aarjaneiro/planetmath huggingface dataset:
residually 𝔛

residually

𝔛

Let

𝔛

be a

property

of groups,
assumed to be an

isomorphic

invariant

(that is, if a group

G

has property

𝔛

,
then every group isomorphic to

G

also has property

𝔛

).
We shall sometimes refer to

groups with

property

𝔛

as

𝔛

-groups.

A group

G

is said to be

residually

X

if for every

x

∈

G

\

{

1

}

there is a

normal subgroup

N

of

G

such that

x

∉

N

and

G

/

N

has property

𝔛

.
Equivalently,

G

is residually

𝔛

if and only if

⋂

N

⁢

⊴

𝔛

⁢

G

N

=

{

1

}

,

where

N

⁢

⊴

𝔛

⁢

G

means that

N

is normal in

G

and

G

/

N

has property

𝔛

.

It can be shown that a group is residually

𝔛

if and only if it is isomorphic to a

subdirect product

of

𝔛

-groups.
If

𝔛

is a hereditary property
(that is, every

subgroup

(

http://planetmath.org/Subgroup

) of an

𝔛

-group is an

𝔛

-group),
then a group is residually

𝔛

if and only if
it can be embedded in an

unrestricted direct product

of

𝔛

-groups.

It can be shown that a group

G

is

residually solvable

if and only if
the

intersection

of the

derived series

of

G

is

trivial

(see

transfinite derived series

).
Similarly, a group

G

is residually nilpotent if and only if
the intersection of the

lower central series

of

G

is trivial.

Title

residually

𝔛

Canonical

name

ResiduallymathfrakX

Date of creation

2013-03-22 14:53:22

Last modified on

2013-03-22 14:53:22

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 20E26

Related topic

AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite

Defines

residually finite

Defines

residually nilpotent

Defines

residually solvable

Defines

residually soluble

Generated on Fri Feb 9 19:27:33 2018 by

LaTeXML
subdirect product of groups

subdirect product of groups

Let

(

G

i

)

i

∈

I

be a family of groups.
A

subgroup

(

http://planetmath.org/Subgroup

)

H

of the

direct product

(

http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups

)

∏

i

∈

I

G

i

is said to be a

subdirect product

(or

subcartesian product

) of

(

G

i

)

i

∈

I

if

π

i

⁢

(

H

)

=

G

i

for each

i

∈

I

,
where

π

i

:

∏

i

∈

I

G

i

→

G

i

is the

i

-th

projection map

.

Title

subdirect product of groups

Canonical

name

SubdirectProductOfGroups

Date of creation

2013-03-22 14:53:25

Last modified on

2013-03-22 14:53:25

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20E26

Synonym

subdirect product

Synonym

subcartesian product

Synonym

subcartesian product of groups

Related topic

ResiduallyCalP

Related topic

DirectProductAndRestrictedDirectProductOfGroups

Generated on Fri Feb 9 19:27:34 2018 by

LaTeXML
existence of maximal subgroups

existence of maximal subgroups

Because every

finite group

is a

finite set

, every chain of

proper subgroups

of a finite group has a

maximal element

and thus every finite group has
a

maximal subgroup

. The same applies to maximal normal subgroups.

However, there are

infinite groups

, even

abelian

, with no maximal subgroups and
no maximal normal subgroups. The Prüfer group

ℤ

p

∞

=

lim

⟵

⁡

ℤ

p

i

(for any prime

p

) is an example of an

abelian group

with no maximal subgroups.
As the group is abelian all

subgroups

are normal so it also has no maximal
normal subgroups. Such groups fail to fit the

hypothesis

of the Jordan-Hölder decomposition

theorem

as they do not have the

ascending chain condition

and so we cannot assign a

composition series

to such groups.

This stands in contrast to the

category

of

unital rings

where if one assumes Zorn’s lemma (

axiom of choice

) then one may prove every unital ring
has a

maximal ideal

.

Title

existence of maximal subgroups

Canonical name

ExistenceOfMaximalSubgroups

Date of creation

2013-03-22 16:24:54

Last modified on

2013-03-22 16:24:54

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Example

Classification

msc 20E28

Related topic

PropertyOfInfiniteSimpleGroup

Related topic

JordanHolderDecomposition

Related topic

EveryRingHasAMaximalIdeal

Generated on Fri Feb 9 19:27:36 2018 by

LaTeXML
maximal subgroup

maximal subgroup

Let

G

be a group.

A

subgroup

H

of

G

is said to be a

maximal subgroup

of

G

if

H

≠

G

and there is no subgroup

K

of

G

such that

H

<

K

<

G

.
Note that a maximal subgroup of

G

is not maximal (

http://planetmath.org/MaximalElement

) among all subgroups of

G

,
but only among all

proper subgroups

of

G

.
For this reason, maximal subgroups are sometimes called

maximal proper subgroups

.

Similarly, a

normal subgroup

N

of

G

is said to be a

maximal normal subgroup

(or

maximal proper normal subgroup

) of

G

if

N

≠

G

and there is no normal subgroup

K

of

G

such that

N

<

K

<

G

.
We have the following theorem:

Theorem

.

A normal subgroup

N

of a group

G

is a maximal normal subgroup
if and only if the

quotient

(

http://planetmath.org/QuotientGroup

)

G

/

N

is simple (

http://planetmath.org/Simple

).

Title

maximal subgroup

Canonical name

MaximalSubgroup

Date of creation

2013-03-22 12:23:46

Last modified on

2013-03-22 12:23:46

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

15

Author

yark (2760)

Entry type

Definition

Classification

msc 20E28

Synonym

maximal proper subgroup

Related topic

MaximalElement

Defines

maximal

Defines

maximal normal subgroup

Defines

maximal proper normal subgroup

Defines

simplicity of quotient group

Generated on Fri Feb 9 19:27:38 2018 by

LaTeXML
example of infinite simple group

example of infinite simple group

This fact that finite

alternating groups

are simple can be extended to a result about an

infinite group

. Let

G

be the

subgroup

of the group of

permutations

on a

countably infinite

set

M

(which we may take to be the set of natural numbers for concreteness) which is

generated by

cycles of length

3

. Note that any since every element of this group is a

product

of a finite number of cycles, the permutations of

G

are such that only a finite number of elements of our set are not mapped to themselves by a given permutation.

We will now show that

G

is simple. Suppose that

π

is an element of

G

other than the

identity

. Let

m

be the set of all

x

such that

π

⁢

(

x

)

≠

x

. By our previous comment,

m

is finite. Consider the

restriction

π

m

of

π

to

m

. By the

theorem

of the parent entry (

http://planetmath.org/SimplicityOfA_n

), the subgroup of

A

m

generated by the

conjugates

of

π

m

is the whole of

A

m

. In particular, this means that there exists a cycle of order

3

in

A

m

which can be expressed as a product of

π

m

and its conjugates. Hence the subgroup of

G

generated by conjugates of

π

contains a cycle of length three as well. However, every cycle of order

3

is conjugate to every other cycle of order

3

so, in fact, the subgroup of

G

generated by the conjugates of

π

is the whole of

G

. Hence, the only

normal subgroups

of

G

are the group consisting of solely the

identity element

and the whole of

G

, so

G

is a

simple group

.

Title

example of

infinite

simple group

Canonical name

ExampleOfInfiniteSimpleGroup

Date of creation

2013-03-22 16:53:31

Last modified on

2013-03-22 16:53:31

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Example

Classification

msc 20E32

Classification

msc 20D06

Generated on Fri Feb 9 19:27:40 2018 by

LaTeXML
homomorphisms of simple groups

homomorphisms of simple groups

If a group

G

is simple, and

H

is an arbitrary group then any

homomorphism

of

G

to

H

must either map all elements of

G

to the

identity

of

H

or be

one-to-one

.

The kernel of a homomorphism must be a

normal subgroup

. Since

G

is
simple, there are only two possibilities: either the kernel is all of

G

of it consists of the identity. In the former case, the
homomorphism will map all elements of

G

to the identity. In the
latter case, we note that a

group homomorphism

is injective iff the kernel
is

trivial

.

This is important in the

context

of

representation theory

. In that
case,

H

is a linear group and this result may be restated as saying
that

representations

of a

simple group

are either trivial or

faithful

.

Title

homomorphisms of simple groups

Canonical name

HomomorphismsOfSimpleGroups

Date of creation

2013-03-22 15:41:59

Last modified on

2013-03-22 15:41:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 20E32

Generated on Fri Feb 9 19:27:46 2018 by

LaTeXML
Jordan-Hölder decomposition theorem

Jordan-Hölder decomposition theorem

Every

finite group

G

has a

filtration

G

⊃

G

0

⊃

⋯

⊃

G

n

=

{

1

}

,

where each

G

i

+

1

is normal in

G

i

and each

quotient group

G

i

/

G

i

+

1

is a

simple group

. Any two such decompositions of

G

have the same

multiset

of simple groups

G

i

/

G

i

+

1

up to

ordering

.

A filtration of

G

satisfying the properties above is called a

Jordan–Hölder decomposition

of

G

.

Title

Jordan-Hölder decomposition

theorem

Canonical name

JordanHolderDecompositionTheorem

Date of creation

2013-03-22 12:08:44

Last modified on

2013-03-22 12:08:44

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Theorem

Classification

msc 20E32

Related topic

SubnormalSeries

Defines

Jordan-Hölder decomposition

Generated on Fri Feb 9 19:27:48 2018 by

LaTeXML
property of infinite simple group

property of infinite simple group

Although much recent work has been done to classify the finite simple groups,

infinite

simple groups

have properties which make the study more difficult. Among them is the following basic result.

Theorem 1

.

If a group is infinite and simple then it has no

proper subgroups

of finite index.

Proof.

Let

G

be an infinite simple group and

H

≤

G

. Then

G

acts on the cosets of

H

and this induces a

homomorphism

form

G

to

S

n

where

n

=

[

G

:

H

]

. If

H

has finite index in

G

then

n

is finite so

G

maps homomorphically into the

finite group

S

n

. Thus the kernel of the homomorphism is non-trivial. As

G

is simple, the kernel is

G

. As

H

contains the kernel,

H

=

G

.
∎

This means that infinite simple groups do not act on

finite sets

so we cannot invoke clever

arguments

about the

configuration

of numbers. However linear

representations

may still apply. For example,

P

⁢

S

⁢

L

⁢

(

2

,

k

)

for an infinite field

k

is simple, infinite, and can be represented in

S

⁢

L

⁢

(

3

,

k

)

through the exponential map of a Chevalley basis of the Lie algebra

𝔰

⁢

𝔩

2

⁢

(

k

)

.

Title

property of infinite simple group

Canonical name

PropertyOfInfiniteSimpleGroup

Date of creation

2013-03-22 16:08:21

Last modified on

2013-03-22 16:08:21

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Result

Classification

msc 20E32

Related topic

ExistenceOfMaximalSubgroups

Generated on Fri Feb 9 19:27:50 2018 by

LaTeXML
simple group

simple group

A non-trivial group

G

is said to be

simple

if the only

normal subgroups

of

G

are

{

1

}

and

G

itself.

Equivalently, a

simple group

is a group in which the

trivial subgroup

is a

maximal normal subgroup

.

Title

simple group

Canonical name

SimpleGroup

Date of creation

2013-03-22 12:23:35

Last modified on

2013-03-22 12:23:35

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 20E32

Related topic

Group

Related topic

NormalSubgroup

Defines

simple

Generated on Fri Feb 9 19:27:52 2018 by

LaTeXML
abelian groups of order 120

abelian groups of order

120

Here we present an application of the

fundamental theorem of
finitely generated abelian groups

.

Example (

Abelian groups

of order

120

)

:

Let

G

be an abelian group of order

n

=

120

. Since the group is
finite it is obviously

finitely generated

, so we can apply the
theorem. There exist

n

1

,

n

2

,

…

,

n

s

with

G

≅

ℤ

/

n

1

⁢

ℤ

⊕

ℤ

/

n

2

⁢

ℤ

⊕

…

⊕

ℤ

/

n

s

⁢

ℤ

∀

i

,

n

i

≥

2

;

n

i

+

1

∣

n

i

for

1

≤

i

≤

s

-

1

Notice that in the case of a

finite group

,

r

,
as in the statement of the theorem, must be equal to

0

. We have

n

=

120

=

2

3

⋅

3

⋅

5

=

∏

i

=

1

s

n

i

=

n

1

⋅

n

2

⋅

…

⋅

n

s

and by the

divisibility

properties of

n

i

we must have that
every

prime divisor

of

n

must divide

n

1

. Thus the
possibilities for

n

1

are the following

2

⋅

3

⋅

5

,

2

2

⋅

3

⋅

5

,

2

3

⋅

3

⋅

5

If

n

1

=

2

3

⋅

3

⋅

5

=

120

then

s

=

1

. In the case that

n

1

=

2

2

⋅

3

⋅

5

then

n

2

=

2

and

s

=

2

. It remains to
analyze the case

n

1

=

2

⋅

3

⋅

5

. Now the only possibility for

n

2

is

2

and

n

3

=

2

as well.

Hence if

G

is an abelian group of order

120

it must be (

up to

isomorphism

) one of the following:

ℤ

/

120

⁢

ℤ

,

ℤ

/

60

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

,

ℤ

/

30

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

Also notice
that they are all non-isomorphic. This is because

ℤ

/

(

n

⋅

m

)

⁢

ℤ

≅

ℤ

/

n

⁢

ℤ

⊕

ℤ

/

m

⁢

ℤ

⇔

gcd

⁡

(

n

,

m

)

=

1

which is due to the

Chinese Remainder theorem

.

Title

abelian groups of order

120

Canonical name

AbelianGroupsOfOrder120

Date of creation

2013-03-22 13:54:17

Last modified on

2013-03-22 13:54:17

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Example

Classification

msc 20E34

Related topic

FundamentalTheoremOfFinitelyGeneratedAbelianGroups

Related topic

AbelianGroup2

Generated on Fri Feb 9 19:27:55 2018 by

LaTeXML
fundamental theorem of finitely generated abelian groups

fundamental theorem of finitely generated abelian groups

Theorem 1

(Fundamental Theorem of Finitely Generated Abelian
Groups).

Let

G

be a

finitely generated

abelian group

. Then there is a
unique expression of the form:

G

≅

ℤ

r

⊕

ℤ

/

n

1

⁢

ℤ

⊕

ℤ

/

n

2

⁢

ℤ

⊕

…

⊕

ℤ

/

n

s

⁢

ℤ

for some integers

r

,

n

i

satisfying:

r

≥

0

;

∀

i

,

n

i

≥

2

;

n

i

+

1

∣

n

i

for

1

≤

i

≤

s

-

1

.

Title

fundamental theorem of finitely generated abelian groups

Canonical name

FundamentalTheoremOfFinitelyGeneratedAbelianGroups

Date of creation

2013-03-22 13:54:12

Last modified on

2013-03-22 13:54:12

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20E34

Synonym

classification of finitely generated abelian groups

Related topic

AbelianGroupsOfOrder120

Related topic

FinitelyGenerated

Related topic

AbelianGroup2

Defines

fundamental theorem of finitely generated abelian groups

Generated on Fri Feb 9 19:28:00 2018 by

LaTeXML
group socle

group socle

The

socle

of a group is the

subgroup

generated by all minimal normal subgroups.
Because the

product

of

normal subgroups

is a subgroup, it follows we can remove the word “generated” and replace it by “product.” So the socle of a group is now the product of its minimal normal subgroups. This description can be further refined with a few observations.

Proposition 1

.

If

M

and

N

are minimal normal subgroups then

M

and

N

centralize
each other.

Proof.

Given two distinct minimal normal subgroup

M

and

N

,

[

M

,

N

]

is contained in

N

and

M

as both are normal. Thus

[

M

,

N

]

≤

M

∩

N

. But

M

and

N

are distinct minimal normal subgroups and

M

∩

N

is normal so

M

∩

N

=

1

thus

[

M

,

N

]

=

1

.
∎

Proposition 2

.

The socle of a

finite group

is a

direct product

of minimal normal subgroups.

Proof.

Let

S

be the socle of

G

. We already know

S

is the product of its minimal normal subgroups, so let us assume

S

=

N

1

⁢

⋯

⁢

N

k

where each

N

i

is a distinct minimal normal subgroup of

G

. Thus

N

1

∩

N

2

=

1

and

N

1

⁢

N

2

clearly contains

N

1

and

N

2

. Now suppose we extend this to a
subsquence

N

i

1

=

N

1

,

N

i

2

=

N

2

,

N

i

3

,

…

,

N

i

j

where

N

i

k

∩

(

N

i

1

⁢

⋯

⁢

N

i

k

-

1

)

=

1

for

1

≤

k

<

j

and

N

i

≤

N

i

1

⁢

⋯

⁢

N

i

j

for all

1

≤

i

≤

i

j

.
Then consider

N

i

j

+

1

.

As

N

i

j

+

1

is a minimal normal subgroup and

N

i

1

⁢

⋯

⁢

N

i

j

is a
normal subgroup,

N

i

j

+

1

is either contained in

N

i

1

⁢

⋯

⁢

N

i

j

or intersects trivially. If

N

i

j

+

1

is contained in

N

i

1

⁢

⋯

⁢

N

i

j

then skip to the next

N

i

, otherwise set it to be

N

i

j

+

1

.
The result is a squence

N

i

1

,

…

,

N

i

j

of minimal normal subgroups
where

S

=

N

i

1

⁢

⋯

⁢

N

i

s

and

N

i

j

∩

(

N

i

1

⁢

⋯

⁢

N

i

j

-

1

)

=

1

,

1

≤

j

≤

s

.

As we have already seen distinct minimal normal subgroups centralize each other
we conclude that

S

=

N

i

1

×

⋯

×

N

i

s

.
∎

Proposition 3

.

A minimal normal subgroup is

characteristically simple

, so if it is finite then it is a product of

isomorphic

simple groups

.

Proof.

If

M

is a minimal normal subgroup of

G

and

1

<

C

<

M

is

characteristic

in

M

, then

C

is normal in

G

which contradicts the minimality of

M

. Thus

M

is characteristically simple.
∎

Corollary 4

.

The socle of a finite group is a direct product of simple groups.

Proof.

As each

N

i

j

is characteristically simple each

N

i

j

is a direct product of isomorphic simple groups, thus

S

is a direct product simple groups.
∎

Title

group socle

Canonical name

GroupSocle

Date of creation

2013-03-22 15:55:12

Last modified on

2013-03-22 15:55:12

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

14

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 20E34

Synonym

socle

Defines

socle

Generated on Fri Feb 9 19:28:03 2018 by

LaTeXML
complete group

complete group

A

complete group

is a group

G

that is

1.

centerless (center

Z

⁢

(

G

)

of

G

is the

trivial group

), and

2.

any of its

automorphism

g

:

G

→

G

is an

inner automorphism

.

If a group

G

is

complete

, then its group of automorphisms,

Aut

⁡

(

G

)

, is

isomorphic

to

G

. Here’s a quick
proof. Define

ϕ

:

G

→

Aut

⁡

(

G

)

by

ϕ

⁢

(

g

)

=

g

#

, where

g

#

⁢

(

x

)

=

g

⁢

x

⁢

g

-

1

. For

g

,

h

∈

G

,

(

g

⁢

h

)

#

⁢

(

x

)

=

(

g

⁢

h

)

⁢

x

⁢

(

g

⁢

h

)

-

1

=

g

⁢

(

h

⁢

x

⁢

h

-

1

)

⁢

g

-

1

=

(

g

#

⁢

h

#

)

⁢

(

x

)

,
so

ϕ

is a

homomorphism

. It is onto because every

α

∈

Aut

⁡

(

G

)

is inner, (=

g

#

for some

g

∈

G

). Finally,
if

g

#

⁢

(

x

)

=

h

#

⁢

(

x

)

, then

g

⁢

x

⁢

g

-

1

=

h

⁢

x

⁢

h

-

1

, which means

(

h

-

1

⁢

g

)

⁢

x

=

x

⁢

(

h

-

1

⁢

g

)

, for all

x

∈

G

. This implies that

h

-

1

⁢

g

∈

Z

⁢

(

G

)

=

⟨

e

⟩

, or

h

=

g

.

ϕ

is

one-to-one

.

It can be shown that all

symmetric groups

on

n

letters are
complete groups, except when

n

=

2

and

6

.

References

1

J. Rotman,

The Theory of Groups, An

Introduction

,
Allyn and Bacon, Boston (1965).

Title

complete group

Canonical

name

CompleteGroup

Date of creation

2013-03-22 15:21:46

Last modified on

2013-03-22 15:21:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20E36

Classification

msc 20F28

Generated on Fri Feb 9 19:28:05 2018 by

LaTeXML
class equation

class equation

The

conjugacy classes

of a group form a

partition

of its elements.
In a

finite group

, this means that the order of the group is
the sum of the number of elements of the distinct conjugacy classes.
For an element

g

of group

G

,
we denote the

centralizer

in

G

of

g

by

C

G

⁢

(

g

)

.
The number of elements in the conjugacy class of

g

is

[

G

:

C

G

⁢

(

g

)

]

,
the index of

C

G

⁢

(

g

)

in

G

.
For an element

g

of the center

Z

⁢

(

G

)

of

G

,
the conjugacy class of

g

consists of the singleton

{

g

}

.
Putting this together gives us the

class equation

|

G

|

=

|

Z

(

G

)

|

+

∑

i

=

1

m

[

G

:

C

G

(

x

i

)

]

where the

x

i

are elements of
the distinct conjugacy classes contained in

G

∖

Z

⁢

(

G

)

.

Title

class equation

Canonical name

ClassEquation

Date of creation

2013-03-22 13:10:41

Last modified on

2013-03-22 13:10:41

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 20E45

Synonym

conjugacy class formula

Related topic

ConjugacyClass

Generated on Fri Feb 9 19:28:07 2018 by

LaTeXML
conjugacy class

conjugacy class

Let

G

a group, and consider its

operation

(action) on itself give by

conjugation

, that is, the

mapping

(

g

,

x

)

↦

g

⁢

x

⁢

g

-

1

Since conjugation is an

equivalence relation

, we obtain a

partition

of

G

into

equivalence classes

, called

conjugacy classes

. So, the conjugacy class of

X

(represented

C

x

or

C

⁢

(

x

)

is given by

C

x

=

{

y

∈

X

:

y

=

g

⁢

x

⁢

g

-

1

⁢

for some

⁢

g

∈

G

}

Title

conjugacy class

Canonical name

ConjugacyClass1

Date of creation

2013-03-22 14:01:39

Last modified on

2013-03-22 14:01:39

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 20E45

Generated on Fri Feb 9 19:28:10 2018 by

LaTeXML
abnormal subgroup

abnormal subgroup

A

subgroup

H

of a group

G

is called an

abnormal subgroup

if

x

∈

⟨

H

,

x

⁢

H

⁢

x

-

1

⟩

for all

x

∈

G

.

Some facts about abnormal subgroups:

•

Abnormal subgroups are

pronormal

and

self-normalizing

.

•

Non-normal (

http://planetmath.org/NormalSubgroup

)

maximal subgroups

are abnormal.

•

The

normalizer

of a pronormal subgroup is abnormal.

Title

abnormal subgroup

Canonical name

AbnormalSubgroup

Date of creation

2013-03-22 16:28:15

Last modified on

2013-03-22 16:28:15

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

4

Author

yark (2760)

Entry type

Definition

Classification

msc 20E99

Defines

abnormal

Defines

abnormality

Generated on Fri Feb 9 19:28:12 2018 by

LaTeXML
Frattini subgroup

Frattini subgroup

Let

G

be a group. The

Φ

⁢

(

G

)

of

G

is the

intersection

of all

maximal subgroups

of

G

.

Equivalently,

Φ

⁢

(

G

)

is the

subgroup

of

non-generators

of

G

.

Title

Frattini subgroup

Canonical name

FrattiniSubgroup

Date of creation

2013-03-22 12:23:41

Last modified on

2013-03-22 12:23:41

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

4

Author

Evandar (27)

Entry type

Definition

Classification

msc 20E99

Related topic

SomethingRelatedToBurnsideBasisTheorem

Generated on Fri Feb 9 19:28:14 2018 by

LaTeXML
non-generator

non-generator

Let

G

be a group.

g

∈

G

is said to be a

non-generator

if whenever

X

is a

generating set

for

G

then

X

∖

{

g

}

is also a generating set for

G

.

Title

non-generator

Canonical name

Nongenerator

Date of creation

2013-03-22 12:23:43

Last modified on

2013-03-22 12:23:43

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

4

Author

Evandar (27)

Entry type

Definition

Classification

msc 20E99

Synonym

nongenerator

Related topic

FrattiniSubgroup

Related topic

Generator

Generated on Fri Feb 9 19:28:15 2018 by

LaTeXML