aarjaneiro/planetmath huggingface dataset:
left and right cosets in a double coset

left and right cosets in a double coset

Let

H

and

K

be

subgroups

of a group

G

. Every

double coset

H

⁢

g

⁢

K

, with

g

∈

G

, is a union of right (

http://planetmath.org/Coset

) or

left cosets

, since

H

⁢

g

⁢

K

=

⋃

k

∈

K

H

⁢

g

⁢

k

=

⋃

h

∈

H

h

⁢

g

⁢

K

,

but these unions need not be

disjoint

. In particular, from the above

equality

we cannot say how many right (or left) cosets fit in a double coset.

The following

proposition

aims to clarify this.

-

Let

H

and

K

be subgroups of a group

G

and

g

∈

G

. We have that

H

⁢

g

⁢

K

=

⋃

[

k

]

∈

(

K

∩

g

-

1

⁢

H

⁢

g

)

\

K

H

⁢

g

⁢

k

=

⋃

[

h

]

∈

H

/

(

H

∩

g

⁢

K

⁢

g

-

1

)

h

⁢

g

⁢

K

hold as

disjoint unions

. In particular, the number of right and left cosets in

H

⁢

g

⁢

K

is respectively given by

#

(

H

\

H

g

K

)

=

[

K

:

K

∩

g

-

1

H

g

]

#

(

H

g

K

/

K

)

=

[

H

:

H

∩

g

K

g

-

1

]

0.1

Remarks

•

The number of right and left cosets in a double coset does not coincide in general, not for double cosets of the form

H

⁢

g

⁢

H

.

References

1

A. Krieg,

, Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.

Title

left and right cosets in a double coset

Canonical name

LeftAndRightCosetsInADoubleCoset

Date of creation

2013-03-22 18:35:10

Last modified on

2013-03-22 18:35:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 20A05

Generated on Fri Feb 9 19:17:57 2018 by

LaTeXML
method of repeated squaring

method of repeated squaring

Theorem 1

.

Given a

semigroup

S

and an

n

∈

Z

+

then the function

f

:

S

→

S

defined by

f

⁢

(

a

)

=

a

n

has an

SLP representations

of
computational length

O

⁢

(

log

2

⁡

n

)

. Inparticular, if we can compute

a

n

using only

O

⁢

(

log

2

⁡

n

)

multiplications

.

Proof.

Let

f

0

:

S

→

S

be the map

f

⁢

(

x

)

=

x

and

f

1

:

S

→

S

be defined by

f

1

⁢

(

x

)

=

x

⁢

x

. So far we
recognize that

f

0

⁢

(

x

)

evaluates to

x

1

=

x

2

0

and

f

1

⁢

(

x

)

evaluates to

x

2

=

x

2

1

.

But we caution that
we do not define these functions with

exponents

because we want
to demonstrate that from an algorithm to multiply we can create
an efficient algorithm to exponentiate.

For

1

<

i

≤

k

define (recalling that

f

i

+

1

may use the output
of any previous

f

i

evaluation)

f

i

+

1

⁢

(

x

)

:=

f

1

⁢

(

f

i

⁢

(

x

)

)

.

Evidently,

f

i

+

1

⁢

(

x

)

evaluates to

f

i

⁢

(

x

)

2

. By induction,

f

i

⁢

(

x

)

evaluates to

x

2

i

and thus

f

i

+

1

⁢

(

x

)

evaluates to

(

x

2

i

)

2

=

x

2

i

+

1

. Therefore we establish that

f

j

⁢

(

x

)

=

x

2

j

for all

nonnegative integers

j

. Furthermore, to evaluate

f

j

⁢

(

x

)

uses

j

multiplications.

Now write

n

in binary:

n

=

a

0

⁢

2

0

+

a

1

⁢

2

1

+

⋯

+

a

k

⁢

2

k

with

a

i

=

0

,

1

for

0

≤

i

<

k

≤

log

2

⁡

n

. Let

0

≤

i

1

<

i

2

<

⋯

<

i

s

≤

k

be
the indices for which

a

i

t

≠

0

,

1

≤

t

≤

s

. Then define

g

n

⁢

(

x

)

=

f

i

1

⁢

(

x

)

⁢

f

i

2

⁢

(

x

)

⁢

⋯

⁢

f

i

s

⁢

(

x

)

Thus,

g

n

⁢

(

x

)

evaluates to:

(

x

2

i

1

)

⁢

(

x

2

i

2

)

⁢

⋯

⁢

(

x

2

i

s

)

=

(

x

2

0

)

a

0

⁢

(

x

2

1

)

a

1

⁢

⋯

⁢

(

x

2

k

)

a

k

=

x

a

0

⁢

2

0

+

a

1

⁢

2

1

+

⋯

+

a

k

⁢

2

k

=

x

n

.

Because each

f

i

+

1

⁢

(

x

)

uses the output of

f

i

⁢

(

x

)

, the cost of
evaluating

g

n

is the cost of evaluating

f

i

s

⁢

(

x

)

plus the final
cost of multiplying

f

i

1

⁢

(

x

)

⁢

⋯

⁢

f

i

s

⁢

(

x

)

. Thus, evaluating

g

n

uses

i

s

+

(

s

-

1

)

multiplications. As

s

≤

log

2

⁡

n

it follows that
evaluating

g

n

uses no more than

2

⁢

log

2

⁡

n

multiplications.
∎

Title

method of repeated squaring

Canonical

name

MethodOfRepeatedSquaring

Date of creation

2013-03-22 16:16:14

Last modified on

2013-03-22 16:16:14

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 08A99

Classification

msc 20-00

Synonym

successive squaring

Generated on Fri Feb 9 19:18:00 2018 by

LaTeXML
natural projection

natural projection

Proposition

.

If

H

is a

normal subgroup

of a group

G

, then the

mapping

φ

:

G

→

G

/

H

where

φ

⁢

(

x

)

=

x

⁢

H

⁢

∀

x

∈

G

is a

surjective

homomorphism

whose kernel is

H

.

Proof.

Because every coset appears as image, the mapping

φ

is surjective.  It is also homomorphic, since for all

elements

x

,

y

of

G

, one has

φ

⁢

(

x

⁢

y

)

=

(

x

⁢

y

)

⁢

H

=

x

⁢

H

⋅

y

⁢

H

=

φ

⁢

(

x

)

⁢

φ

⁢

(

y

)

.

The

identity element

of the

factor group

G

/

H

is the coset

e

⁢

H

=

H

,  whence

ker

(

φ

)

=

{

x

∈

G

⋮

φ

(

x

)

=

H

}

=

{

x

∈

G

⋮

x

H

=

H

}

=

H

.

The mapping

φ

in the proposition is called

natural projection

or

canonical homomorphism

.

Title

natural projection

Canonical

name

NaturalProjection

Date of creation

2013-03-22 19:10:16

Last modified on

2013-03-22 19:10:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Definition

Classification

msc 20A05

Synonym

canonical homomorphism

Synonym

natural homomorphism

Related topic

QuotientGroup

Related topic

KernelOfAGroupHomomorphismIsANormalSubgroup

Generated on Fri Feb 9 19:18:01 2018 by

LaTeXML
non-isomorphic groups of given order

non-isomorphic groups of given order

Theorem.

For every

positive integer

n

, there exists only a finite amount of non-isomorphic groups of order

n

.

This assertion follows from Cayley’s theorem, according to which any group of order

n

is

isomorphic

with a

subgroup

of the

symmetric group

𝔖

n

.  The number of non-isomorphic subgroups of

𝔖

n

cannot be greater than

(

n

!

-

1

n

-

1

)

.

The above theorem may be used in proving the following Landau’s theorem:

Theorem (Landau).

For every positive integer

n

, there exists only a finite amount of finite non-isomorphic groups which contain exactly

n

conjugacy classes

of elements.

One needs also the

Lemma.

If

n

∈

ℤ

+

and

0

<

r

∈

ℝ

,  then there is at most a finite amount of the vectors

(

m

1

,

m

2

,

…

,

m

n

)

consisting of positive integers such that

∑

j

=

1

n

1

m

j

=

r

.

The lemma is easily proved by induction on

n

.

Title

non-isomorphic groups of given order

Canonical name

NonisomorphicGroupsOfGivenOrder

Date of creation

2013-03-22 18:56:38

Last modified on

2013-03-22 18:56:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Related topic

BinomialCoefficient

Related topic

PropertiesOfConjugacy

Defines

Landau’s theorem

Generated on Fri Feb 9 19:18:07 2018 by

LaTeXML
normal closure

normal closure

Let

S

be a subset of a group

G

.
The

normal closure

of

S

in

G

is the

intersection

of all

normal subgroups

of

G

that contain

S

, that is

⋂

S

⊆

N

⁢

⊴

⁢

G

N

.

The normal closure of

S

is the smallest normal subgroup of

G

that contains

S

, and so is also called the

normal subgroup generated by

S

.

It is not difficult to show that the normal closure of

S

is the

subgroup

generated by all the

conjugates

of

elements of

S

.

The normal closure of

S

in

G

is variously denoted by

⟨

S

G

⟩

or

⟨

S

⟩

G

or

S

G

.

If

H

is a subgroup of

G

,
and

H

is of finite index in its normal closure,
then

H

is said to be

nearly normal

.

Title

normal closure

Canonical name

NormalClosure1

Date of creation

2013-03-22 14:41:50

Last modified on

2013-03-22 14:41:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Synonym

normal subgroup generated by

Synonym

conjugate closure

Synonym

smallest normal subgroup containing

Related topic

Normalizer

Related topic

CoreOfASubgroup

Defines

nearly normal

Generated on Fri Feb 9 19:18:08 2018 by

LaTeXML
normal is not transitive

normal is not transitive

The phrase “

normal is not transitive

” can be used as a mnemonic for two statements.

The first is: “The

relation

‘is a

normal subgroup

of’ is not

transitive

.” This means that, if

H

◁

N

◁

G

, it does not follow that

H

◁

G

. See

normality of subgroups is not transitive

for more details.

The second is: “The relation ‘is a

normal extension

of’ is not transitive.” This means that, if

K

/

F

and

L

/

K

are normal extensions, it does not follow that

L

/

F

is normal. See example of normal extension for more details.

Title

normal is not transitive

Canonical name

NormalIsNotTransitive

Date of creation

2013-03-22 16:00:34

Last modified on

2013-03-22 16:00:34

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 20A05

Classification

msc 12F10

Related topic

ExampleOfNormalExtension

Related topic

NormalityOfSubgroupsIsNotTransitive

Generated on Fri Feb 9 19:18:10 2018 by

LaTeXML
normal subgroup

normal subgroup

A

subgroup

H

of a group

G

is

normal

if

a

⁢

H

=

H

⁢

a

for all

a

∈

G

. Equivalently,

H

⊂

G

is normal if and only if

a

⁢

H

⁢

a

-

1

=

H

for all

a

∈

G

, i.e., if and only if each

conjugacy class

of

G

is either entirely inside

H

or entirely outside

H

.

The notation

H

⁢

⊴

⁢

G

or

H

◁

G

is often used to denote that

H

is a

normal subgroup

of

G

.

The kernel

ker

⁡

(

f

)

of any

group homomorphism

f

:

G

⟶

G

′

is a normal subgroup of

G

. More surprisingly, the

converse

is also true: any normal subgroup

H

⊂

G

is the kernel of some

homomorphism

(one of these being the

projection map

ρ

:

G

⟶

G

/

H

, where

G

/

H

is the

quotient group

).

Title

normal subgroup

Canonical name

NormalSubgroup

Date of creation

2013-03-22 12:08:07

Last modified on

2013-03-22 12:08:07

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 20A05

Synonym

normal

Related topic

QuotientGroup

Related topic

Normalizer

Defines

normality

Generated on Fri Feb 9 19:18:11 2018 by

LaTeXML
normality of subgroups is not transitive

normality of subgroups is not transitive

Let

G

be a group.
A

subgroup

K

of a subgroup

H

of

G

is obviously a subgroup of

G

.
It seems plausible that a

similar

situation would also hold for

normal subgroups

, but in fact it does not:
even when

K

⁢

⊴

⁢

H

and

H

⁢

⊴

⁢

G

, it is possible that

K

⋬

G

. Here are two examples:

1.

Let

G

be the subgroup of

orientation-preserving

isometries

(

http://planetmath.org/Isometry

) of the plane

ℝ

2

(

G

is just all

rotations

and

translations

), let

H

be the subgroup of

G

of translations, and let

K

be the subgroup of

H

of integer translations

τ

i

,

j

⁢

(

x

,

y

)

=

(

x

+

i

,

y

+

j

)

, where

i

,

j

∈

ℤ

.

Any element

g

∈

G

may be represented as

g

=

r

1

∘

t

1

=

t

2

∘

r

2

, where

r

1

,

2

are rotations and

t

1

,

2

are translations. So for any translation

t

∈

H

we may write

g

-

1

∘

t

∘

g

=

r

-

1

∘

t

′

∘

r

,

where

t

′

∈

H

is some other translation and

r

is some rotation. But this is an orientation-preserving isometry of the plane that does not rotate, so it too must be a translation. Thus

G

-

1

⁢

H

⁢

G

=

H

, and

H

⁢

⊴

⁢

G

.

H

is an

abelian group

, so all its subgroups,

K

included, are normal.

We claim that

K

⋬

G

. Indeed, if

ρ

∈

G

is rotation by

45

∘

about the origin, then

ρ

-

1

∘

τ

1

,

0

∘

ρ

is not an integer translation.

2.

A related example uses

finite subgroups

. Let

G

=

D

4

be the

dihedral group

with eight elements (the group of

automorphisms

of the graph of the square). Then

D

4

=

⟨

r

,

f

∣

f

2

=

1

,

r

4

=

1

,

f

⁢

r

=

r

-

1

⁢

f

⟩

is generated by

r

, rotation, and

f

, flipping.

The subgroup

H

=

⟨

r

⁢

f

,

f

⁢

r

⟩

=

{

1

,

r

⁢

f

,

r

2

,

f

⁢

r

}

≅

C

2

×

C

2

is

isomorphic

to the

Klein 4-group

– an

identity

and 3 elements of order 2.

H

⁢

⊴

⁢

G

since

[

G

:

H

]

=

2

. Finally, take

K

=

⟨

r

⁢

f

⟩

=

{

1

,

r

⁢

f

}

⁢

⊴

⁢

H

.

We claim that

K

⋬

G

. And indeed,

f

∘

r

⁢

f

∘

f

=

f

⁢

r

∉

K

.

Title

normality of subgroups is not transitive

Canonical name

NormalityOfSubgroupsIsNotTransitive

Date of creation

2013-03-22 12:49:27

Last modified on

2013-03-22 12:49:27

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Example

Classification

msc 20A05

Related topic

NormalIsNotTransitive

Generated on Fri Feb 9 19:18:14 2018 by

LaTeXML
normality of subgroups of prime index

normality of subgroups of prime index

Proposition

.

If

H

is a

subgroup

of a

finite group

G

of index

p

, where

p

is the smallest prime dividing the order of

G

, then

H

is normal in

G

.

Proof.

Suppose

H

≤

G

with

|

G

|

finite and

|

G

:

H

|

=

p

, where

p

is the smallest

prime divisor

of

|

G

|

, let

G

act on the set

L

of

left cosets

of

H

in

G

by left , and let

φ

:

G

→

S

p

be the

http://planetmath.org/node/3820

homomorphism

induced by this action. Now, if

g

∈

ker

⁡

φ

, then

g

⁢

x

⁢

H

=

x

⁢

H

for each

x

∈

G

, and in particular,

g

⁢

H

=

H

, whence

g

∈

H

. Thus

K

=

ker

⁡

φ

is a

normal subgroup

of

H

(being

contained

in

H

and normal in

G

). By the

First Isomorphism Theorem

,

G

/

K

is

isomorphic

to a subgroup of

S

p

, and consequently

|

G

/

K

|

=

|

G

:

K

|

must

http://planetmath.org/node/923

divide

p

!

; moreover, any

divisor

of

|

G

:

K

|

must also

|

G

|

=

|

G

:

K

|

|

K

|

, and because

p

is the smallest divisor of

|

G

|

different from

1

, the only possibilities are

|

G

:

K

|

=

p

or

|

G

:

K

|

=

1

. But

|

G

:

K

|

=

|

G

:

H

|

|

H

:

K

|

=

p

|

H

:

K

|

≥

p

, which

|

G

:

K

|

=

p

, and consequently

|

H

:

K

|

=

1

, so that

H

=

K

, from which it follows that

H

is normal in

G

.
∎

Title

normality of subgroups of prime index

Canonical name

NormalityOfSubgroupsOfPrimeIndex

Date of creation

2013-03-22 17:26:38

Last modified on

2013-03-22 17:26:38

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

13

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 20A05

Related topic

Coset

Related topic

GroupAction

Related topic

ASubgroupOfIndex2IsNormal

Generated on Fri Feb 9 19:18:16 2018 by

LaTeXML
normalizer

normalizer

Definitions

Let

G

be a group, and let

H

⊆

G

.
The

normalizer

of

H

in

G

, written

N

G

⁢

(

H

)

, is the set

{

g

∈

G

∣

g

⁢

H

⁢

g

-

1

=

H

}

.

A

subgroup

H

of

G

is said to be

self-normalizing

if

N

G

⁢

(

H

)

=

H

.

Properties

N

G

⁢

(

H

)

is always a subgroup of

G

,
as it is the

stabilizer

of

H

under the action

(

g

,

H

)

↦

g

⁢

H

⁢

g

-

1

of

G

on the set of all subsets of

G

(or on the set of all subgroups of

G

, if

H

is a subgroup).

If

H

is a subgroup of

G

, then

H

≤

N

G

⁢

(

H

)

.

If

H

is a subgroup of

G

, then

H

is a

normal subgroup

of

N

G

⁢

(

H

)

;
in fact,

N

G

⁢

(

H

)

is the largest subgroup of

G

of which

H

is a normal subgroup.
In particular, if

H

is a subgroup of

G

,
then

H

is normal in

G

if and only if

N

G

⁢

(

H

)

=

G

.

Title

normalizer

Canonical name

Normalizer

Date of creation

2013-03-22 12:36:53

Last modified on

2013-03-22 12:36:53

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Synonym

normaliser

Related topic

Centralizer

Related topic

NormalSubgroup

Related topic

NormalClosure2

Defines

self-normalizing

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

LaTeXML
one-sided normality of subsemigroup

one-sided normality of subsemigroup

Let

S

be a

semigroup

. A

subsemigroup

N

of

S

is said to be

left-normal

if

g

⁢

N

⊆

N

⁢

g

for all

g

∈

S

and
it is said to be

right-normal

if

g

⁢

N

⊇

N

⁢

g

for all

g

∈

S

.
One may similarly define

left-normalizers

LN

S

⁢

(

N

)

:=

{

g

∈

S

∣

g

⁢

N

⊆

N

⁢

g

}

and

right-normalizers

RN

S

⁢

(

N

)

:=

{

g

∈

S

∣

N

⁢

g

⊆

g

⁢

N

}

⁢

.

A left-normal

subgroup

N

of a

group

S

is automatically
normal, since

g

⁢

N

⊆

N

⁢

g

=

g

⁢

g

-

1

⁢

N

⁢

g

⊆

g

⁢

N

⁢

g

-

1

⁢

g

=

g

⁢

N

⁢

.

In is similarly shown for general

S

and

N

that if some

g

∈

LN

S

⁢

(

N

)

has an

inverse

g

-

1

then

g

-

1

∈

RN

S

⁢

(

N

)

and vice versa. Left- and right-normalizers
are always

closed under

multiplication

(hence subsemigroups)
and contain the

identity element

of

S

if there is one.

An example of a left-normal but not right-normal

N

⊆

S

can be constructed using matrices under multiplication, if one takes

S

=

{

(

k

m

0

1

)

|

k

,

m

∈

ℤ

}

and

N

=

{

(

1

n

0

1

)

|

n

∈

ℤ

}

⁢

,

where one may note that

N

is a group and

S

is a monoid.
Since

(

k

m

0

1

)

⁢

(

1

n

0

1

)

=

(

k

k

⁢

n

+

m

0

1

)

and

(

1

n

0

1

)

⁢

(

k

m

0

1

)

=

(

k

n

+

m

0

1

)

it follows that

g

⁢

N

⊆

N

⁢

g

for all

g

∈

S

, with
proper

inclusion

(

http://planetmath.org/ProperSubset

)
when

k

≠

±

1

.

The definition of left and

is somewhat arbitrary in the choice of whether to call
something the or left form.
A reference supporting the choice documented here is:

References

1

Karl Heinrich

Hofmann

and Michael

Mislove

:
The centralizing

theorem

for left normal groups of units in
compact monoids,

Semigroup Forum

3

(1971/72), no. 1, 31–42.

It may also be observed that the

combination

‘left normal’ in semigroup
theory frequently occurs as part of the phrase ‘left normal band’,
but in that case the etymology rather seems to be that ‘left’ qualifies
the phrase ‘normal band’.

Title

one-sided normality of subsemigroup

Canonical name

OnesidedNormalityOfSubsemigroup

Date of creation

2013-03-22 16:10:41

Last modified on

2013-03-22 16:10:41

Owner

lars_h (9802)

Last modified by

lars_h (9802)

Numerical id

6

Author

lars_h (9802)

Entry type

Definition

Classification

msc 20A05

Defines

left-normal

Defines

right-normal

Defines

left-normalizer

Defines

right-normalizer

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

LaTeXML
order of elements in finite groups

order of elements in finite groups

This article proves two elementary results regarding the orders of group elements in

finite groups

.

Theorem 1

Let

G

be a finite group, and let

a

∈

G

and

b

∈

G

be elements of

G

that commute with each other. Let

m

=

|

a

|

,

n

=

|

b

|

. If

gcd

⁡

(

m

,

n

)

=

1

, then

m

⁢

n

=

|

a

⁢

b

|

.

Proof.

Note first that

(

a

⁢

b

)

m

⁢

n

=

a

m

⁢

n

⁢

b

m

⁢

n

=

(

a

m

)

n

⁢

(

b

n

)

m

=

e

G

since

a

and

b

commute with each other. Thus

|

a

⁢

b

|

≤

m

⁢

n

. Now suppose

|

a

⁢

b

|

=

k

. Then

e

G

=

(

a

⁢

b

)

k

=

(

a

⁢

b

)

k

⁢

m

=

a

k

⁢

m

⁢

b

k

⁢

m

=

b

k

⁢

m

and thus

n

|

k

m

. But

gcd

⁡

(

m

,

n

)

=

1

, so

n

|

k

. Similarly,

m

|

k

and thus

m

n

|

k

=

|

a

b

|

. These two results together imply that

m

⁢

n

=

k

.

Theorem 2

Let

G

be a finite abelian group. If

G

contains elements of orders

m

and

n

, then it contains an element of order

lcm

⁢

(

m

,

n

)

.

Proof.

Choose

a

and

b

of orders

m

and

n

respectively, and write

lcm

⁢

(

m

,

n

)

=

∏

p

i

k

i

where the

p

i

are distinct primes. Thus for each

i

, either

p

i

k

i

∣

m

or

p

i

k

i

∣

n

. Thus either

a

m

/

p

i

k

i

or

b

n

/

p

i

k

i

has order

p

i

k

i

. Let this element be

c

i

. Now, the orders of the

c

i

are

pairwise coprime

by construction, so

|

∏

c

i

|

=

∏

|

c

i

|

=

lcm

⁢

(

m

,

n

)

and thus

∏

c

i

is the required element.

Title

order of elements in finite groups

Canonical name

OrderOfElementsInFiniteGroups

Date of creation

2013-03-22 16:34:02

Last modified on

2013-03-22 16:34:02

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20A05

Generated on Fri Feb 9 19:18:23 2018 by

LaTeXML
order of products

order of products

If

a

and

b

are elements of a group, then both

a

⁢

b

and

b

⁢

a

have always the same order.

Proof.

Let

e

be the indentity element of the group.  For

n

>

1

,  we have the

equivalent

(

http://planetmath.org/Equivalent3

) conditions

e

=

(

a

⁢

b

)

n

=

(

a

⁢

b

)

⁢

(

a

⁢

b

)

⁢

⋯

⁢

(

a

⁢

b

)

⏟

n

=

a

⁢

(

b

⁢

a

)

n

-

1

⁢

b

,

a

-

1

⁢

b

-

1

=

(

b

⁢

a

)

n

-

1

,

(

b

⁢

a

)

-

1

=

(

b

⁢

a

)

n

-

1

,

e

=

(

b

⁢

a

)

n

.

As for the

infinite order

, it makes the conditions false.

Note.

More generally, all elements of any

conjugacy class

have the same order.

Title

order of products

Canonical name

OrderOfProducts

Date of creation

2013-03-22 18:56:43

Last modified on

2013-03-22 18:56:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Related topic

InverseFormingInProportionToGroupOperation

Generated on Fri Feb 9 19:18:25 2018 by

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order (of a group)

order (of a group)

The

order

of a group

G

is the number of elements of

G

, denoted

|

G

|

; if

|

G

|

is finite, then

G

is said to be a

finite group

.

The

order

of an element

g

∈

G

is the smallest

positive integer

n

such that

g

n

=

e

, where

e

is the

identity element

; if there is no such

n

, then

g

is said to be of

infinite order

. By Lagrange’s theorem, the order of any element in a finite group divides the order of the group.

Title

order (of a group)

Canonical name

OrderofAGroup

Date of creation

2013-03-22 12:36:47

Last modified on

2013-03-22 12:36:47

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20A05

Synonym

order

Related topic

Group

Related topic

Cardinality

Related topic

OrdersOfElementsInIntegralDomain

Related topic

OrderRing

Related topic

IdealOfElementsWithFiniteOrder

Defines

finite group

Defines

infinite order

Defines

order (of a group element)

Generated on Fri Feb 9 19:18:27 2018 by

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