aarjaneiro/planetmath huggingface dataset:
length function

length function

Let

G

be a group.
A

length function

on

G

is a function

L

:

G

→

ℝ

+

satisfying:

L

⁢

(

e

)

=

0

,

L

⁢

(

g

)

=

L

⁢

(

g

-

1

)

,

∀

g

∈

G

,

L

⁢

(

g

1

⁢

g

2

)

≤

L

⁢

(

g

1

)

+

L

⁢

(

g

2

)

,

∀

g

1

,

g

2

∈

G

.

Title

length function

Canonical name

LengthFunction

Date of creation

2013-03-22 13:41:29

Last modified on

2013-03-22 13:41:29

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

6

Author

mhale (572)

Entry type

Definition

Classification

msc 20-02

Generated on Fri Feb 9 19:16:31 2018 by

LaTeXML
a characterization of groups

a characterization of groups

\PMlinkescapephrase

idempotent element

Theorem

.

A non-empty

semigroup

S

is a group
if and only if
for every

x

∈

S

there is a unique

y

∈

S

such that

x

⁢

y

⁢

x

=

x

.

Proof.

Suppose that

S

is a non-empty semigroup,
and for every

x

∈

S

there is a unique

y

∈

S

such that

x

⁢

y

⁢

x

=

x

.
For each

x

∈

S

,
let

x

′

denote the unique element of

S

such that

x

⁢

x

′

⁢

x

=

x

.
Note that

x

⁢

(

x

′

⁢

x

⁢

x

′

)

⁢

x

=

(

x

⁢

x

′

⁢

x

)

⁢

x

′

⁢

x

=

x

⁢

x

′

⁢

x

=

x

,
so, by uniqueness,

x

′

⁢

x

⁢

x

′

=

x

′

,
and therefore

x

′′

=

x

.

For any

x

∈

S

, the

element

x

⁢

x

′

is

idempotent

(

http://planetmath.org/Idempotency

),
because

(

x

⁢

x

′

)

2

=

(

x

⁢

x

′

⁢

x

)

⁢

x

′

=

x

⁢

x

′

.
As

S

is nonempty, this means that

S

has at least one idempotent element.
If

i

∈

S

is idempotent,
then

i

⁢

x

=

i

⁢

x

⁢

(

i

⁢

x

)

′

⁢

i

⁢

x

=

i

⁢

x

⁢

(

i

⁢

x

)

′

⁢

i

⁢

i

⁢

x

, and so

(

i

⁢

x

)

′

⁢

i

=

(

i

⁢

x

)

′

,
and therefore

(

i

⁢

x

)

′

=

(

i

⁢

x

)

′

⁢

(

i

⁢

x

)

′′

⁢

(

i

⁢

x

)

′

=

(

i

⁢

x

)

′

⁢

i

⁢

x

⁢

(

i

⁢

x

)

′

=

(

i

⁢

x

)

′

⁢

x

⁢

(

i

⁢

x

)

′

,
which means that

i

⁢

x

=

(

i

⁢

x

)

′′

=

x

.
So every idempotent is a

left identity

,
and, by a

symmetric

argument

, a right identity.
Therefore,

S

has at most one idempotent element.
Combined with the previous result,
this means that

S

has exactly one idempotent element,
which we will denote by

e

.
We have shown that

e

is an

identity

,
and that

x

⁢

x

′

=

e

for each

x

∈

S

, so

S

is a group.

Conversely

, if

S

is a group
then

x

⁢

y

⁢

x

=

x

clearly has a unique

solution

, namely

y

=

x

-

1

.
∎

Note.

Note that

inverse semigroups

do not in general

satisfy

the

hypothesis

of this

theorem

:
in an inverse semigroup there is for each

x

a unique

y

such that

x

⁢

y

⁢

x

=

x

and

y

⁢

x

⁢

y

=

y

,
but this

y

need not be unique as a solution of

x

⁢

y

⁢

x

=

x

alone.

Title

a

characterization of groups

Canonical name

ACharacterizationOfGroups

Date of creation

2013-03-22 14:45:08

Last modified on

2013-03-22 14:45:08

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 20A05

Related topic

Group

Related topic

RegularSemigroup

Related topic

AlternativeDefinitionOfGroup

Generated on Fri Feb 9 19:16:33 2018 by

LaTeXML
a group of even order contains an element of order 2

a group of even order contains an element of order 2

Proposition

.

Every group of even order contains an element of order

2

.

Proof.

Let

G

be a group of even order, and consider the set

S

=

{

g

∈

G

:

g

≠

g

-

1

}

. We claim that

|

S

|

is even; to see this, let

a

∈

S

, so that

a

≠

a

-

1

; since

(

a

-

1

)

-

1

=

a

≠

a

-

1

, we see that

a

-

1

∈

S

as well. Thus the elements of

S

may be exhausted by repeatedly selecting an

element

and it with its

inverse

, from which it follows that

|

S

|

is a

multiple

(

http://planetmath.org/Divisibility

) of

2

(i.e., is even). Now, because

S

∩

(

G

∖

S

)

=

∅

and

S

∪

(

G

∖

S

)

=

G

, it must be that

|

S

|

+

|

G

∖

S

|

=

|

G

|

, which, because

|

G

|

is even, implies that

|

G

∖

S

|

is also even. The

identity element

e

of

G

is in

G

∖

S

, being its own inverse, so the set

G

∖

S

is nonempty, and consequently must contain at least two distinct elements; that is, there must exist some

b

≠

e

∈

G

∖

S

, and because

b

∉

S

, we have

b

=

b

-

1

, hence

b

2

=

1

. Thus

b

is an element of order

2

in

G

.
∎

Notice that the above is

logically equivalent

to the assertion that a group of even order has a non-identity element that is its own inverse.

Title

a group of even order contains an element of order 2

Canonical name

AGroupOfEvenOrderContainsAnElementOfOrder2

Date of creation

2013-03-22 17:11:37

Last modified on

2013-03-22 17:11:37

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

20

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 20A05

Generated on Fri Feb 9 19:16:35 2018 by

LaTeXML
a subgroup of index 2 is normal

a subgroup of index 2 is normal

Lemma

.

Let

(

G

,

⋅

)

be a group and let

H

be a

subgroup

of

G

of index 2. Then

H

is normal in

G

.

Proof.

Let

G

be a group and let

H

be an index 2 subgroup of

G

. By definition of index, there are only two

left cosets

of

H

in

G

, namely:

H

,

g

1

⁢

H

where

g

1

is any element of

G

which is not in

H

. Notice that if

g

1

,

g

2

are two elements in

G

which are not in

H

then

g

1

⋅

g

2

belongs to

H

. Indeed, the coset

g

1

⁢

g

2

⁢

H

≠

g

1

⁢

H

(because

g

1

⁢

g

2

=

g

1

⁢

h

would immediately yield

g

2

=

h

∈

H

) and so

g

1

⁢

g

2

⁢

H

=

H

and

g

1

⁢

g

2

∈

H

.

Let

h

∈

H

be an arbitrary element of

H

and let

g

∈

G

. If

g

∈

H

then

g

⁢

h

⁢

g

-

1

∈

H

and we are done. Otherwise, assume that

g

∉

H

. Thus

g

⁢

h

∉

H

and by the remark above

g

⁢

h

⁢

g

-

1

=

(

g

⁢

h

)

⁢

g

-

1

∈

H

, as desired.
∎

Title

a

subgroup of index 2 is normal

Canonical name

ASubgroupOfIndex2IsNormal

Date of creation

2013-03-22 15:09:25

Last modified on

2013-03-22 15:09:25

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20A05

Related topic

Coset

Related topic

QuotientGroup

Related topic

NormalityOfSubgroupsOfPrimeIndex

Generated on Fri Feb 9 19:16:37 2018 by

LaTeXML
alternative definition of group

alternative definition of group

The below

theorem

gives three conditions that form alternative

postulates

.  It is not hard to show that they hold in the group defined ordinarily.

Theorem

.

Let the non-empty set

G

satisfy

the following three conditions:

I.     For every two

elements

a

,

b

of

G

there is a unique element

a

⁢

b

of

G

.

II.   For every three elements

a

,

b

,

c

of

G

the

equation

(

a

⁢

b

)

⁢

c

=

a

⁢

(

b

⁢

c

)

holds.

III. For every two elements

a

and

b

of

G

there exists at least one such element

x

and at least one such element

y

of

G

that

x

⁢

a

=

a

⁢

y

=

b

.

Then the set

G

forms a group.

Proof.

If

a

and

b

are arbitrary elements, then there are at least one such

e

a

and such

e

b

that

e

a

⁢

a

=

a

and

b

⁢

e

b

=

b

.  There are also such

x

and

y

that

x

⁢

b

=

e

a

and

a

⁢

y

=

e

b

.  Thus we have

e

a

=

x

⁢

b

=

x

⁢

(

b

⁢

e

b

)

=

(

x

⁢

b

)

⁢

e

b

=

e

a

⁢

e

b

=

e

a

⁢

(

a

⁢

y

)

=

(

e

a

⁢

a

)

⁢

y

=

a

⁢

y

=

e

b

,

i.e. there is a unique

neutral element

e

in

G

.  Moreover, for any element

a

there is at least one couple

a

′

,

a

′′

such that

a

′

⁢

a

=

a

⁢

a

′′

=

e

.  We then see that

a

′

=

a

′

⁢

e

=

a

′

⁢

(

a

⁢

a

′′

)

=

(

a

′

⁢

a

)

⁢

a

′′

=

e

⁢

a

′′

=

a

′′

,

i.e.

a

has a unique

neutralizing element

a

′

.

Title

alternative definition of group

Canonical name

AlternativeDefinitionOfGroup

Date of creation

2013-03-22 15:07:58

Last modified on

2013-03-22 15:07:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 20-00

Classification

msc 08A99

Related topic

Characterization

Related topic

ACharacterizationOfGroups

Related topic

DivisionInGroup

Related topic

MoreOnDivisionInGroups

Related topic

LoopAndQuasigroup

Generated on Fri Feb 9 19:16:39 2018 by

LaTeXML
automorphism group of a cyclic group

automorphism group of a cyclic group

Theorem 1

.

The

automorphism group

of the

cyclic group

Z

/

n

⁢

Z

is

(

Z

/

n

⁢

Z

)

×

, which is of order

ϕ

⁢

(

n

)

(here

ϕ

is the

Euler totient function

).

Proof.

Choose a

generator

x

for

ℤ

/

n

⁢

ℤ

. If

ρ

∈

Aut

⁡

(

ℤ

/

n

⁢

ℤ

)

, then

ρ

⁢

(

x

)

=

x

a

for some

integer

a

(defined up to

multiples

of

n

); further, since

x

generates

ℤ

/

n

⁢

ℤ

, it is clear that

a

uniquely determines

ρ

. Write

ρ

a

for this

automorphism

. Since

ρ

a

is an automorphism,

x

a

is also a generator, and thus

a

and

n

are relatively prime

1

1

If they were not, say

(

a

,

n

)

=

d

, then

(

x

a

)

n

/

d

=

(

x

a

/

d

)

n

=

1

so that

x

a

would not generate.

. Clearly, then, every

a

relatively prime to

n

induces an automorphism. We can therefore define a surjective map

Φ

:

Aut

⁡

(

ℤ

/

n

⁢

ℤ

)

→

(

ℤ

/

n

⁢

ℤ

)

×

:

ρ

a

↦

a

(

mod

n

)

Φ

is also obviously injective, so all that remains is to show that it is a

group homomorphism

. But for every

a

,

b

∈

(

ℤ

/

n

⁢

ℤ

)

×

, we have

(

ρ

a

∘

ρ

b

)

⁢

(

x

)

=

ρ

a

⁢

(

x

b

)

=

(

x

b

)

a

=

x

a

⁢

b

=

ρ

a

⁢

b

⁢

(

x

)

and thus

Φ

⁢

(

ρ

a

∘

ρ

b

)

=

Φ

⁢

(

ρ

a

⁢

b

)

=

a

⁢

b

(

mod

n

)

=

Φ

⁢

(

ρ

a

)

⁢

Φ

⁢

(

ρ

b

)

∎

References

1

Dummit, D., Foote, R.M.,

Abstract Algebra, Third Edition

, Wiley, 2004.

Title

automorphism group of a cyclic group

Canonical name

AutomorphismGroupOfACyclicGroup

Date of creation

2013-03-22 18:42:35

Last modified on

2013-03-22 18:42:35

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 20F28

Generated on Fri Feb 9 19:16:41 2018 by

LaTeXML
center of a group

center of a group

The

center

of a group

G

is the

subgroup

consisting of those elements that commute with every other element. Formally,

Z

⁡

(

G

)

=

{

x

∈

G

∣

x

⁢

g

=

g

⁢

x

⁢

for all

⁢

g

∈

G

}

.

It can be shown that the center has the following

properties

:

•

It is a

normal subgroup

(in fact, a

characteristic subgroup

).

•

It consists of those

conjugacy classes

containing just one element.

•

The center of an

abelian group

is the entire group.

•

For every prime

p

, every non-trivial finite

p

-group (

http://planetmath.org/PGroup4

) has a non-trivial center.
(Proof of a stronger version of this

theorem

. (

http://planetmath.org/ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection

))

A subgroup of the center of a group

G

is called a

central subgroup

of

G

.
All central subgroups of

G

are normal in

G

.

For any group

G

, the

quotient

(

http://planetmath.org/QuotientGroup

)

G

/

Z

⁡

(

G

)

is called the

central quotient

of

G

,
and is

isomorphic

to the

inner automorphism group

Inn

⁡

(

G

)

.

Title

center of a group

Canonical

name

CenterOfAGroup

Date of creation

2013-03-22 12:23:38

Last modified on

2013-03-22 12:23:38

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Synonym

center

Synonym

centre

Related topic

CenterOfARing

Related topic

Centralizer

Defines

central quotient

Generated on Fri Feb 9 19:16:43 2018 by

LaTeXML
characteristic subgroup

characteristic subgroup

If

(

G

,

*

)

is a group, then

H

is a

characteristic subgroup

of

G

(written

H

⁢

char

⁡

G

) if every

automorphism

of

G

maps

H

to itself. That is, if

f

∈

Aut

⁢

(

G

)

and

h

∈

H

then

f

⁢

(

h

)

∈

H

.

A few

properties

of characteristic subgroups:

•

If

H

⁢

char

⁡

G

then

H

is a

normal subgroup

of

G

.

•

If

G

has only one

subgroup

of a given

cardinality

then that subgroup is characteristic.

•

If

K

⁢

char

⁡

H

and

H

⁢

⊴

⁢

G

then

K

⁢

⊴

⁢

G

. (Contrast with

normality of subgroups is not transitive

.)

•

If

K

⁢

char

⁡

H

and

H

⁢

char

⁡

G

then

K

⁢

char

⁡

G

.

Proofs of these properties:

•

Consider

H

⁢

char

⁡

G

under the

inner automorphisms

of

G

. Since every automorphism

preserves

H

, in particular every inner automorphism preserves

H

, and therefore

g

*

h

*

g

-

1

∈

H

for any

g

∈

G

and

h

∈

H

. This is precisely the

definition

of a normal subgroup.

•

Suppose

H

is the only subgroup of

G

of order

n

. In general,

homomorphisms

(

http://planetmath.org/GroupHomomorphism

) take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of

G

of order

n

, any automorphism must take

H

to

H

, and so

H

⁢

char

⁡

G

.

•

Take

K

⁢

char

⁡

H

and

H

⁢

⊴

⁢

G

, and consider the inner automorphisms of

G

(automorphisms of the form

h

↦

g

*

h

*

g

-

1

for some

g

∈

G

). These all preserve

H

, and so are automorphisms of

H

. But any automorphism of

H

preserves

K

, so for any

g

∈

G

and

k

∈

K

,

g

*

k

*

g

-

1

∈

K

.

•

Let

K

⁢

char

⁡

H

and

H

⁢

char

⁡

G

, and let

ϕ

be an automorphism of

G

. Since

H

⁢

char

⁡

G

,

ϕ

⁢

[

H

]

=

H

, so

ϕ

H

, the

restriction

of

ϕ

to

H

is an automorphism of

H

. Since

K

⁢

char

⁡

H

, so

ϕ

H

⁢

[

K

]

=

K

. But

ϕ

H

is just a restriction of

ϕ

, so

ϕ

⁢

[

K

]

=

K

. Hence

K

⁢

char

⁡

G

.

Title

characteristic subgroup

Canonical

name

CharacteristicSubgroup

Date of creation

2013-03-22 12:50:56

Last modified on

2013-03-22 12:50:56

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Related topic

FullyInvariantSubgroup

Related topic

NormalSubgroup

Related topic

SubnormalSubgroup

Defines

characteristic

Generated on Fri Feb 9 19:16:45 2018 by

LaTeXML
class function

class function

Given a field

K

, a

K

–valued

class function

on a group

G

is a

function

f

:

G

⟶

K

such that

f

⁢

(

g

)

=

f

⁢

(

h

)

whenever

g

and

h

are elements of the same

conjugacy class

of

G

.

An important example of a class function is the

character

of a

group representation

. Over the

complex numbers

, the set of characters of the

irreducible representations

of

G

form a basis for the

vector space

of all

ℂ

–valued class functions, when

G

is a

compact Lie group

.

Relation to the convolution algebra

Class functions are also known as central functions, because they correspond to functions

f

in the

convolution algebra

C

*

⁢

(

G

)

that have the property

f

*

g

=

g

*

f

for all

g

∈

C

*

⁢

(

G

)

(i.e., they commute with everything under the convolution operation). More precisely, the set of measurable

complex

valued class functions

f

is equal to the set of

central elements

of the convolution algebra

C

*

⁢

(

G

)

, for

G

a

locally compact group

admitting a Haar measure.

Title

class function

Canonical name

ClassFunction

Date of creation

2013-03-22 12:18:06

Last modified on

2013-03-22 12:18:06

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 20A05

Synonym

central function

Generated on Fri Feb 9 19:16:46 2018 by

LaTeXML
C_{m⁢n}≅C_m×C_n when m,n are relatively prime

C

m

⁢

n

≅

C

m

×

C

n

when

m

,

n

are relatively prime

We show that

C

m

⁢

n

, gcd

(

m

,

n

)

=

1

, is

isomorphic

to

C

m

×

C

n

, where

C

r

denotes the

cyclic group

of order

r

for any

positive integer

r

.

Let

C

m

=

⟨

x

⟩

and

C

n

=

⟨

y

⟩

. Then the

external direct product

C

m

×

C

n

consists of elements

(

x

i

,

y

j

)

, where

0

≤

i

≤

m

-

1

and

0

≤

j

≤

n

-

1

.

Next, we show that the group

C

m

×

C

n

is cyclic. We do so by showing that it is generated by an element, namely

(

x

,

y

)

:
if

(

x

,

y

)

generates

C

m

×

C

n

, then for each

(

x

i

,

y

j

)

∈

C

m

×

C

n

, we must have

(

x

i

,

y

j

)

=

(

x

,

y

)

k

for some

k

∈

{

0

,

1

,

2

,

…

,

m

⁢

n

-

1

}

. Such

k

, if exists, would satisfy

k

≡

i

⁢

(

m

⁢

o

⁢

d

⁢

m

)

k

≡

j

⁢

(

m

⁢

o

⁢

d

⁢

n

)

.

Indeed, by the

Chinese Remainder Theorem

, such

k

exists and is unique modulo

m

⁢

n

. (Here is where the relative

primality

of

m

,

n

comes into play.) Thus,

C

m

×

C

n

is generated by

(

x

,

y

)

, so it is cyclic.

The order of

C

m

×

C

n

is

m

⁢

n

, so is the order of

C

m

⁢

n

. Since cyclic groups of the same order are isomorphic, we finally have

C

m

⁢

n

≅

C

m

×

C

n

.

Title

C

m

⁢

n

≅

C

m

×

C

n

when

m

,

n

are relatively prime

Canonical name

CmncongCmtimesCnWhenMNAreRelativelyPrime

Date of creation

2013-03-22 17:59:46

Last modified on

2013-03-22 17:59:46

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

8

Author

yesitis (13730)

Entry type

Proof

Classification

msc 20A05

Generated on Fri Feb 9 19:16:48 2018 by

LaTeXML
conjugacy class

conjugacy class

Two

elements

g

and

g

′

of a group

G

are said to be

conjugate

if there exists

h

∈

G

such that

g

′

=

h

⁢

g

⁢

h

-

1

.

Conjugacy

of elements is an

equivalence relation

, and the

equivalence classes

of

G

are called

conjugacy classes

.

Two subsets

S

and

T

of

G

are said to be

conjugate

if there exists

g

∈

G

such that

T

=

{

g

⁢

s

⁢

g

-

1

∣

s

∈

S

}

⊂

G

.

In this situation, it is common to write

g

⁢

S

⁢

g

-

1

for

T

to denote the fact that everything in

T

has the form

g

⁢

s

⁢

g

-

1

for some

s

∈

S

. We say that two

subgroups

of

G

are conjugate if they are conjugate as subsets.

Title

conjugacy class

Canonical name

ConjugacyClass

Date of creation

2013-03-22 12:18:09

Last modified on

2013-03-22 12:18:09

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 20A05

Synonym

conjugate

Synonym

conjugate set

Synonym

conjugate subgroup

Related topic

ConjugacyClassFormula

Generated on Fri Feb 9 19:16:50 2018 by

LaTeXML
conjugate stabilizer subgroups

conjugate stabilizer subgroups

Let

⋅

be a right group action of

G

on a set

M

. Then

G

α

⋅

g

=

g

-

1

⁢

G

α

⁢

g

for any

α

∈

M

and

g

∈

G

.

1

1

G

α

is the

stabilizer

subgroup

of

α

∈

M

.

Proof:

x

∈

G

α

⋅

g

↔

α

⋅

(

g

⁢

x

)

=

α

⋅

g

↔

α

⋅

(

g

⁢

x

⁢

g

-

1

)

=

α

↔

g

⁢

x

⁢

g

-

1

∈

G

α

↔

x

∈

g

-

1

⁢

α

⁢

g

and therefore

G

α

⋅

g

=

g

-

1

⁢

G

α

⁢

g

.

Thus all stabilizer subgroups for elements of the orbit

G

⁢

(

α

)

of

α

are

conjugate

to

G

α

.

Title

conjugate stabilizer subgroups

Canonical name

ConjugateStabilizerSubgroups

Date of creation

2013-03-22 13:21:44

Last modified on

2013-03-22 13:21:44

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

7

Author

Thomas Heye (1234)

Entry type

Derivation

Classification

msc 20A05

Related topic

Orbit

Generated on Fri Feb 9 19:16:52 2018 by

LaTeXML
core of a subgroup

core of a subgroup

Let

H

be a

subgroup

of a group

G

.

The

core

(or

normal interior

, or

normal core

) of

H

in

G

is the

intersection

of all

conjugates

of

H

in

G

:

core

G

⁡

(

H

)

=

⋂

x

∈

G

x

-

1

⁢

H

⁢

x

.

It is not hard to show that

core

G

⁡

(

H

)

is the largest

normal subgroup

of

G

contained

in

H

,
that is,

core

G

⁡

(

H

)

⁢

⊴

⁢

G

and
if

N

⁢

⊴

⁢

G

and

N

⊆

H

then

N

⊆

core

G

⁡

(

H

)

.
For this reason, some authors denote the core by

H

G

rather than

core

G

⁡

(

H

)

,
by

analogy

with the notation

H

G

for the

normal closure

.

If

core

G

⁡

(

H

)

=

{

1

}

,
then

H

is said to be

core-free

.

If

core

G

⁡

(

H

)

is of finite index in

H

,
then

H

is said to be

normal-by-finite

.

Let

ℒ

be the set of

left cosets

of

H

in

G

.
By considering the action of

G

on

ℒ

it can be shown that
the

quotient

(

http://planetmath.org/QuotientGroup

)

G

/

core

G

⁡

(

H

)

embeds in the

symmetric group

Sym

⁡

(

ℒ

)

.
A

consequence

of this is that if

H

is of finite index in

G

,
then

core

G

⁡

(

H

)

is also of finite index in

G

,
and

[

G

:

core

G

⁡

(

H

)

]

divides

[

G

:

H

]

!

(the

factorial

of

[

G

:

H

]

).
In particular, if a

simple group

S

has a

proper subgroup

of finite index

n

,
then

S

must be of finite order dividing

n

!

,
as the core of the subgroup is

trivial

.
It also follows that
a group is

virtually abelian

if and only if it is abelian-by-finite,
because the core of an

abelian

subgroup of finite index
is a normal abelian subgroup of finite index
(and the same

argument

applies if ‘abelian’ is replaced by
any other

property

that is inherited by subgroups).

Title

core of a subgroup

Canonical

name

CoreOfASubgroup

Date of creation

2013-03-22 15:37:22

Last modified on

2013-03-22 15:37:22

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Synonym

core

Synonym

normal core

Synonym

normal interior

Related topic

NormalClosure2

Defines

core-free

Defines

corefree

Defines

normal-by-finite

Defines

core-free subgroup

Defines

corefree subgroup

Defines

normal-by-finite subgroup

Generated on Fri Feb 9 19:16:54 2018 by

LaTeXML
coset

coset

Let

H

be a

subgroup

of a group

G

, and let

a

∈

G

. The

left coset

of

a

with respect to

H

in

G

is defined to be the set

a

⁢

H

:=

{

a

⁢

h

∣

h

∈

H

}

.

The

right coset

of

a

with respect to

H

in

G

is defined to be the set

H

⁢

a

:=

{

h

⁢

a

∣

h

∈

H

}

.

Two left cosets

a

⁢

H

and

b

⁢

H

of

H

in

G

are either identical or

disjoint

. Indeed, if

c

∈

a

⁢

H

∩

b

⁢

H

, then

c

=

a

⁢

h

1

and

c

=

b

⁢

h

2

for some

h

1

,

h

2

∈

H

, whence

b

-

1

⁢

a

=

h

2

⁢

h

1

-

1

∈

H

. But then, given any

a

⁢

h

∈

a

⁢

H

, we have

a

⁢

h

=

(

b

⁢

b

-

1

)

⁢

a

⁢

h

=

b

⁢

(

b

-

1

⁢

a

)

⁢

h

∈

b

⁢

H

, so

a

⁢

H

⊂

b

⁢

H

, and similarly

b

⁢

H

⊂

a

⁢

H

. Therefore

a

⁢

H

=

b

⁢

H

.

Similarly, any two right cosets

H

⁢

a

and

H

⁢

b

of

H

in

G

are either identical or disjoint. Accordingly, the

collection

of left cosets (or right cosets)

partitions

the group

G

; the corresponding

equivalence relation

for left cosets can be described succintly by the

relation

a

∼

b

if

a

-

1

⁢

b

∈

H

, and for right cosets by

a

∼

b

if

a

⁢

b

-

1

∈

H

.

The

index

of

H

in

G

, denoted

[

G

:

H

]

, is the

cardinality

of the set

G

/

H

of left cosets of

H

in

G

.

Title

coset

Canonical name

Coset

Date of creation

2013-03-22 12:08:10

Last modified on

2013-03-22 12:08:10

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 20A05

Defines

index

Defines

left coset

Defines

right coset

Generated on Fri Feb 9 19:16:58 2018 by

LaTeXML