aarjaneiro/planetmath huggingface dataset:
pronormal subgroup

pronormal subgroup

A

subgroup

H

of a group

G

is called a

pronormal subgroup

if for all

x

∈

G

the subgroups

H

and

x

⁢

H

⁢

x

-

1

are

conjugate

in

⟨

H

,

x

⁢

H

⁢

x

-

1

⟩

.

Some facts about pronormal subgroups:

•

Normal subgroups

are pronormal.

•

Maximal subgroups

are pronormal.

•

Abnormal subgroups

are pronormal.

•

Sylow subgroups

of

finite groups

are pronormal.

•

The

normalizer

of a pronormal subgroup is abnormal.

•

A pronormal subgroup is normal if and only if it is

subnormal

.

Title

pronormal subgroup

Canonical name

PronormalSubgroup

Date of creation

2013-03-22 16:28:13

Last modified on

2013-03-22 16:28:13

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 20E99

Defines

pronormal

Defines

pronormality

Generated on Fri Feb 9 19:28:17 2018 by

LaTeXML
Gromov’s theorem

Gromov’s theorem

Let

G

be a

finitely generated group

.

Then,

G

has

polynomial growth

if and only if it is

virtually nilpotent

.

References

1

M. Gromov,

Groups of polynomial growth and expanding maps

,
Publications Mathématique d’IHÉS 53 (1981), pages 53 to 78.

Title

Gromov’s theorem

Canonical name

GromovsTheorem

Date of creation

2013-03-22 14:36:40

Last modified on

2013-03-22 14:36:40

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 20Exx

Classification

msc 20Fxx

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

LaTeXML
Baumslag-Solitar group

Baumslag-Solitar group

A

Baumslag-Solitar group

is a

group with

presentation

⟨

b

,

t

∣

t

-

1

⁢

b

m

⁢

t

=

b

n

⟩

for some non-zero integers

m

and

n

.

These groups were studied by Baumslag and Solitar

[

1

]

,
who were interested in finding examples
of

finitely generated

non-Hopfian (

http://planetmath.org/HopfianGroup

) groups.
In particular, the group

⟨

b

,

t

∣

t

-

1

⁢

b

2

⁢

t

=

b

3

⟩

is non-Hopfian.

References

1

G. Baumslag, D. Solitar,

Some two-generator one-relator non-Hopfian groups

,
Bull. Amer. Math. Soc., 68 (1962) 199–201.
(This paper is

http://projecteuclid.org/euclid.bams/1183524561

available on-line from Project Euclid.)

2

D. J. Collins,

http://eom.springer.de/b/b130070.htm

Baumslag–Solitar group

,
in the

Online Encyclopaedia of Mathematics

.

Title

Baumslag-Solitar group

Canonical name

BaumslagSolitarGroup

Date of creation

2013-03-22 17:25:12

Last modified on

2013-03-22 17:25:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 20F05

Synonym

Solitar-Baumslag group

Generated on Fri Feb 9 19:28:21 2018 by

LaTeXML
triangle groups

triangle groups

Consider the following

group presentation

:

Δ

(

l

,

m

,

n

)

=

⟨

a

,

b

,

c

:

a

2

,

b

2

,

c

2

,

(

a

b

)

l

,

(

b

c

)

n

,

(

c

a

)

m

⟩

where

l

,

m

,

n

∈

ℕ

.

A

group with

this

presentation

corresponds to a triangle; roughly, the

generators

are

reflections

in its sides and its angles are

π

/

l

,

π

/

m

,

π

/

n

.

Denote by

D

⁢

(

l

,

m

,

n

)

the

subgroup

of index (

http://planetmath.org/Coset

) 2 in

Δ

⁢

(

l

,

m

,

n

)

, corresponding to preservation of of the triangle.

The

D

⁢

(

l

,

m

,

n

)

are defined by the following presentation:

D

(

l

,

m

,

n

)

=

⟨

x

,

y

:

x

l

,

y

m

,

(

x

y

)

n

⟩

Note that

D

⁢

(

l

,

m

,

n

)

≅

D

⁢

(

m

,

l

,

n

)

≅

D

⁢

(

n

,

m

,

l

)

, so

D

⁢

(

l

,

m

,

n

)

is of the

l

,

m

,

n

.

Arising from the geometrical nature of these groups,

1

/

l

+

1

/

m

+

1

/

n

>

1

is called the

spherical case

,

1

/

l

+

1

/

m

+

1

/

n

=

1

is called the

Euclidean case

, and

1

/

l

+

1

/

m

+

1

/

n

<

1

is called the

hyperbolic case

Groups either of the form

Δ

⁢

(

l

,

m

,

n

)

or

D

⁢

(

l

,

m

,

n

)

are referred to as

triangle groups

; groups of the form

D

⁢

(

l

,

m

,

n

)

are sometimes refered to as

von Dyck groups

.

Title

triangle groups

Canonical name

TriangleGroups

Date of creation

2013-03-22 14:25:07

Last modified on

2013-03-22 14:25:07

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

10

Author

rmilson (146)

Entry type

Definition

Classification

msc 20F05

Related topic

ExamplesOfGroups

Defines

von Dyck groups

Generated on Fri Feb 9 19:28:23 2018 by

LaTeXML
Cayley graph of S_3

Cayley graph of

S

3

In this entry, a

Cayley graph

of

S

3

, the group of

permutations

of

{

1

,

2

,

3

}

, will be investigated.

Let

ε

∈

S

3

denote the

identity element

,

τ

∈

S

3

be a

transposition

, and

γ

∈

S

3

be a three-cycle (

http://planetmath.org/SymmetricGroup

). Then

{

τ

,

γ

}

is a generating set of

S

3

and

⟨

τ

,

γ

|

τ

2

=

ε

,

γ

3

=

ε

,

τ

⁢

γ

=

γ

2

⁢

τ

⟩

is a

presentation

of

S

3

. The corresponding Cayley graph

Γ

⁢

(

S

3

,

{

τ

,

γ

}

)

is:

If each of the edges is assigned a length (

http://planetmath.org/BasicLength

) of one, then the Cayley graph is a

hyperbolic metric space

, as it is 2 hyperbolic. Thus,

S

3

is a

hyperbolic group

.

Title

Cayley graph of

S

3

\metatable

Generated on Fri Feb 9 19:28:26 2018 by

LaTeXML
automatic group

automatic group

Let

G

be a

finitely generated group

. Let

A

be a finite

generating set

for

G

under

inverses

.

G

is an

automatic group

if there is a

language

L

⊆

A

*

and a surjective map

f

:

L

→

G

such that

•

L

can be checked by a finite automaton (

http://planetmath.org/DeterministicFiniteAutomaton

)

•

The language of all convolutions of

x

,

y

where

f

⁢

(

x

)

=

f

⁢

(

y

)

can be checked by a

•

For each

a

∈

A

, the language of all convolutions of

x

,

y

where

f

⁢

(

x

)

.

a

=

f

⁢

(

y

)

can be checked by a

(

A

,

L

)

is said to be an

automatic structure

for

G

.

Note that by taking a

finitely generated

semigroup

S

, and some finite generating set

A

, these conditions define an

automatic semigroup

.

Title

automatic group

Canonical name

AutomaticGroup

Date of creation

2013-03-22 14:16:54

Last modified on

2013-03-22 14:16:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20F10

Related topic

AutomaticPresentation

Defines

automatic semigroup

Defines

automatic structure

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

LaTeXML
Tietze transform

Tietze transform

Tietze transforms

are the following four

transformations

whereby one
can transform a

presentation

of a group into another presentation of
the same group:

1.

If a

relation

W

=

V

, where

W

and

V

are some word in the

generators

of the group, can be derived from the

defining relations

of
a group, add

W

=

V

to the list of relations.

2.

If a relation

W

=

V

can be derived from the remaining
generators, remove

W

=

V

fronm the list of relations.

3.

If

W

is a word in the generators and

W

=

x

, then add

x

to
the list of generators and

W

=

x

to the list of relations.

4.

If a relation takes the form

W

=

x

, where

x

is a generator and

W

is a word in generators other than

x

, then remove

W

=

x

from the
list of relations, replace all occurences of

x

in the remaining
relations by

W

and remove

x

from the list of generators.

Note that transforms 1 and 2 are

inverse

to each other and likewise 3
and 4 are inverses. More generally, the term “Tietze transform”
referes to a transform which can be expressed as the compositon of a
finite number of the four transforms listed above. By way of
contrast, the term “

elementary Tietze transformation

” is used
to denote the four transformations given above and the term
“

general Tietze transform

” could be used to indicate a member
of the larger class.

Tieze showed that any two presentations of the same finitely presented
group differ by a general Tietze transform.

Title

Tietze transform

Canonical name

TietzeTransform

Date of creation

2013-03-22 15:42:40

Last modified on

2013-03-22 15:42:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 20F10

Defines

elementary Tietze transformation

Defines

general Tietze transform

Generated on Fri Feb 9 19:28:41 2018 by

LaTeXML
abelianization

abelianization

The

abelianization

of a group

G

is

G

/

[

G

,

G

]

, the

quotient

(

http://planetmath.org/QuotientGroup

) of

G

by its

derived subgroup

.

The abelianization of

G

is the largest

abelian

quotient of

G

, in the sense that if

N

is a

normal subgroup

of

G

then

G

/

N

is abelian if and only if

[

G

,

G

]

⊆

N

.
In particular, every abelian quotient of

G

is a

homomorphic image

of

G

/

[

G

,

G

]

.

If

A

is an

abelian group

and

ϕ

:

G

→

A

is a

homomorphism

(

http://planetmath.org/GroupHomomorphism

),
then there is a unique homomorphism

ψ

:

G

/

[

G

,

G

]

→

A

such that

ψ

∘

π

=

ϕ

, where

π

:

G

→

G

/

[

G

,

G

]

is the

canonical projection

.

Title

abelianization

Canonical name

Abelianization

Date of creation

2013-03-22 14:52:57

Last modified on

2013-03-22 14:52:57

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20F14

Synonym

abelianisation

Related topic

DerivedSubgroup

Generated on Fri Feb 9 19:28:49 2018 by

LaTeXML
derived subgroup

derived subgroup

Let

G

be a group.
For any

a

,

b

∈

G

, the element

a

-

1

⁢

b

-

1

⁢

a

⁢

b

is called the

commutator

of

a

and

b

.

The commutator

a

-

1

⁢

b

-

1

⁢

a

⁢

b

is sometimes written

[

a

,

b

]

.
(Usage varies, however, and some authors instead use

[

a

,

b

]

to represent the commutator

a

⁢

b

⁢

a

-

1

⁢

b

-

1

.)
If

A

and

B

are subsets of

G

, then

[

A

,

B

]

denotes the

subgroup

of

G

generated by

{

[

a

,

b

]

∣

a

∈

A

⁢

and

⁢

b

∈

B

}

.
This notation can be further extended by recursively defining

[

X

1

,

…

,

X

n

+

1

]

=

[

[

X

1

,

…

,

X

n

]

,

X

n

+

1

]

for subsets

X

1

,

…

,

X

n

+

1

of

G

.

The subgroup of

G

generated by all the commutators in

G

(that is, the smallest subgroup of

G

containing all the commutators)
is called the

derived subgroup

,
or the

commutator subgroup

, of

G

.
Using the notation of the previous paragraph, the derived subgroup is denoted by

[

G

,

G

]

.
Alternatively, it is often denoted by

G

′

, or sometimes

G

(

1

)

.

Note that

a

and

b

commute if and only if the commutator of

a

,

b

∈

G

is trivial, i.e.,

a

-

1

⁢

b

-

1

⁢

a

⁢

b

=

1

.

Thus, in a fashion, the derived subgroup measures the degree to which a group fails to be

abelian

.

Proposition 1

The derived subgroup

[

G

,

G

]

is normal (in fact,

fully invariant

) in

G

,
and the

factor group

G

/

[

G

,

G

]

is abelian.
Moreover,

G

is abelian if and only if

[

G

,

G

]

is the trivial subgroup.

The factor group

G

/

[

G

,

G

]

is the largest abelian quotient (

http://planetmath.org/QuotientGroup

) of

G

,
and is called the

abelianization

of

G

.

One can of course form the derived subgroup of the derived subgroup;
this is called the

second derived subgroup

, and denoted by

G

′′

or

G

(

2

)

. Proceeding inductively one defines the

n

th

derived
subgroup

G

(

n

)

as the derived subgroup of

G

(

n

-

1

)

. In this fashion one
obtains a sequence of subgroups, called the

derived series

of

G

:

G

=

G

(

0

)

⊇

G

(

1

)

⊇

G

(

2

)

⊇

⋯

Proposition 2

The group

G

is

solvable

if and only if the derived series
terminates in the

trivial group

{

1

}

after a finite (

http://planetmath.org/Finite

) number of steps.

The derived series can also be continued transfinitely—see the article on the

transfinite derived series

.

Title

derived subgroup

Canonical name

DerivedSubgroup

Date of creation

2013-03-22 12:33:53

Last modified on

2013-03-22 12:33:53

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Definition

Classification

msc 20F14

Classification

msc 20E15

Classification

msc 20A05

Synonym

commutator subgroup

Related topic

JordanHolderDecomposition

Related topic

Solvable

Related topic

TransfiniteDerivedSeries

Related topic

Abelianization

Defines

commutator

Defines

derived series

Defines

second derived subgroup

Generated on Fri Feb 9 19:28:52 2018 by

LaTeXML
metacyclic group

metacyclic group

Definition

A

metacyclic group

is a group

G

that possesses a

normal subgroup

N

such that

N

and

G

/

N

are both cyclic.

Examples

•

All

cyclic groups

, and

direct products

of two cyclic groups.

•

All

dihedral groups

(including the

infinite dihedral group

).

•

All

finite groups

whose

Sylow subgroups

are cyclic (and so, in particular, all finite groups of squarefree (

http://planetmath.org/SquareFreeNumber

) order).

Properties

Subgroups

(

http://planetmath.org/Subgroup

) and

quotients

(

http://planetmath.org/QuotientGroup

) of metacyclic groups are also metacyclic.

Metacyclic groups are obviously

supersolvable

, with

Hirsch length

at most

2

.

Title

metacyclic group

Canonical

name

MetacyclicGroup

Date of creation

2013-03-22 15:36:39

Last modified on

2013-03-22 15:36:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 20F16

Defines

metacyclic

Generated on Fri Feb 9 19:28:54 2018 by

LaTeXML
polycyclic group

polycyclic group

\PMlinkescapephrase

amenable groups

A group

G

is said to be

polycyclic

if it has a

subnormal series

{

1

}

=

G

0

◁

G

1

◁

…

◁

G

n

-

1

◁

G

n

=

G

such that

G

i

+

1

/

G

i

is cyclic for each

i

=

0

,

…

,

n

-

1

.
(Note that this differs from the definition of a

supersolvable group

in that it does not require each

G

i

to be normal in

G

.)
A subnormal series of this form is called a

polycyclic series

.

Polycyclic groups

are obviously

solvable

.
In fact, the polycyclic groups are precisely those

solvable groups

that satisfy the

maximal condition

(that is, those solvable groups all of whose

subgroups

(

http://planetmath.org/Subgroup

) are

finitely generated

).
In particular, a

finite group

is polycyclic if and only if it is solvable.

The

Hirsch length

(or

Hirsch number

, named after

http://www-history.mcs.st-and.ac.uk/history//Biographies/Hirsch.html

Kurt Hirsch)
of a polycyclic group

G

is the number of infinite factors in a polycyclic series of

G

.
This is

independent

of the choice of polycyclic series, as a consequence of the

Schreier Refinement Theorem

.
More generally, the Hirsch length of a

polycyclic-by-finite

group

G

is the Hirsch length of a polycyclic

normal subgroup

of finite index in

G

(all such subgroups having the same Hirsch length).
J. A. Hillman

[

1

]

has further extended the concept of Hirsch length to cover all elementary amenable groups.

References

1

Jonathan A. Hillman,
,
J. Austral. Math. Soc. (Series A) 50 (1991), 160–170.
(This paper can be viewed

http://anziamj.austms.org.au/JAMSA/V50/Part1/Hillman/p0160.html

on the Australian Mathematical Society website.)

Title

polycyclic group

Canonical name

PolycyclicGroup

Date of creation

2013-03-22 14:40:50

Last modified on

2013-03-22 14:40:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 20F16

Related topic

SupersolvableGroup

Defines

polycyclic

Defines

polycyclic series

Defines

Hirsch number

Defines

Hirsch length

Generated on Fri Feb 9 19:28:56 2018 by

LaTeXML
supersolvable group

supersolvable group

A group

G

is

supersolvable

if it has a finite

normal series

G

=

G

0

⊳

G

1

⊳

⋯

⊳

G

n

=

1

with the property that each

factor group

G

i

-

1

/

G

i

is cyclic.

A supersolvable group is

solvable

.

Finitely generated

nilpotent groups

are supersolvable.

Title

supersolvable group

Canonical name

SupersolvableGroup

Date of creation

2013-03-22 13:58:44

Last modified on

2013-03-22 13:58:44

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Definition

Classification

msc 20F16

Classification

msc 20D10

Related topic

PolycyclicGroup

Defines

supersolvable

Generated on Fri Feb 9 19:28:58 2018 by

LaTeXML
enumerating groups

enumerating groups

1

How many finite groups are there?

The current tables list the number of groups up to order 2000 [Besche, Eick, O’Brien] (2000).

The graph is chaotic – both figuratively and mathematically.
Most groups are distributed along the

interval

at values

2

i

⁢

m

where

m

is odd and

i

large, for instance

i

>

5

. Indeed most groups are actually of order

2

10

=

1024

. We see this by connecting the dots of certain families of groups.

Most

integers

are

square-free

, most groups are not [Mays 1980; Miller 1930;
Balas 1966].

An explanation for this distribution is offered by considering

nilpotent groups

.
Nilpotent groups are the product of their

Sylow subgroups

. So enumerating
nilpotent groups asks to enumerating

p

-groups.

2

How many nilpotent groups are there?

Theorem 1

(Pyber, 1993).

If

g

n

⁢

i

⁢

l

⁢

(

N

)

is the number of nilpotent groups of order

≤

N

and

g

⁢

(

N

)

the number
of groups of order

≤

N

then

lim

N

→

∞

⁡

log

⁡

g

n

⁢

i

⁢

l

⁢

(

N

)

log

⁡

g

⁢

(

N

)

=

1

.

The proof bounds the number of

groups with

a given set of Sylow subgroups and involves
the Classification of Finite Simple Groups.

Conjecture 2

(Pyber, 1993).

lim

N

→

∞

⁡

g

n

⁢

i

⁢

l

⁢

(

N

)

g

⁢

(

N

)

=

1

.

If the conjecture is true, then most groups are 2-groups.

3

The Higman and Sims bounds

Theorem 3

(Higman 1960, Sims 1964).

The number of

p

-groups of order

p

n

, denoted,

f

⁢

(

p

n

)

, satisfies

2

27

⁢

n

3

+

C

1

⁢

n

2

≤

log

p

⁡

f

⁢

(

p

n

)

≤

2

27

⁢

n

3

+

C

2

⁢

n

8

/

3

for constants

C

1

and

C

2

.

This result should be compared to the later work of Neretin on enumerating algebras. The

lower bound

is the work of Higman and is achieved by constructing a large family of class 2

p

-groups (called

Φ

-class 2 groups as

Φ

⁢

(

Φ

⁢

(

P

)

)

=

1

where

Φ

is the

Frattini subgroup

of

P

).

The

n

8

/

3

factor has been improved to

o

⁢

(

n

5

/

2

)

by M. Newman and
Seeley. Sims’ suggests that it should be possible to show

log

p

⁡

f

⁢

(

p

,

n

)

∈

2

27

⁢

n

3

+

O

⁢

(

n

2

)

(with a

positive

leading coefficient

) which would prove Pyber’s conjecture [Shalev].

S. R. Blackburn’s work (1992) on the number of class 3

p-groups

provides strong evidence that this claim is true as he demonstrates that class 3 groups also attain this lower bound. Since class 3 groups involve the

Jacobi identity

(

Hall-Witt identity

) it is plausible to expect class c, for c less than some fixed bound, will asymptotically achieve the lower bound as well.

Title

enumerating groups

\metatable

Generated on Fri Feb 9 19:29:02 2018 by

LaTeXML
nilpotent group

nilpotent group

We define the

lower central series

of a group

G

to be the

filtration

of

subgroups

G

=

G

1

⊃

G

2

⊃

⋯

defined inductively by:

G

1

:=

G

,

G

i

:=

[

G

i

-

1

,

G

]

,

i

>

1

,

where

[

G

i

-

1

,

G

]

denotes the subgroup of

G

generated by all

commutators

of the form

h

⁢

k

⁢

h

-

1

⁢

k

-

1

where

h

∈

G

i

-

1

and

k

∈

G

. The group

G

is said to be

nilpotent

if

G

i

=

1

for some

i

.

Nilpotent groups

can also be equivalently defined by means of upper central series. For a group

G

, the

upper central series

of

G

is the filtration of subgroups

C

0

⊂

C

1

⊂

C

2

⊂

⋯

defined by setting

C

0

to be the trivial subgroup of

G

, and inductively taking

C

i

to be the unique subgroup of

G

such that

C

i

/

C

i

-

1

is the center of

G

/

C

i

-

1

, for each

i

>

1

. The group

G

is nilpotent if and only if

G

=

C

i

for some

i

. Moreover, if

G

is nilpotent, then the length of the upper central series (i.e., the smallest

i

for which

G

=

C

i

) equals the length of the lower central series (i.e., the smallest

i

for which

G

i

+

1

=

1

).

The

nilpotency class

or

nilpotent class

of a nilpotent group is the length of the lower central series (equivalently, the length of the upper central series).

Nilpotent groups are related to

nilpotent Lie algebras

in that a Lie group is nilpotent as a group if and only if its corresponding

Lie algebra

is nilpotent. The

analogy

extends to

solvable groups

as well: every nilpotent group is solvable, because the upper central series is a filtration with

abelian

quotients

.

Title

nilpotent group

Canonical name

NilpotentGroup

Date of creation

2013-03-22 12:47:50

Last modified on

2013-03-22 12:47:50

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 20F18

Defines

nilpotent

Defines

upper central series

Defines

lower central series

Defines

nilpotency class

Defines

nilpotent class

Generated on Fri Feb 9 19:29:05 2018 by

LaTeXML
locally nilpotent group

locally nilpotent group

Definition

A

locally nilpotent group

is
a group in which every

finitely generated subgroup

is

nilpotent

.

Examples

All

nilpotent groups

are locally nilpotent,
because

subgroups

of nilpotent groups are nilpotent.

An example of a locally nilpotent group that is not nilpotent
is

Dih

⁡

(

ℤ

⁢

(

2

∞

)

)

, the

generalized dihedral group

formed from the

quasicyclic

2

-group (

http://planetmath.org/PGroup4

)

ℤ

⁢

(

2

∞

)

.

The

Fitting subgroup

of any group is locally nilpotent.

All

N-groups

are locally nilpotent. More generally, all

Gruenberg groups

are locally nilpotent.

Properties

Any subgroup or

quotient

(

http://planetmath.org/QuotientGroup

) of a locally nilpotent group is locally nilpotent.

Restricted direct products

of locally nilpotent groups are locally nilpotent.

For each prime

p

,
the elements of

p

-power order in a locally nilpotent group
form a

fully invariant subgroup

(the

maximal

p

-subgroup (

http://planetmath.org/PGroup4

)).
The elements of finite order in a locally nilpotent group
also form a fully invariant subgroup (the

torsion subgroup

),
which is the restricted direct product of the maximal

p

-subgroups.
(This generalizes the fact that a

finite nilpotent group

is the

direct product

of its

Sylow subgroups

.)

Every group

G

has a unique maximal locally nilpotent

normal subgroup

.
This subgroup is called the

Hirsch-Plotkin radical

,
or

locally nilpotent radical

, and is often denoted

HP

⁡

(

G

)

.
If

G

is finite (or, more generally, satisfies the

maximal condition

),
then the Hirsch-Plotkin radical is the same as the Fitting subgroup,
and is nilpotent.

Title

locally nilpotent group

Canonical name

LocallyNilpotentGroup

Date of creation

2013-03-22 15:40:42

Last modified on

2013-03-22 15:40:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20F19

Related topic

LocallyCalP

Related topic

NilpotentGroup

Related topic

NormalizerCondition

Defines

locally nilpotent

Defines

Hirsch-Plotkin radical

Defines

locally nilpotent radical

Generated on Fri Feb 9 19:29:07 2018 by

LaTeXML