aarjaneiro/planetmath huggingface dataset:
normalizer condition

normalizer condition

A group is said to satisfy the

normalizer condition

if every

proper subgroup

is properly contained in its own

normalizer

.
That is, a group

G

satisfies the normalizer condition if and only if

H

<

N

G

⁢

(

H

)

for all

H

<

G

.
A group that satisfies the normalizer condition is sometimes called an

N-group

.

Every

nilpotent group

is an N-group, and every N-group is

locally nilpotent

.
In particular, a

finitely generated group

is an N-group
if and only if it is

nilpotent

.

A group satisfies the normalizer condition if and only if all its

subgroups

are

ascendant

.

Title

normalizer condition

Canonical name

NormalizerCondition

Date of creation

2013-03-22 16:14:41

Last modified on

2013-03-22 16:14:41

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 20F19

Synonym

normaliser condition

Related topic

LocallyNilpotentGroup

Defines

N-group

Generated on Fri Feb 9 19:29:09 2018 by

LaTeXML
transfinite derived series

transfinite derived series

The

transfinite derived series

of a group is
an

extension

of its

derived series

, defined as follows.
Let

G

be a group and let

G

(

0

)

=

G

.
For each

ordinal

α

let

G

(

α

+

1

)

be the derived subgroup of

G

(

α

)

.
For each

limit ordinal

δ

let

G

(

δ

)

=

⋂

α

∈

δ

G

(

α

)

.

Every member of the transfinite derived series of

G

is a

fully invariant subgroup

of

G

.

The transfinite derived series eventually terminates, that is,
there is some ordinal

α

such that

G

(

α

+

1

)

=

G

(

α

)

.
All remaining terms of the series are then equal to

G

(

α

)

,
which is called the

perfect radical

or

maximum perfect subgroup

of

G

, and is denoted

𝒫

⁢

G

.
As the name suggests,

𝒫

⁢

G

is

perfect

,
and every perfect

subgroup

(

http://planetmath.org/Subgroup

) of

G

is a subgroup of

𝒫

⁢

G

.
A group in which the perfect radical is

trivial

(that is, a group without any non-trivial perfect subgroups)
is called a

hypoabelian group

.
For any group

G

, the

quotient

(

http://planetmath.org/QuotientGroup

)

G

/

𝒫

⁢

G

is hypoabelian, and is sometimes called the

hypoabelianization

of

G

(by

analogy

with the

abelianization

).

A group

G

for which

G

(

n

)

is trivial for some finite

n

is called a

solvable group

.
A group

G

for which

G

(

ω

)

(the

intersection

of the derived series)
is trivial is called a

residually solvable

group.

Free groups

(

http://planetmath.org/FreeGroup

) of rank greater than

1

are examples of residually solvable groups that are not solvable.

Title

transfinite derived series

Canonical

name

TransfiniteDerivedSeries

Date of creation

2013-03-22 14:16:33

Last modified on

2013-03-22 14:16:33

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 20F19

Classification

msc 20F14

Related topic

DerivedSubgroup

Defines

perfect radical

Defines

maximum perfect subgroup

Defines

hypoabelianization

Defines

hypoabelianisation

Generated on Fri Feb 9 19:29:12 2018 by

LaTeXML
virtually cyclic group

virtually cyclic group

A

virtually cyclic group

is
a group that has a

cyclic subgroup

of finite index (

http://

planetmath

.org/Coset

).
Every virtually cyclic group in fact
has a normal cyclic subgroup of finite index
(namely, the core of any cyclic subgroup of finite index),
and virtually cyclic groups are therefore also known as

cyclic-by-finite groups

.

A finite-by-cyclic group
(that is, a group

G

with a finite

normal subgroup

N

such that

G

/

N

is cyclic)
is always virtually cyclic.
To see this, note that a finite-by-cyclic group is either finite,
in which case it is certainly virtually cyclic,
or it is finite-by-

ℤ

,
in which case the

extension

(

http://planetmath.org/GroupExtension

)
splits (

http://planetmath.org/SemidirectProductOfGroups

).

Finite-by-dihedral (

http://planetmath.org/DihedralGroup

) groups
are also virtually cyclic.
In fact, we have the following classification theorem:

[

1

]

[

2

]

Theorem

.

Groups of the following three types are all virtually cyclic.
Moreover, every virtually cyclic group is of exactly one of these three types.

•

finite

•

finite-by-(infinite cyclic)

•

finite-by-(

infinite dihedral

)

As an immediate

corollary

we have the following result:

[

3

]

Corollary

.

Every

torsion-free

virtually cyclic group is either trivial or infinite cyclic.

References

1

Lemma 11.4 (pages 102–103) in:
John Hempel,

3-Manifolds

,

American Mathematical Society

, 2004,
ISBN 0821836951.

2

Page 137 of:
Alejandro Adem, Jesus Gonzalez, Guillermo Pastor (eds.),
Recent developments in algebraic topology — A conference to celebrate
Sam Gitler’s 70th birthday, San Miguel de Allende, Mexico, December 3–6, 2003.

3

Lemma 3.2 (pages 225–226) of:
Dugald Macpherson,

Permutation Groups

Whose

Subgroups

Have Just Finitely Many Orbits

(pages 221–230 in: W. Charles Holland (ed.)

Ordered Groups

and Infinite Permutation Groups

,
Kluwer Academic Publishers, 1996).

Title

virtually cyclic group

Canonical

name

VirtuallyCyclicGroup

Date of creation

2013-03-22 15:47:15

Last modified on

2013-03-22 15:47:15

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 20F19

Classification

msc 20E22

Synonym

cyclic-by-finite group

Related topic

VirtuallyAbelian

Defines

virtually cyclic

Defines

cyclic-by-finite

Defines

finite-by-cyclic group

Defines

finite-by-cyclic

Generated on Fri Feb 9 19:29:14 2018 by

LaTeXML
Hamiltonian group

Hamiltonian group

A

Hamiltonian group

is a

non-abelian group

in which all

subgroups

(

http://planetmath.org/Subgroup

) are normal.

Richard Dedekind investigated finite Hamiltonian groups in 1895, and proved that they all contain a copy of the

quaternion group

Q

8

of order

8

(see the

structure

theorem

below). He named them in honour of William Hamilton, the discoverer of quaternions.

Groups in which all subgroups are normal (that is, groups that are either

abelian

or

Hamiltonian

) are sometimes called

Dedekind groups

, or

quasi-Hamiltonian groups

.

The following structure theorem was proved in its full form by Baer

[

1

]

, but Dedekind already came close to it in his original paper

[

2

]

.

Theorem

.

A group is Hamiltonian if and only if it is

isomorphic

to

Q

8

×

P

for some

periodic

abelian group

P

that has no element of order

4

.

In particular, Hamiltonian groups are always periodic (in fact,

locally finite

),

nilpotent

of class

2

, and

solvable

of length

2

.

From the structure theorem one can also see that the only Hamiltonian

p

-groups (

http://planetmath.org/PGroup4

) are

2

-groups of the form

Q

8

×

B

,
where

B

is an

elementary abelian

2

-group.

References

1

R. Baer,

Situation der Untergruppen und Struktur der Gruppe

,
S. B. Heidelberg. Akad. Wiss. 2 (1933), 12–17.

2

R. Dedekind,

Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind

,
Mathematische Annalen 48 (1897), 548–561. (This paper is

http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002256258

available from GDZ.)

Title

Hamiltonian group

Canonical name

HamiltonianGroup

Date of creation

2013-03-22 15:36:14

Last modified on

2013-03-22 15:36:14

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 20F24

Classification

msc 20F18

Classification

msc 20F50

Synonym

Hamilton group

Defines

Dedekind group

Defines

quasi-Hamiltonian group

Defines

Hamiltonian

Generated on Fri Feb 9 19:29:22 2018 by

LaTeXML
examples of outer automorphism group

examples of outer automorphism group

It is easy to understand that

Out

⁢

ℤ

=

Aut

⁢

ℤ

=

ℤ

/

2

⁢

ℤ

, since

ℤ

is

abelian

and there are no inner-automorphisms, save the trivial one.

Also, it is known that

Out

⁢

S

⁢

L

⁢

(

2

,

ℤ

)

=

ℤ

/

2

⁢

ℤ

Another example is that, at least for orientable surfaces, the extended

mapping class group

(or the zeroth homeotopy group) of a

surface

F

is related to its

fundamental group

via

ℳ

*

⁢

(

F

)

=

Out

⁢

(

π

1

⁢

(

F

)

)

.

References

1. L. K. Hua, I Reiner

Automorphisms

of

unimodular group

, Trans Amer. Math. Soc. 71 (1951), 331-348.

2. H. Zieschang, E. Vogt, H. D. Coldewey,

Surfaces and planar

discontinuous

groups

, L.N.M. 875 (1981) Springer-Verlag.

Title

examples of outer automorphism group

Canonical name

ExamplesOfOuterAutomorphismGroup

Date of creation

2013-03-22 16:30:25

Last modified on

2013-03-22 16:30:25

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

8

Author

juanman (12619)

Entry type

Example

Classification

msc 20F28

Generated on Fri Feb 9 19:29:25 2018 by

LaTeXML
outer automorphism group

outer automorphism group

The

outer automorphism group

of a group is the

quotient

(

http://planetmath.org/QuotientGroup

) of its

automorphism group

by its

inner automorphism group

:

Out

⁢

(

G

)

=

Aut

⁢

(

G

)

/

Inn

⁢

(

G

)

.

There is some variance in terminology about “an

outer automorphism

.” Some authors define an outer automorphism as any

automorphism

ϕ

:

G

→

G

which is not an

inner automorphism

. In this way an outer automorphism still permutes the group

G

. However, an equally common

definition

is to declare an outer automorphism as an element of

Out

⁢

(

G

)

and consequently the

elements

are cosets of

Inn

⁢

(

G

)

⁢

ϕ

, and not a map

ϕ

:

G

→

G

. In this definition it is not generally possible to treat the element as a

permutation

of

G

. In particular, the outer automorphism group of a general group

G

does not act on the group

G

in a natural way. An exception is when

G

is

abelian

so that

Inn

⁢

(

G

)

=

1

; thus, the elements of

Out

⁢

(

G

)

are canonically identified with those of

Aut

⁢

(

G

)

so we can speak of the action by outer automorphisms.

Title

outer automorphism group

Canonical

name

OuterAutomorphismGroup

Date of creation

2013-03-22 14:01:26

Last modified on

2013-03-22 14:01:26

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

13

Author

Thomas Heye (1234)

Entry type

Definition

Classification

msc 20F28

Related topic

InnerAutomorphism

Defines

outer automorphism group

Generated on Fri Feb 9 19:29:28 2018 by

LaTeXML
homeotopy

homeotopy

Let

X

be a topological

Hausdorff space

. Let

Homeo

⁢

(

X

)

be the group of

homeomorphisms

X

→

X

, which can be also turn into a

topological space

by means of the

compact-open topology

. And let

π

k

be the k-th

homotopy group

functor.

Then the

k-th homeotopy

is defined as:

ℋ

k

⁢

(

X

)

=

π

k

⁢

(

Homeo

⁢

(

X

)

)

that is, the group of

homotopy classes

of maps

S

k

→

Homeo

⁢

(

X

)

.
Which is different from

π

k

⁢

(

X

)

, the group of homotopy classes of maps

S

k

→

X

.

One important result for any low dimensional topologist is that for a

surface

F

ℋ

0

⁢

(

F

)

=

Out

⁢

(

π

1

⁢

(

F

)

)

which is the

F

’s extended

mapping class group

.

Reference

G.S. McCarty,

Homeotopy groups

, Trans. A.M.S. 106(1963)293-304.

Title

homeotopy

Canonical name

Homeotopy

Date of creation

2013-03-22 15:41:54

Last modified on

2013-03-22 15:41:54

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

17

Author

juanman (12619)

Entry type

Definition

Classification

msc 20F38

Synonym

mapping class group

Related topic

isotopy

Related topic

group

Related topic

homeomorphism

Related topic

Group

Related topic

Isotopy

Related topic

Homeomorphism

Generated on Fri Feb 9 19:29:34 2018 by

LaTeXML
Chernikov group

Chernikov group

A

Chernikov group

is a group

G

that has a

normal subgroup

N

such that

G

/

N

is finite
and

N

is a

direct product

of finitely many

quasicyclic groups

.

The significance of this somewhat arbitrary-looking definition is that all such groups

satisfy

the

minimal condition

, and for a long time they were the only known

groups with

this property.

Chernikov groups are named after

http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Chernikov.html

Sergei Chernikov,
who proved that every

solvable group

that satisfies the minimal condition
is a Chernikov group.
We can state this result in the form of the following

theorem

.

Theorem

.

The following are

equivalent

for a group

G

:

•

G

is a Chernikov group.

•

G

is

virtually abelian

and satisfies the minimal condition.

•

G

is virtually solvable and satisfies the minimal condition.

Title

Chernikov group

Canonical name

ChernikovGroup

Date of creation

2013-03-22 15:48:21

Last modified on

2013-03-22 15:48:21

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Synonym

Černikov group

Related topic

MinimalCondition

Generated on Fri Feb 9 19:29:36 2018 by

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elementary abelian group

elementary abelian group

An

elementary abelian group

is an

abelian group

in which every

non-trivial element

has the same finite order. It is easy to see that the non-trivial elements must in fact be of prime order, so every elementary abelian group is a

p

-group (

http://planetmath.org/PGroup4

) for some prime

p

.

Elementary abelian

2

-groups are sometimes called

Boolean groups

.
A group in which every non-trivial element has order

2

is necessarily

Boolean

, because abelianness is automatic:

x

⁢

y

=

(

x

⁢

y

)

-

1

=

y

-

1

⁢

x

-

1

=

y

⁢

x

.
There is no analogous result for odd primes, because for every odd prime

p

there is a

non-abelian group

of order

p

3

and

exponent

p

.

Let

p

be a

prime number

.
Any elementary abelian

p

-group can be considered as a

vector space

over the field of order

p

, and is therefore

isomorphic

to the

direct sum

of

κ

copies of the

cyclic group

of order

p

, for some

cardinal number

κ

.

Conversely

, any such direct sum is obviously an elementary abelian

p

-group.
So, in particular, for every

infinite

cardinal

κ

there is, up to

isomorphism

, exactly one elementary abelian

p

-group of order

κ

.

Title

elementary abelian group

Canonical name

ElementaryAbelianGroup

Date of creation

2013-03-22 14:53:11

Last modified on

2013-03-22 14:53:11

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Classification

msc 20K10

Defines

elementary abelian

Defines

Boolean group

Generated on Fri Feb 9 19:29:38 2018 by

LaTeXML
locally finite group

locally finite group

A group

G

is

locally finite

if any

finitely generated subgroup

of

G

is finite.

A locally finite group is a

torsion group

. The

converse

, also known as the

Burnside Problem

, is not true. Burnside, however, did show that if a

matrix group

is torsion, then it is locally finite.

(Kaplansky) If

G

is a group such that for a

normal subgroup

N

of

G

,

N

and

G

/

N

are locally finite, then

G

is locally finite.

A

solvable

torsion group is locally finite. To see this, let

G

=

G

0

⊃

G

1

⊃

⋯

⊃

G

n

=

(

1

)

be a

composition series

for

G

. We have that each

G

i

+

1

is normal in

G

i

and the

factor group

G

i

/

G

i

+

1

is

abelian

. Because

G

is a torsion group, so is the factor group

G

i

/

G

i

+

1

. Clearly an abelian torsion group is locally finite. By applying the fact in the previous paragraph for each step in the composition series, we see that

G

must be locally finite.

References

1

E. S. Gold and I. R. Shafarevitch,

On towers of class fields

, Izv. Akad. Nauk SSR, 28 (1964) 261-272.

2

I. N. Herstein,

Noncommutative Rings

, The Carus Mathematical Monographs, Number 15, (1968).

3

I. Kaplansky,

Notes on Ring Theory

, University of Chicago, Math Lecture Notes, (1965).

4

C. Procesi,

On the Burnside problem

, Journal of Algebra, 4 (1966) 421-426.

Title

locally finite group

Canonical name

LocallyFiniteGroup

Date of creation

2013-03-22 14:18:44

Last modified on

2013-03-22 14:18:44

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20F50

Related topic

LocallyCalP

Related topic

PeriodicGroup

Related topic

ProofThatLocalFinitenessIsClosedUnderExtension

Defines

locally finite

Generated on Fri Feb 9 19:29:44 2018 by

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periodic group

periodic group

A group

G

is said to be

periodic

(or

torsion

)
if every element of

G

is of finite order.

All

finite groups

are periodic.
More generally, all

locally finite groups

are periodic.
Examples of periodic groups that are not

locally finite

include

Tarski groups

,
and

Burnside groups

B

⁢

(

m

,

n

)

of odd

exponent

n

≥

665

on

m

>

1

generators

.

Some easy results on periodic groups:

Theorem 1

.

Every

subgroup

(

http://planetmath.org/Subgroup

) of a periodic group is periodic.

Theorem 2

.

Every

quotient

(

http://planetmath.org/QuotientGroup

) of a periodic group is periodic.

Theorem 3

.

Every

extension

(

http://planetmath.org/GroupExtension

) of a periodic group by a periodic group is periodic.

Theorem 4

.

Every

restricted direct product

of periodic groups is periodic.

Note that (unrestricted)

direct products

of periodic groups are not necessarily periodic. For example, the direct product of all finite

cyclic groups

ℤ

/

n

⁢

ℤ

is not periodic, as the

element

that is

1

in every coordinate has infinite order.

Some further results on periodic groups:

Theorem 5

.

Every

solvable

periodic group is locally finite.

Theorem 6

.

Every periodic

abelian group

is the

direct sum

of its

maximal

p

-groups (

http://planetmath.org/PGroup4

) over all primes

p

.

Title

periodic group

Canonical name

PeriodicGroup

Date of creation

2013-03-22 15:35:50

Last modified on

2013-03-22 15:35:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Synonym

torsion group

Related topic

LocallyFiniteGroup

Related topic

Torsion3

Defines

periodic

Defines

torsion

Generated on Fri Feb 9 19:29:46 2018 by

LaTeXML
p-group

p

-group

Primary groups

Let

p

be a

prime number

.
A

p

-group

(or

p

-

primary group

) is a group in which the order of every element is a power of

p

.
A group that is a

p

-group for some prime

p

is also called a

primary group

.

Using Lagrange’s Theorem and Cauchy’s Theorem one may show that a

finite group

G

is a

p

-group if and only if

|

G

|

is a power of

p

.

Primary subgroups

A

p

-

subgroup

(or

p

-primary subgroup

) of a group

G

is a subgroup (

http://planetmath.org/Subgroup

)

H

of

G

such that

H

is also a

p

-group.
A group that is a

p

-subgroup for some prime

p

is also called a

primary subgroup

.

It follows from Zorn’s Lemma that every group has a

maximal

p

-subgroup, for every prime

p

. The maximal

p

-subgroup need not be unique (though for

abelian groups

it is always unique, and is called the

p

-

primary component

of the abelian group). A maximal

p

-subgroup may, of course, be trivial. Non-trivial maximal

p

-subgroups of finite groups are called Sylow

p

-subgroups (

http://planetmath.org/SylowPSubgroups

).

Title

p

-group

Canonical name

Pgroup

Date of creation

2013-03-22 14:53:08

Last modified on

2013-03-22 14:53:08

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Synonym

p-group

Synonym

p-primary group

Synonym

primary group

Related topic

PGroup

Related topic

PExtension

Related topic

ProPGroup

Related topic

QuasicyclicGroup

Related topic

Subgroup

Defines

p-subgroup

Defines

primary component

Defines

p-primary

Defines

p-primary subgroup

Defines

primary subgroup

Generated on Fri Feb 9 19:29:48 2018 by

LaTeXML
quasicyclic group

quasicyclic group

Let

p

be a

prime number

.
The

p

-

quasicyclic group

(or

Prüfer

p

-group

, or

p

∞

group

) is the

p

-

primary component

of

ℚ

/

ℤ

,
that is, the unique

maximal

p

-

subgroup

(

http://planetmath.org/PGroup4

) of

ℚ

/

ℤ

.
Any group (

http://planetmath.org/Group

)

isomorphic

to this will also be called a

p

-quasicyclic group.

The

p

-quasicyclic group will be denoted by

ℤ

⁢

(

p

∞

)

.
Other notations in use include

ℤ

⁢

[

p

∞

]

,

ℤ

/

p

∞

⁢

ℤ

,

ℤ

p

∞

and

C

p

∞

.

ℤ

⁢

(

p

∞

)

may also be defined in a number of other (

equivalent

) ways
(again, up to

isomorphism

):

•

ℤ

⁢

(

p

∞

)

is
the group of all

p

n

-th complex roots of

1

, for

n

∈

ℕ

.

•

ℤ

⁢

(

p

∞

)

is
the injective hull of

ℤ

/

p

⁢

ℤ

(viewing

abelian groups

as

ℤ

-modules (

http://planetmath.org/Module

)).

•

ℤ

⁢

(

p

∞

)

is the

direct limit

of the groups

ℤ

/

p

n

⁢

ℤ

.

A

quasicyclic group

(or

Prüfer group

) is
a group that is

p

-quasicyclic for some prime

p

.

The subgroup (

http://planetmath.org/Subgroup

)

structure

of

ℤ

⁢

(

p

∞

)

is particularly simple:
all

proper subgroups

are finite and cyclic,
and there is exactly one of order

p

n

for each non-negative integer

n

.
In particular,
this means that the subgroups are

linearly ordered

by

inclusion

,
and all subgroups are

fully invariant

.
The quasicyclic groups are
the only

infinite groups

with a linearly ordered

subgroup lattice

.
They are also
the only

infinite

solvable groups

whose proper subgroups are all finite.

Quasicyclic groups are

locally cyclic

, divisible (

http://planetmath.org/DivisibleGroup

) and

co-Hopfian

.

Every infinite locally cyclic

p

-group is isomorphic to

ℤ

⁢

(

p

∞

)

.

Title

quasicyclic group

Canonical name

QuasicyclicGroup

Date of creation

2013-03-22 15:35:22

Last modified on

2013-03-22 15:35:22

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

19

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Classification

msc 20K10

Synonym

quasi-cyclic group

Synonym

Prüfer group

Defines

quasicyclic

Defines

quasi-cyclic

Defines

Prüfer

p-group

Generated on Fri Feb 9 19:29:50 2018 by

LaTeXML
Tarski group

Tarski group

A

Tarski group

is an

infinite group

G

such that every non-trivial

proper subgroup

of

G

is of prime order.

Tarski groups are also called

Tarski monsters

,
especially in the case when
all the proper non-trivial

subgroups

are of the same order
(that is, when the Tarski group is
a

p

-group (

http://planetmath.org/PGroup4

) for some prime

p

).

Alexander Ol’shanskii

[

1

,

2

]

showed that Tarski groups exist,
and that there is a Tarski

p

-group for every prime

p

>

10

75

.

From the definition one can easily deduce
a number of properties of Tarski groups.
For example,
every Tarski group is a

simple group

,
it satisfies the

minimal condition

and the

maximal condition

,
it can be generated by just two elements,
it is

periodic

but not

locally finite

,
and its

subgroup lattice

(

http://planetmath.org/LatticeOfSubgroups

) is modular (

http://planetmath.org/ModularLattice

).

References

1

A. Yu. Olshanskii,

An infinite group with subgroups of prime orders

,
Math. USSR Izv. 16 (1981), 279–289;

translation

of Izvestia Akad. Nauk SSSR Ser. Matem. 44 (1980), 309–321.

2

A. Yu. Olshanskii,

Groups of bounded period with subgroups of prime order

,

Algebra

and Logic 21 (1983), 369–418;
translation of Algebra i Logika 21 (1982), 553–618.

Title

Tarski group

Canonical name

TarskiGroup

Date of creation

2013-03-22 15:46:00

Last modified on

2013-03-22 15:46:00

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 20F50

Defines

Tarski monster

Generated on Fri Feb 9 19:29:53 2018 by

LaTeXML