date of creation

2013-03-22 17:21:56

Last modified on

2013-03-22 17:21:56

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Example

Classification

msc 20B20

Generated on Fri Feb 9 19:20:30 2018 by

LaTeXML
Jordan’s theorem (multiply transitive groups)

Jordan’s theorem (multiply transitive groups)

Let

G

be a sharply

n

-

transitive

permutation group

, with

n

≥

4

. Then

1.

G

is

similar

to

S

n

with the standard action or

2.

G

is similar to

A

n

+

2

with the standard action or

3.

n

=

4

and

G

is similar to

M

11

, the

Mathieu group

of degree

10

or

4.

n

=

5

and

G

is similar to

M

12

, the Mathieu group of degree

11

.

Title

Jordan’s

theorem

(

multiply transitive

groups)

Canonical name

JordansTheoremmultiplyTransitiveGroups

Date of creation

2013-03-22 13:16:42

Last modified on

2013-03-22 13:16:42

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20B20

Generated on Fri Feb 9 19:20:32 2018 by

LaTeXML
multiply transitive

multiply transitive

Let

G

be a group,

X

a set on which it acts. Let

X

(

n

)

be the set of order

n

-tuples
of distinct elements of

X

. This is a

G

-set by the diagonal action:

g

⋅

(

x

1

,

…

,

x

n

)

=

(

g

⋅

x

1

,

…

,

g

⋅

x

n

)

The action of

G

on

X

is said to be

n

-

transitive

if it acts transitively on

X

(

n

)

.

For example, the standard action of

S

n

, the

symmetric group

, is

n

-transitive, and the
standard action of

A

n

, the

alternating group

, is

(

n

-

2

)

-transitive.

Title

multiply transitive

Canonical name

MultiplyTransitive

Date of creation

2013-03-22 13:16:37

Last modified on

2013-03-22 13:16:37

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 20B20

Synonym

n

-transitive

Generated on Fri Feb 9 19:20:37 2018 by

LaTeXML
sharply multiply transitive

sharply multiply transitive

Let

G

be a group, and

X

a set that

G

acts on, and let

X

(

n

)

be the set of ordered

n

-tuples of distinct elements of

X

. Then the action of

G

on

X

is

sharply

n

-

transitive

if

G

acts regularly on

X

(

n

)

.

Title

sharply multiply transitive

Canonical

name

SharplyMultiplyTransitive

Date of creation

2013-03-22 13:16:39

Last modified on

2013-03-22 13:16:39

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20B20

Synonym

sharply

n

-transitive

Generated on Fri Feb 9 19:20:39 2018 by

LaTeXML
symmetric group

symmetric group

Let

X

be a set.
Let

Sym

⁢

(

X

)

be the set of

permutations

of

X

(i.e. the set of

bijective functions

from

X

to itself).
Then the act of taking the

composition

of two permutations
induces a group

structure

on

Sym

⁢

(

X

)

.
We call this group the

symmetric group

.

The group

Sym

⁢

(

{

1

,

2

,

…

,

n

}

)

is often denoted

S

n

or

𝔖

n

.

S

n

is generated by the

transpositions

{

(

1

,

2

)

,

(

2

,

3

)

,

…

,

(

n

-

1

,

n

)

}

,
and by any pair of a 2-cycle and

n

-cycle.

S

n

is the

Weyl group

of the

A

n

-

1

root system (and hence of the

special linear group

S

⁢

L

n

-

1

).

Title

symmetric group

Canonical

name

SymmetricGroup

Date of creation

2013-03-22 12:01:53

Last modified on

2013-03-22 12:01:53

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

11

Author

bwebste (988)

Entry type

Definition

Classification

msc 20B30

Related topic

Group

Related topic

Cycle2

Related topic

CayleyGraphOfS_3

Related topic

Symmetry2

Generated on Fri Feb 9 19:20:57 2018 by

LaTeXML
symmetric group

symmetric group

Let

X

be a set.
Let

S

⁢

(

X

)

be the set of

permutations

of

X

(i.e. the set of

bijective functions

on

X

).
Then the act of taking the

composition

of two permutations
induces a group

structure

on

S

⁢

(

X

)

.
We call this group the

symmetric group

and it is often denoted

Sym

⁢

(

X

)

.

When

X

has a finite number

n

of

elements

,
we often refer to the symmetric group as

S

n

,
and describe the elements by using

cycle notation

.

Title

symmetric group

Canonical

name

SymmetricGroup1

Date of creation

2013-03-22 14:03:53

Last modified on

2013-03-22 14:03:53

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

5

Author

antizeus (11)

Entry type

Definition

Classification

msc 20B30

Related topic

Symmetry2

Generated on Fri Feb 9 19:20:58 2018 by

LaTeXML
symmetric group is generated by adjacent transpositions

symmetric group is generated by adjacent transpositions

Theorem 1

.

The

symmetric group

on

{

1

,

2

,

…

,

n

}

is generated by the

permutations

(

1

,

2

)

,

(

2

,

3

)

,

…

,

(

n

-

1

,

n

)

.

Proof.

We proceed by

induction

on

n

. If

n

=

2

, the

theorem

is trivially true
because the the group only consists of the

identity

and a single

transposition

.

Suppose, then, that we know permutations of

n

numbers are generated
by transpositions of successive numbers. Let

ϕ

be a permutation of

{

1

,

2

,

…

,

n

+

1

}

. If

ϕ

⁢

(

n

+

1

)

=

n

+

1

, then the

restriction

of

ϕ

to

{

1

,

2

,

…

,

n

}

is a permutation of

n

numbers, hence,
by

hypothesis

, it can be expressed as a

product

of transpositions.

Suppose that, in

addition

,

ϕ

⁢

(

n

+

1

)

=

m

with

m

≠

n

+

1

. Consider
the following product of transpositions:

(

n

⁢

n

+

1

)

⁢

(

n

-

1

⁢

n

)

⁢

⋯

⁢

(

m

+

1

⁢

m

+

1

)

⁢

(

m

⁢

m

+

1

)

It is easy to see that acting upon

m

with this product of transpositions
produces

+

1

. Therefore, acting upon

n

+

1

with the permutation

(

n

⁢

n

+

1

)

⁢

(

n

-

1

⁢

n

)

⁢

⋯

⁢

(

m

+

1

⁢

m

+

1

)

⁢

(

m

⁢

m

+

1

)

⁢

ϕ

produces

n

+

1

. Hence, the restriction of this permutation to

{

1

,

2

,

…

,

n

}

is a permutation of

n

numbers, so,
by hypothesis, it can be expressed as a product of transpositions.
Since a transposition is its own

inverse

, it follows that

ϕ

may also be expressed as a product of transpositions.
∎

Title

symmetric group is generated by adjacent transpositions

Canonical

name

SymmetricGroupIsGeneratedByAdjacentTranspositions

Date of creation

2013-03-22 16:49:02

Last modified on

2013-03-22 16:49:02

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 20B30

Generated on Fri Feb 9 19:21:01 2018 by

LaTeXML