aarjaneiro/planetmath huggingface dataset:
free product with amalgamated subgroup

free product with amalgamated subgroup

Definition 1

.

Let

G

k

,

k

=

0

,

1

,

2

be groups and

i

k

:

G

0

→

G

i

,

k

=

1

,

2

be

monomorphisms

.
The

free product

of

G

1

and

G

2

with amalgamated

subgroup

G

0

,
is defined to be a group

G

that has the following
two properties

1.

there are

homomorphisms

j

k

:

G

k

→

G

,

k

=

1

,

2

that make the following

diagram

commute

Generated on Fri Feb 9 19:25:37 2018 by

LaTeXML
free products and group actions

free products and group actions

Theorem 1

.

(See Lang, Exercise 54 p. 81) Suppose

G

1

,

…

⁢

G

n

are

subgroups

of

G

that

generate

G

. Suppose further that

G

acts on a set

S

and that there are subsets

S

1

,

S

2

,

…

⁢

S

n

⊂

S

, and some

s

∈

S

-

∪

S

i

such that for each

1

≤

i

≤

n

, the following holds for each

g

∈

G

i

,

g

≠

e

:

•

g

⁢

(

S

j

)

⊂

S

i

if

j

≠

i

, and

•

g

⁢

(

s

)

∈

S

i

.

Then

G

=

G

1

⋆

…

⋆

G

n

(where

⋆

denotes the

free product

).

Proof:

Any

g

∈

G

can be written

g

=

g

1

⁢

g

2

⁢

…

⁢

g

k

with

g

i

∈

G

j

i

,

j

i

≠

j

i

+

1

,

g

i

≠

e

, since the

G

i

generate

G

. Thus there is a

surjective

homomorphism

ϕ

:

∐

G

i

↠

G

(since

∐

G

i

, as the

coproduct

, has this

universal property

). We must show

ker

⁡

ϕ

is

trivial

. Choose

g

1

⁢

g

2

⁢

…

⁢

g

k

as above. Then

g

k

⁢

(

s

)

∈

S

j

k

,

g

k

-

1

⁢

(

g

k

⁢

(

s

)

)

∈

S

j

k

-

1

, and so forth, so that

g

1

(

g

2

(

…

(

g

k

(

s

)

…

)

∈

S

j

1

. But

e

⁢

(

s

)

=

s

∉

S

j

1

. Thus

ϕ

⁢

(

g

1

⁢

g

2

⁢

…

⁢

g

k

)

≠

e

, and

ϕ

is

injective

.

Title

free products and group actions

Canonical name

FreeProductsAndGroupActions

Date of creation

2013-03-22 17:34:56

Last modified on

2013-03-22 17:34:56

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20E06

Generated on Fri Feb 9 19:25:40 2018 by

LaTeXML
HNN extension

HNN extension

The

HNN extension

group

G

for a group

A

, is constructed from a pair of

isomorphic

subgroups

B

≅

ϕ

C

in

A

, according to formula

G

=

A

*

⟨

t

|

-

⟩

N

where

⟨

t

|

-

⟩

is a cyclic

free group

,

*

is the

free product

and

N

is the

normal closure

of

{

t

⁢

b

⁢

t

-

1

⁢

ϕ

⁢

(

b

)

-

1

:

b

∈

B

}

.

As an example take a surface bundle

F

⊂

E

→

S

1

, hence the

homotopy

long exact sequence

of this bundle implies that the

fundamental group

π

1

⁢

(

E

)

is given by

π

1

⁢

(

E

)

=

⟨

x

1

,

…

,

x

k

,

t

|

Π

=

1

,

t

⁢

x

i

⁢

t

-

1

=

ϕ

⁢

(

x

i

)

⟩

where

k

is the genus of the

surface

and the

relation

Π

is

[

x

1

,

x

2

]

⁢

[

x

3

,

x

4

]

⁢

⋯

⁢

[

x

k

-

1

,

x

k

]

for an orientable surface or

x

1

2

⁢

x

2

2

⁢

⋯

⁢

x

k

2

is for a non-orientable one.

ϕ

is an

isomorphism

induced by a self

homeomorphism

of

F

.

Title

HNN extension

Canonical name

HNNExtension

Date of creation

2013-03-22 16:04:03

Last modified on

2013-03-22 16:04:03

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Definition

Classification

msc 20E06

Related topic

GroupExtension

Generated on Fri Feb 9 19:25:43 2018 by

LaTeXML
Scott-Wiegold conjecture

Scott-Wiegold conjecture

The

Scott-Wiegold conjecture

(1976) is stated as follows:

Given distinct

prime numbers

p

,

q

and

r

, the

free product

of

cyclic groups

C

p

*

C

q

*

C

r

is not the

normal closure

of any single element.

In 1992 this was included as problem 5.53 of The Kourovka Notebook:

Unsolved Problems in

[

1

]

.

The conjecture was proven to be true in 2001 by James Howie

[

2

]

. Despite remaining an unsolved problem for 25 years, the proof is both brief and fairly elementary.

Whilst the question is group theoretic and involves only , the proof does not use any combinatorial but instead depends on basic notions from topology.

References

1

V.D.Mazurov, E.I. Khukhro (Eds.),

Unsolved Problems in

Group Theory

: The Kourovka Notebook,

12

𝑡ℎ

Edition, Russian Academy of Sciences, Novosibirsk, 1992.

2

James Howie,

A proof of the Scott-Wiegold conjecture on free products of cyclic groups

, Journal of Pure and Applied Algebra 173, 2002 pp.167–176

Title

Scott-Wiegold conjecture

Canonical name

ScottWiegoldConjecture

Date of creation

2013-03-22 18:29:34

Last modified on

2013-03-22 18:29:34

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

8

Author

whm22 (2009)

Entry type

Theorem

Classification

msc 20E06

Synonym

one

relator

products of cyclic groups

Generated on Fri Feb 9 19:26:02 2018 by

LaTeXML
A finitely generated group has only finitely many subgroups of a given index

A finitely generated group has only finitely many subgroups of a given index

Let

G

be a

finitely generated group

and let

n

be a positive
integer. Let

H

be a

subgroup

of

G

of index

n

and consider the
action of

G

on the coset space

(

G

:

H

)

by right

multiplication

.
Label the cosets

1

,

…

,

n

, with the coset

H

labelled by

1

.
This gives a

homomorphism

ϕ

:

G

→

S

n

. Now,

x

∈

H

if and
only if

H

⁢

x

=

H

, that is,

G

fixes the coset

H

. Therefore,

H

=

Stab

G

⁢

(

1

)

=

{

g

∈

G

∣

1

⁢

(

g

⁢

ϕ

)

=

1

}

, and this is completely
determined by

ϕ

. Now let

X

be a finite

generating set

for

G

.
Then

ϕ

is determined by the images

x

⁢

ϕ

of the

generators

x

∈

X

. There are

|

S

n

|

=

n

!

choices for the image of each

x

∈

X

, so there are at most

n

!

|

X

|

homomorphisms

G

→

S

n

. Hence,
there are only finitely many possibilities for

H

.

References

1

M. Hall, Jr.,

A

topology

for

free groups

and related groups

, Ann. of
Math.

52

(1950), no. 1, 127–139.

Title

A finitely generated group has only finitely many subgroups of a given index

Canonical name

AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex

Date of creation

2013-03-22 15:16:03

Last modified on

2013-03-22 15:16:03

Owner

avf (9497)

Last modified by

avf (9497)

Numerical id

6

Author

avf (9497)

Entry type

Theorem

Classification

msc 20E07

Related topic

Group

Related topic

FinitelyGenerated

Generated on Fri Feb 9 19:26:07 2018 by

LaTeXML
permutable subgroup

permutable subgroup

\PMlinkescapephrase

permutable subgroups

Let

G

be a group.
A

subgroup

H

of

G

is said to be

permutable

if it permutes with all subgroups of

G

,
that is,

K

⁢

H

=

H

⁢

K

for all

K

≤

G

.
We sometimes write

H

⁢

per

⁡

G

to indicate that

H

is a permutable subgroup of

G

.

Permutable subgroups were introduced by Øystein Ore (

http://planetmath.org/OysteinOre

),
who called them

quasinormal

subgroups.

Normal subgroups

are clearly permutable.

Permutable subgroups are

ascendant

.
This is a result of Stonehewer

[

1

]

,
who also showed that in a

finitely generated group

,
all permutable subgroups are

subnormal

.

References

1

Stewart E. Stonehewer,

Permutable subgroups of

infinite groups

,
Math. Z. 125 (1972), 1–16. (This paper is

http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002410435

available from GDZ.)

Title

permutable subgroup

Canonical name

PermutableSubgroup

Date of creation

2013-03-22 16:15:47

Last modified on

2013-03-22 16:15:47

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 20E07

Synonym

quasinormal subgroup

Synonym

quasi-normal subgroup

Defines

permutable

Defines

quasinormal

Defines

quasi-normal

Generated on Fri Feb 9 19:26:16 2018 by

LaTeXML
proof of Cauchy’s theorem

proof of Cauchy’s theorem

Let

G

be a

finite group

, and suppose

p

is a

prime divisor

of

|

G

|

.
Consider the set

X

of all

p

-tuples

(

x

1

,

…

,

x

p

)

for which

x

1

⁢

⋯

⁢

x

p

=

1

.
Note that

|

X

|

=

|

G

|

p

-

1

is a

multiple

of

p

.
There is a natural

group action

of
the

cyclic group

ℤ

/

p

⁢

ℤ

on

X

under which

m

∈

ℤ

/

p

⁢

ℤ

sends
the tuple

(

x

1

,

…

,

x

p

)

to

(

x

m

+

1

,

…

,

x

p

,

x

1

,

…

,

x

m

)

.
By the

Orbit-Stabilizer Theorem

, each orbit contains exactly

1

or

p

tuples.
Since

(

1

,

…

,

1

)

has an orbit of

cardinality

1

,
and the orbits

partition

X

,
the cardinality of which is

divisible

by

p

,
there must exist at least one other tuple

(

x

1

,

…

,

x

p

)

which is left fixed by every element of

ℤ

/

p

⁢

ℤ

.
For this tuple we have

x

1

=

…

=

x

p

,
and so

x

1

p

=

x

1

⁢

⋯

⁢

x

p

=

1

,
and

x

1

is therefore an element of order

p

.

References

1

James H. McKay.

Another Proof of Cauchy’s Group

Theorem

,
American Math. Monthly, 66 (1959), p119.

Title

proof of Cauchy’s theorem

Canonical name

ProofOfCauchysTheorem

Date of creation

2013-03-22 12:23:30

Last modified on

2013-03-22 12:23:30

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Proof

Classification

msc 20E07

Classification

msc 20D99

Generated on Fri Feb 9 19:26:18 2018 by

LaTeXML
metabelian group

metabelian group

Definition

A

metabelian group

is a group

G

that possesses a

normal subgroup

N

such that

N

and

G

/

N

are both

abelian

.
Equivalently,

G

is metabelian if and only if the

commutator subgroup

[

G

,

G

]

is abelian.
Equivalently again,

G

is metabelian if and only if it is

solvable

of length at most

2

.

(Note that in older literature the term tends to be used in the

stronger

sense that the

central quotient

G

/

Z

⁢

(

G

)

is abelian. This is

equivalent

to being

nilpotent

of class at most

2

. We shall not use this sense here.)

Examples

•

All

abelian groups

.

•

All

generalized dihedral groups

.

•

All groups of order less than

24

.

•

All

metacyclic groups

.

Properties

Subgroups

(

http://planetmath.org/Subgroup

),

quotients

(

http://planetmath.org/QuotientGroup

) and (unrestricted)

direct products

of metabelian groups are also metabelian.
In other words, metabelian groups form a

variety

(

http://planetmath.org/VarietyOfGroups

);
they are, in fact, the groups in which

(

w

-

1

⁢

x

-

1

⁢

w

⁢

x

)

⁢

(

y

-

1

⁢

z

-

1

⁢

y

⁢

z

)

=

(

y

-

1

⁢

z

-

1

⁢

y

⁢

z

)

⁢

(

w

-

1

⁢

x

-

1

⁢

w

⁢

x

)

for all

elements

w

,

x

,

y

and

z

.

Title

metabelian group

Canonical name

MetabelianGroup

Date of creation

2013-03-22 15:36:42

Last modified on

2013-03-22 15:36:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 20E10

Classification

msc 20F16

Synonym

meta-abelian group

Related topic

AbelianGroup2

Defines

metabelian

Defines

meta-abelian

Generated on Fri Feb 9 19:26:26 2018 by

LaTeXML
variety of groups

variety of groups

\PMlinkescapephrase

closed under

\PMlinkescapephrase

free group

Definition

A

variety

of groups is the class of groups

G

that

satisfy

a given set of equationally defined

relations

{

r

i

⁢

(

x

1

,

…

,

x

n

i

)

=

1

∣

i

∈

I

}

for all

elements

x

1

,

x

2

,

x

3

,

…

of

G

,
where

I

is some

index set

.

Examples

Abelian groups

are a variety defined by the

equations

{

[

x

1

,

x

2

]

=

1

}

,

where

[

x

,

y

]

=

x

⁢

y

⁢

x

-

1

⁢

y

-

1

.

Nilpotent groups

of class less than

c

form a variety defined by

{

[

[

⋯

⁢

[

[

x

1

,

x

2

]

,

x

3

]

⁢

⋯

]

,

x

c

]

=

1

}

.

Similarly,

solvable groups

of length less than

c

form a variety.
(Abelian groups are a special case of both of these.)
Note, however, that the class of all nilpotent groups is not a variety,
nor is the class of all solvable groups.

For any positive integer

n

,
the variety defined by

{

x

1

n

=

1

}

consists of all groups of finite

exponent

dividing

n

.
For

n

=

1

this gives the variety containing only the

trivial groups

,
which is the smallest variety.

The largest variety is the variety of all groups,
given by an

empty set

of relations.

Notes

By a

theorem

of Birkhoff

[

1

]

,
a class of groups is a variety if and only if
it is closed under taking

subgroups

,

homomorphic images

and

unrestricted direct products

(that is,
every unrestricted direct product of members
of the class

𝒞

is in

𝒞

,
and all subgroups and homomorphic images of members of

𝒞

are also in

𝒞

).

A variety of groups is a

full subcategory

of the category of groups,
and there is a free group on any set of elements in the variety,
which is the usual free group (

http://planetmath.org/FreeGroup

)
modulo the relations of the variety applied to all elements.
This satisfies the usual

universal property

of the free group on groups in the variety,
and is thus

adjoint

(

http://planetmath.org/AdjointFunctor

) to the

forgetful functor

in the

category of sets

.
In the variety of abelian groups,
the free groups are the usual

free abelian groups

.
In the variety of groups satisfying

x

n

=

1

,
the free groups are called

Burnside groups

,
and are commonly denoted by

B

⁢

(

m

,

n

)

, where

m

is the number of

generators

.

References

1

G. Birkhoff,

On the

structure

of

abstract algebras

,
Proc. Cambridge Philos. Soc., 31 (1935), 433–454.

Title

variety of groups

Canonical name

VarietyOfGroups

Date of creation

2013-03-22 13:12:02

Last modified on

2013-03-22 13:12:02

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

27

Author

yark (2760)

Entry type

Definition

Classification

msc 20E10

Classification

msc 20J15

Synonym

variety

Related topic

GroupVariety

Related topic

EquationalClass

Defines

Burnside group

Generated on Fri Feb 9 19:26:29 2018 by

LaTeXML
ascending series

ascending series

Let

G

be a group.

An

ascending series

of

G

is a family

(

H

α

)

α

≤

β

of

subgroups

of

G

,
where

β

is an

ordinal

,
such that

H

0

=

{

1

}

and

H

β

=

G

,
and

H

α

⁢

⊴

⁢

H

α

+

1

for all

α

<

β

,
and

⋃

α

<

δ

H

α

=

H

δ

whenever

δ

≤

β

is a

limit ordinal

.

Note that this is a

generalization

of the

concept

of a

subnormal series

.
Compare also the dual concept of a

descending series

.

Given an ascending series

(

H

α

)

α

≤

β

,
the subgroups

H

α

are called the

terms

of the series
and the

quotients

(

http://planetmath.org/QuotientGroup

)

H

α

+

1

/

H

α

are called the

factors

of the series.

A subgroup of

G

that is a term of some ascending series of

G

is called an

ascendant subgroup

of

G

.
The notation

H

⁢

asc

⁡

G

is sometimes used
to indicate that

H

is an ascendant subgroup of

G

.

The groups in which every subgroup is ascendant
are precisely the groups that

satisfy

the

normalizer condition

.
Groups in which every

cyclic subgroup

is ascendant
are called

Gruenberg groups

.
It can be shown that in a Gruenberg group,
every

finitely generated subgroup

is ascendant and

nilpotent

(and so, in particular, Gruenberg groups are

locally nilpotent

).

An ascending series of

G

in which all terms are normal in

G

is called an

ascending normal series

.

Let

𝔛

be a property of groups.
A group is said to be

hyper-

X

if it has an ascending normal series
whose factors all have property

𝔛

.
So, for example, a

hyperabelian group

is a group that has an ascending normal series with

abelian

factors.
Hyperabelian groups are sometimes called

S

⁢

I

*

-groups

.

Title

ascending series

Canonical name

AscendingSeries

Date of creation

2013-03-22 16:14:55

Last modified on

2013-03-22 16:14:55

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 20E15

Classification

msc 20F22

Related topic

DescendingSeries

Related topic

SubnormalSeries

Related topic

SubnormalSubgroup

Defines

ascending normal series

Defines

ascendant subgroup

Defines

ascendant

Defines

hyperabelian group

Defines

hyperabelian

Defines

Gruenberg group

Generated on Fri Feb 9 19:26:31 2018 by

LaTeXML
descending series

descending series

Let

G

be a group.

A

descending series

of

G

is a family

(

H

α

)

α

≤

β

of

subgroups

of

G

,
where

β

is an

ordinal

,
such that

H

0

=

G

and

H

β

=

{

1

}

,
and

H

α

+

1

⁢

⊴

⁢

H

α

for all

α

<

β

,
and

⋂

α

<

δ

H

α

=

H

δ

whenever

δ

≤

β

is a

limit ordinal

.

Note that this is a

generalization

of the

concept

of a

subnormal series

.
Compare also the dual concept of an

ascending series

.

Given a descending series

(

H

α

)

α

≤

β

,
the subgroups

H

α

are called the

terms

of the series
and the

quotients

(

http://planetmath.org/QuotientGroup

)

H

α

/

H

α

+

1

are called the

factors

of the series.

A subgroup of

G

that is a term of some descending series of

G

is called a

descendant subgroup

of

G

.

A descending series of

G

in which all terms are normal in

G

is called a

descending normal series

.

Let

𝔛

be a property of groups.
A group is said to be

hypo-

X

if it has a descending normal series
whose factors all have property

𝔛

.
So, for example, a

hypoabelian group

is a group that has a descending normal series with

abelian

factors.
Hypoabelian groups are sometimes called

SD-groups

;
they are precisely the groups that have no non-trivial

perfect

subgroups,
and they are also precisely the groups
in which the

transfinite derived series

eventually reaches

{

1

}

.

Title

descending series

Canonical name

DescendingSeries

Date of creation

2013-03-22 16:14:52

Last modified on

2013-03-22 16:14:52

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20E15

Classification

msc 20F22

Related topic

AscendingSeries

Related topic

SubnormalSeries

Related topic

SubnormalSubgroup

Defines

descending normal series

Defines

descendant subgroup

Defines

descendant

Defines

hypoabelian group

Defines

hypoabelian

Defines

SD-group

Generated on Fri Feb 9 19:26:33 2018 by

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example of a Jordan-Hölder decomposition

example of a Jordan-Hölder decomposition

A group that has a

composition series

will often have several different composition series.

For example, the

cyclic group

C

12

has

(

E

,

C

2

,

C

6

,

C

12

)

, and

(

E

,

C

2

,

C

4

,

C

12

)

, and

(

E

,

C

3

,

C

6

,

C

12

)

as different composition series.
However, the result of the Jordan-Hölder

Theorem

is that any two composition series of a group are

equivalent

, in the sense that the

sequence

of

factor groups

in each series are the same, up to rearrangement of their order in the sequence

A

i

+

1

/

A

i

. In the above example, the factor groups are

isomorphic

to

(

C

2

,

C

3

,

C

2

)

,

(

C

2

,

C

2

,

C

3

)

, and

(

C

3

,

C

2

,

C

2

)

, respectively.

This is taken from the

http://en.wikipedia.org/wiki/Solvable_group

Wikipedia article on

solvable groups

.

Title

example of a Jordan-Hölder decomposition

Canonical name

ExampleOfAJordanHolderDecomposition

Date of creation

2013-03-22 14:24:33

Last modified on

2013-03-22 14:24:33

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Example

Classification

msc 20E15

Synonym

example of

Jordan-Holder decomposition

Generated on Fri Feb 9 19:26:35 2018 by

LaTeXML
Jordan-Hölder decomposition

Jordan-Hölder decomposition

A

Jordan–Hölder decomposition

of a group

G

is a

filtration

G

=

G

1

⊃

G

2

⊃

⋯

⊃

G

n

=

{

1

}

such that

G

i

+

1

is a

normal subgroup

of

G

i

and the

quotient

G

i

/

G

i

+

1

is a

simple group

for each

i

.

Title

Jordan-Hölder decomposition

Canonical name

JordanHolderDecomposition

Date of creation

2013-03-22 12:08:41

Last modified on

2013-03-22 12:08:41

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 20E15

Synonym

composition series

Related topic

DerivedSubgroup

Related topic

JordanHolderDecompositionTheorem

Generated on Fri Feb 9 19:26:37 2018 by

LaTeXML
normal subgroups form sublattice of a subgroup lattice

normal subgroups form sublattice of a subgroup lattice

Consider

L

⁢

(

G

)

, the

subgroup lattice

of a group

G

. Let

N

⁢

(

G

)

be the subset of

L

⁢

(

G

)

, consisting of all

normal subgroups

of

G

.

First, we show that

N

⁢

(

G

)

is

closed under

∧

. Suppose

H

and

K

are normal subgroups of

G

. If

x

∈

H

∧

K

=

H

∩

K

, then for any

g

∈

G

,

g

⁢

x

⁢

g

-

1

∈

H

since

H

is normal, and

g

⁢

x

⁢

g

-

1

∈

K

likewise. So

g

⁢

x

⁢

g

-

1

∈

H

∩

K

=

H

∧

K

, implying that

H

∧

K

is normal in

G

, or

H

∧

K

∈

N

⁢

(

G

)

.

To see that

N

⁢

(

G

)

is closed under

∨

, let

H

,

K

be normal subgroups of

G

, and consider an

element

x

=

x

1

⁢

x

2

⁢

⋯

⁢

x

n

∈

H

∨

K

,

where

x

i

∈

H

or

x

i

∈

K

. If

g

∈

G

, then

g

⁢

x

⁢

g

-

1

=

g

⁢

x

1

⁢

x

2

⁢

⋯

⁢

x

n

⁢

g

-

1

=

(

g

⁢

x

1

⁢

g

-

1

)

⁢

(

g

⁢

x

2

⁢

g

-

1

)

⁢

⋯

⁢

(

g

⁢

x

n

⁢

g

-

1

)

,

where each

g

⁢

x

i

⁢

g

-

1

∈

H

or

K

. Therefore,

g

⁢

x

⁢

g

-

1

∈

H

∨

K

, so

H

∨

K

is normal in

G

and

H

∨

K

∈

N

⁢

(

G

)

.

Since

N

⁢

(

G

)

is closed under

∧

and

∨

,

N

⁢

(

G

)

is a

sublattice

of

L

⁢

(

G

)

.

Remark

. If

G

is finite, it can be shown (Wielandt) that the

subnormal subgroups

of

G

form a sublattice of

L

⁢

(

G

)

.

References

1

H. Wielandt

Eine Verallgemeinerung der invarianten Untergruppen

, Math. Zeit. 45, pp. 209-244 (1939)

Title

normal subgroups form sublattice of a subgroup lattice

Canonical name

NormalSubgroupsFormSublatticeOfASubgroupLattice

Date of creation

2013-03-22 15:48:24

Last modified on

2013-03-22 15:48:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Example

Classification

msc 20E15

Generated on Fri Feb 9 19:26:43 2018 by

LaTeXML
Schreier refinement theorem

Schreier refinement theorem

The

Schreier Refinement Theorem

states that any two

subnormal series

for a group have

equivalent

refinements

.
Here, two subnormal series are considered equivalent if they have the same factors (up to

isomorphism

), not necessarily in the same order.

This

theorem

can be used to prove the Jordan-Hölder Theorem (

http://planetmath.org/JordanHolderDecompositionTheorem

), and can also be used to prove that the

Hirsch number

of a

polycyclic group

is well-defined.

Title

Schreier refinement theorem

Canonical name

SchreierRefinementTheorem

Date of creation

2013-03-22 14:40:53

Last modified on

2013-03-22 14:40:53

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Theorem

Classification

msc 20E15

Generated on Fri Feb 9 19:26:45 2018 by

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