aarjaneiro/planetmath huggingface dataset:
groups of order pq

groups of order pq

We can use Sylow’s

theorems

to examine a group

G

of order

p

⁢

q

, where

p

and

q

are primes (

http://planetmath.org/Prime

) and

p

<

q

.

Let

n

p

and

n

q

denote, respectively, the number of Sylow

p

-

subgroups

and Sylow

q

-subgroups of

G

.

Sylow’s theorems tell us that

n

q

=

1

+

k

⁢

q

for some integer

k

and

n

q

divides

p

⁢

q

.
But

p

and

q

are prime and

p

<

q

, so this implies that

n

q

=

1

.
So there is exactly one Sylow

q

-subgroup, which is therefore normal (indeed,

fully invariant

) in

G

.

Denoting the Sylow

q

-subgroup by

Q

, and letting

P

be a Sylow

p

-subgroup, then

Q

∩

P

=

{

1

}

and

Q

⁢

P

=

G

, so

G

is a

semidirect product

of

Q

and

P

. In particular, if there is only one Sylow

p

-subgroup, then

G

is a

direct product

of

Q

and

P

, and is therefore cyclic.

Given

G

=

Q

⋊

P

, it remains to determine the action of

P

on

Q

by

conjugation

. There are two cases:

Case 1: If

p

does not divide

q

-

1

, then since

n

p

=

1

+

m

⁢

p

cannot equal

q

we must have

n

p

=

1

, and so

P

is a

normal subgroup

of

G

. This gives

G

=

C

p

×

C

q

a direct product, which is

isomorphic

to the

cyclic group

C

p

⁢

q

.

Case 2: If

p

divides

q

-

1

,
then

Aut

⁡

(

Q

)

≅

C

q

-

1

has a unique subgroup (

http://planetmath.org/Subgroup

)

P

′

of order

p

,
where

P

′

=

{

x

↦

x

i

∣

i

∈

ℤ

/

q

⁢

ℤ

,

i

p

=

1

}

.
Let

a

and

b

be

generators

for

P

and

Q

respectively,
and suppose the action of

a

on

Q

by conjugation is

x

↦

x

i

0

,
where

i

0

≠

1

in

ℤ

/

q

⁢

ℤ

.
Then

G

=

⟨

a

,

b

∣

a

p

=

b

q

=

1

,

a

⁢

b

⁢

a

-

1

=

b

i

0

⟩

.
Choosing a different

i

0

amounts to choosing a different generator

a

for

P

,
and hence does not result in a new isomorphism class.
So there are exactly two isomorphism classes of groups of order

p

⁢

q

.

Title

groups of order pq

Canonical name

GroupsOfOrderPq

Date of creation

2013-03-22 12:51:05

Last modified on

2013-03-22 12:51:05

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Example

Classification

msc 20D20

Related topic

SylowTheorems

Related topic

SemidirectProductOfGroups

Generated on Fri Feb 9 19:23:50 2018 by

LaTeXML
Hall subgroup

Hall subgroup

Let

G

be a

finite group

. A

subgroup

H

of

G

is said to
be a

Hall subgroup

if

gcd

⁡

(

|

H

|

,

|

G

/

H

|

)

=

1

.

In other
words,

H

is a Hall subgroup if the order of

H

and its index
in

G

are coprime. These subgroups are
name after Philip Hall who used them to characterize

solvable groups

.

Hall subgroups are a

generalization

of

Sylow subgroups

. Indeed,
every Sylow subgroup is a Hall subgroup. According to Sylow’s

theorem

, this means that any group of order

p

k

⁢

m

,

gcd

⁡

(

p

,

m

)

=

1

,
has a Hall subgroup (of order

p

k

).

A common notation used with Hall subgroups is to use the notion of

π

-groups (

http://planetmath.org/PiGroupsAndPiGroups

). Here

π

is a set
of primes and a Hall

π

-subgroup of a group is a subgroup which is
also a

π

-group, and

maximal

with this

property

.

Theorem 1

(Hall (1928)).

A finite group

G

is solvable iff

G

has a Hall

π

-subgroup for
any set of primes

π

.

The sets of primes

π

in Hall’s theorem can be restricted to
the subsets of primes which divide

|

G

|

.
However, this result fails for non-solvable groups.

Example 2

.

The group

A

5

has no Hall

{

2

,

5

}

-subgroup. That is,

A

5

has no subgroup of order

20

.

Proof.

Suppose that

A

5

has a Hall

{

2

,

5

}

-subgroup

H

.
As

|

A

5

|

=

60

, it follows that

|

H

|

=

20

. Thus, there
are three cosets of

H

. Since a group always acts
on the cosets of a subgroup, it follows that

A

5

acts
on the three member set

C

of cosets of

H

. This induces
a non-trivial

homomorphism

from

A

5

to

S

C

≅

S

3

(here,

S

C

is the

symmetric group

on

C

, see
this (

http://planetmath.org/GroupActionsAndHomomorphisms

) for more detail).
Since

A

5

is simple, this homomorphism must be

one-to-one

,
implying that its image must have order at most

6

, an
impossibility.
∎

This example can also be proved by direct inspection of the subgroups of

A

5

.
In any case,

A

5

is

non-abelian

simple and therefore it is not a
solvable group. Thus, Hall’s theorem does not apply to

A

5

.

Title

Hall subgroup

Canonical

name

HallSubgroup

Date of creation

2013-03-22 14:02:02

Last modified on

2013-03-22 14:02:02

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

22

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 20D20

Related topic

VeeCuhininsTheorem

Related topic

SylowTheorems

Defines

Hall’s theorem

Defines

Hall

π

-subgroup

Generated on Fri Feb 9 19:23:53 2018 by

LaTeXML
p-primary component

p-primary component

Definition 1

.

Let

G

be a finite abelian group and let

p

∈

N

be a prime.
The

p

-

primary component

of

G

,

Π

p

, is the

subgroup

of all elements whose order is a power of

p

.

Note: The

p

-primary component of an

abelian group

G

coincides
with the unique Sylow

p

-subgroup of

G

.

Title

p-primary component

Canonical name

PprimaryComponent

Date of creation

2013-03-22 13:52:05

Last modified on

2013-03-22 13:52:05

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 20D20

Synonym

primary component

Related topic

SylowPSubgroup

Generated on Fri Feb 9 19:23:55 2018 by

LaTeXML
proof of class equation theorem

proof of class equation theorem

X

is a finite

disjoint union

of finite orbits:

X

=

∪

i

G

⁢

x

i

. We can separate this union by considerating first only the orbits of 1

element

and then the rest:

X

=

∪

j

=

1

l

{

x

i

j

}

∪

∪

k

=

1

s

G

x

i

k

=

G

X

∪

k

=

1

s

G

x

i

k

Then using the

orbit-stabilizer theorem

, we have

#

X

=

#

G

X

+

∑

k

=

1

s

[

G

:

G

x

i

k

]

where for every

k

,

[

G

:

G

x

i

k

]

≥

2

, because if one of them were 1, then it would be associated to an orbit of 1 element, but we counted those orbits first. Then this

stabilizers

are not

G

. This finishes the proof.

Title

proof of class equation theorem

Canonical name

ProofOfClassEquationTheorem

Date of creation

2013-03-22 14:20:52

Last modified on

2013-03-22 14:20:52

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

4

Author

gumau (3545)

Entry type

Proof

Classification

msc 20D20

Generated on Fri Feb 9 19:23:57 2018 by

LaTeXML
proof of Frattini argument

proof of Frattini argument

Let

g

∈

G

be any element. Since

H

is normal,

g

⁢

S

⁢

g

-

1

⊂

H

.
Since

S

is a

Sylow subgroup

of

H

,

g

⁢

S

⁢

g

-

1

=

h

⁢

S

⁢

h

-

1

for some

h

∈

H

, by Sylow’s theorems.
Thus

n

=

h

-

1

⁢

g

normalizes

S

, and so

g

=

h

⁢

n

for

h

∈

H

and

n

∈

N

G

⁢

(

S

)

.

Title

proof of Frattini argument

Canonical name

ProofOfFrattiniArgument

Date of creation

2013-03-22 13:16:05

Last modified on

2013-03-22 13:16:05

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Proof

Classification

msc 20D20

Generated on Fri Feb 9 19:23:59 2018 by

LaTeXML
proof that a nontrivial normal subgroup of a finite p-group G and the center of G have nontrivial intersection

proof that a nontrivial normal subgroup of a finite

p

-group

G

and the center of

G

have nontrivial intersection

Define

G

to act on

H

by

conjugation

; that is, for

g

∈

G

,

h

∈

H

, define

g

⋅

h

=

g

⁢

h

⁢

g

-

1

Note that

g

⋅

h

∈

H

since

H

◁

G

. This is easily seen to be a well-defined

group action

.

Now, the set of

invariants

of

H

under this action are

G

H

=

{

h

∈

H

|

g

⋅

h

=

h

∀

g

∈

G

}

=

{

h

∈

H

|

g

h

g

-

1

=

h

∀

g

∈

G

}

=

H

∩

Z

(

G

)

The

class equation theorem

states that

|

H

|

=

|

G

H

|

+

∑

i

=

1

r

[

G

:

G

x

i

]

where the

G

x

i

are

proper subgroups

of

G

, and thus that

|

G

H

|

=

|

H

|

-

∑

i

=

1

r

[

G

:

G

x

i

]

We now use elementary group theory to show that

p

divides each term on the right, and conclude as a result that

p

divides

|

G

H

|

, so that

G

H

=

H

∩

Z

⁢

(

G

)

cannot be

trivial

.

As

G

is a nontrivial finite

p

-group, it is obvious from Cauchy’s

theorem

that

|

G

|

=

p

n

for

n

>

0

. Since

H

and the

G

x

i

are

subgroups

of

G

, each either is trivial or has order a power of

p

, by Lagrange’s theorem. Since

H

is nontrivial, its order is a nonzero power of

p

. Since each

G

x

i

is a proper subgroup of

G

and has order a power of

p

, it follows that

[

G

:

G

x

i

]

also has order a nonzero power of

p

.

Title

proof that a nontrivial

normal subgroup

of a finite

p

-group

G

and the center of

G

have nontrivial

intersection

Canonical name

ProofThatANontrivialNormalSubgroupOfAFinitePgroupGAndTheCenterOfGHaveNontrivialIntersection

Date of creation

2013-03-22 14:21:07

Last modified on

2013-03-22 14:21:07

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

9

Author

rm50 (10146)

Entry type

Proof

Classification

msc 20D20

Generated on Fri Feb 9 19:24:02 2018 by

LaTeXML
p-subgroup

p

-subgroup

Let

G

be a

finite group

with order

n

, and let

p

be a

prime integer

.
We can write

n

=

p

k

⁢

m

for some

k

,

m

integers

, such that

k

and

m

are

coprimes

(that is,

p

k

is the highest power of

p

that divides

n

).
Any

subgroup

of

G

whose order is

p

k

is called a Sylow

p

-subgroup.

While there is no reason for Sylow

p

-subgroups to exist for any finite group, the fact is that all groups have Sylow

p

-subgroups for every prime

p

that divides

|

G

|

. This statement is the First

Sylow theorem

When

|

G

|

=

p

k

we simply say that

G

is a

p

-group.

Title

p

-subgroup

Canonical name

Psubgroup

Date of creation

2013-03-22 14:02:14

Last modified on

2013-03-22 14:02:14

Owner

drini (3)

Last modified by

drini (3)

Numerical id

8

Author

drini (3)

Entry type

Definition

Classification

msc 20D20

Related topic

PGroup4

Defines

Sylow

p

-subgroup

Defines

Sylow p-subgroup

Defines

p

-group

Defines

p-group

Generated on Fri Feb 9 19:24:04 2018 by

LaTeXML
subgroups containing the normalizers of Sylow subgroups normalize themselves

subgroups containing the normalizers of Sylow subgroups normalize themselves

Let

G

be a

finite group

, and

S

a

Sylow subgroup

. Let

M

be a

subgroup

such that

N

G

⁢

(

S

)

⊂

M

. Then

M

=

N

G

⁢

(

M

)

.

Proof.

By order considerations,

S

is a Sylow subgroup of

M

. Since

M

is normal in

N

G

⁢

(

M

)

,
by the

Frattini argument

,

N

G

⁢

(

M

)

=

N

G

⁢

(

S

)

⁢

M

=

M

.
∎

Title

subgroups containing the

normalizers

of Sylow subgroups normalize themselves

Canonical name

SubgroupsContainingTheNormalizersOfSylowSubgroupsNormalizeThemselves

Date of creation

2013-03-22 13:16:47

Last modified on

2013-03-22 13:16:47

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Corollary

Classification

msc 20D20

Generated on Fri Feb 9 19:24:06 2018 by

LaTeXML
Sylow p-subgroup

Sylow p-subgroup

If

(

G

,

*

)

is a group then any

subgroup

of order

p

a

for any integer a is called a

p-subgroup

. If

|

G

|

=

p

a

⁢

m

, where

p

∤

m

then any subgroup

S

of

G

with

|

S

|

=

p

a

is a

Sylow p-subgroup

. We use

Syl

p

⁢

(

G

)

for the set of Sylow

p-groups

of

G

.

More generally, if

G

is any group (not necessarily finite), a Sylow p-subgroup is a

maximal

p

-subgroup of

G

.

Title

Sylow p-subgroup

Canonical name

SylowPsubgroup

Date of creation

2013-03-22 12:50:59

Last modified on

2013-03-22 12:50:59

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

8

Author

Henry (455)

Entry type

Definition

Classification

msc 20D20

Synonym

Sylow subgroup

Synonym

Sylow group

Related topic

SylowTheorems

Related topic

ProofOfSylowTheorems

Related topic

PPrimaryComponent

Related topic

SylowsThirdTheorem

Defines

Sylow p-subgroup

Defines

p-subgroup

Generated on Fri Feb 9 19:24:08 2018 by

LaTeXML
Sylow theorems

Sylow theorems

Let

G

be a

finite group

whose order is

divisible

by the prime

p

. Suppose

p

m

is the highest power of

p

which is a factor of

|

G

|

and set

k

=

|

G

|

p

m

.

Then

1.

the group

G

contains at least one

subgroup

of order

p

m

,

2.

any two subgroups of

G

of order

p

m

are

conjugate

, and

3.

the number of subgroups of

G

of order

p

m

is

congruent

to

1

modulo

p

and is a factor of

k

.

Title

Sylow theorems

Canonical name

SylowTheorems

Date of creation

2013-03-22 12:24:12

Last modified on

2013-03-22 12:24:12

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D20

Related topic

SylowPSubgroup

Related topic

ApplicationOfSylowsTheoremsToGroupsOfOrderPq

Related topic

SylowsFirstTheorem

Related topic

SylowsThirdTheorem

Related topic

SylowPSubgroups

Generated on Fri Feb 9 19:24:10 2018 by

LaTeXML
Sylow’s first theorem

Sylow’s first theorem

If

G

is a

finite group

and

p

is a prime (

http://planetmath.org/Prime

) such that

p

k

divides

|

G

|

, then there is a

subgroup

(

http://planetmath.org/Subgroup

)

H

of

G

such that

|

H

|

=

p

k

.

This is the first part of several results usually called the

Sylow theorems

.

Title

Sylow’s first theorem

Canonical name

SylowsFirstTheorem

Date of creation

2013-03-22 14:00:46

Last modified on

2013-03-22 14:00:46

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

11

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20D20

Related topic

SylowTheorems

Related topic

ProofOfSylowTheorems

Generated on Fri Feb 9 19:24:16 2018 by

LaTeXML
Sylow’s third theorem

Sylow’s third theorem

Let

G

be a

finite group

, and let

n

be the number of Sylow

p

-

subgroups

of

G

. Then

n

≡

1

(

mod

p

)

, and any
two Sylow

p

-subgroups of

G

are

conjugate

to one another.

Title

Sylow’s third theorem

Canonical name

SylowsThirdTheorem

Date of creation

2013-03-22 14:00:41

Last modified on

2013-03-22 14:00:41

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

19

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20D20

Related topic

SylowTheorems

Related topic

ProofOfSylowTheorems

Related topic

SylowPSubgroup

Generated on Fri Feb 9 19:24:18 2018 by

LaTeXML
π-groups and π′-groups

π

-groups and

π

′

-groups

Let

π

be a set of primes. A

torsion group

G

is called a

π

-group if each prime dividing the order of an element of

G

is in

π

and a

π

′

-group if none of them are. Typically, if

π

is a

singleton

π

=

{

p

}

, we write

p

-group and

p

′

-group for these.

Remark

. If

G

is finite, then

G

is a

π

-group if every prime dividing

|

G

|

is in

π

.

Title

π

-groups and

π

′

-groups

Canonical

name

pigroupsAndpigroups

Date of creation

2013-03-22 13:17:51

Last modified on

2013-03-22 13:17:51

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

9

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 20D20

Classification

msc 20F50

Defines

π

-group

Defines

π

′

-group

Generated on Fri Feb 9 19:24:20 2018 by

LaTeXML
characteristically simple group

characteristically simple group

A group

G

is called

characteristically simple

if its only

characteristic subgroups

are

{

1

}

and

G

. Any finite characteristically simple group is
the

direct product

of several copies of

isomorphic

simple groups

.

Title

characteristically simple group

Canonical name

CharacteristicallySimpleGroup

Date of creation

2013-03-22 13:16:58

Last modified on

2013-03-22 13:16:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20D25

Synonym

characteristically simple

Generated on Fri Feb 9 19:24:22 2018 by

LaTeXML
Fitting’s theorem

Fitting’s theorem

Fitting’s Theorem

states that if

G

is a group
and

M

and

N

are normal

nilpotent

subgroups

(

http://planetmath.org/Subgroup

) of

G

,
then

M

⁢

N

is also a normal nilpotent subgroup
(of nilpotency class less than or equal to
the sum of the nilpotency classes of

M

and

N

).

Thus, any

finite group

has a unique largest normal nilpotent subgroup, called its

Fitting subgroup

.
More generally, the Fitting subgroup of a group

G

is defined to be the subgroup of

G

generated by the normal nilpotent subgroups of

G

;
Fitting’s Theorem shows that the Fitting subgroup is always

locally nilpotent

.
A group that is equal to its own Fitting subgroup is sometimes called a

Fitting group

.

Title

Fitting’s theorem

Canonical name

FittingsTheorem

Date of creation

2013-03-22 13:51:39

Last modified on

2013-03-22 13:51:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D25

Defines

Fitting subgroup

Defines

Fitting group

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Frattini subgroup of a finite group is nilpotent, the

Frattini subgroup of a finite group is nilpotent, the

The

Frattini subgroup

of a

finite group

is

nilpotent

(

http://planetmath.org/NilpotentGroup

).

Proof.

Let

Φ

⁢

(

G

)

denote the Frattini subgroup of a finite group

G

.
Let

S

be a

Sylow subgroup

of

Φ

⁢

(

G

)

.
Then by the

Frattini argument

,

G

=

Φ

⁢

(

G

)

⁢

N

G

⁢

(

S

)

=

⟨

Φ

⁢

(

G

)

∪

N

G

⁢

(

S

)

⟩

.
But the Frattini subgroup is finite and formed of

non-generators

,
so it follows that

G

=

⟨

N

G

⁢

(

S

)

⟩

=

N

G

⁢

(

S

)

.
Thus

S

is normal in

G

, and therefore normal in

Φ

⁢

(

G

)

.
The result now follows, as any finite group whose Sylow subgroups are all normal is nilpotent (

http://planetmath.org/ClassificationOfFiniteNilpotentGroups

).
∎

In fact, the same proof shows that for any group

G

,
if

Φ

⁢

(

G

)

is finite then

Φ

⁢

(

G

)

is nilpotent.

Title

Frattini subgroup of a finite group is nilpotent, the

Canonical name

FrattiniSubgroupOfAFiniteGroupIsNilpotentThe

Date of creation

2013-03-22 13:16:44

Last modified on

2013-03-22 13:16:44

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

16

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D25

Generated on Fri Feb 9 19:24:25 2018 by

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maximal condition

maximal condition

A group is said to

satisfy

the

maximal condition

if every strictly ascending chain of

subgroups

G

1

⊂

G

2

⊂

G

3

⊂

⋯

is finite.

This is also called the

ascending chain condition

.

A group satisfies the maximal condition if and only if the group and all its subgroups are

finitely generated

.

Similar

properties are useful in other classes of

algebraic structures

: see for example the

Noetherian

condition for rings and modules.

Title

maximal condition

Canonical name

MaximalCondition

Date of creation

2013-03-22 13:58:47

Last modified on

2013-03-22 13:58:47

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 20D30

Synonym

ascending chain condition

Generated on Fri Feb 9 19:24:27 2018 by

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minimal condition

minimal condition

A group is said to

satisfy

the

minimal condition

if every

strictly

descending chain of

subgroups

G

1

⊃

G

2

⊃

G

3

⊃

⋯

is finite.

This is also called the

descending chain condition

.

A group which satisfies the minimal condition is necessarily

periodic

. For if it

contained

an

element

x

of

infinite order

, then

⟨

x

⟩

⊃

⟨

x

2

⟩

⊃

⟨

x

4

⟩

⊃

⋯

⊃

⟨

x

2

n

⟩

⊃

⋯

is an infinite descending chain of subgroups.

Similar

properties

are useful in other classes of

algebraic structures

: see for example the

Artinian

condition for rings and modules.

Title

minimal condition

Canonical

name

MinimalCondition

Date of creation

2013-03-22 13:58:49

Last modified on

2013-03-22 13:58:49

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Definition

Classification

msc 20D30

Synonym

descending chain condition

Related topic

ChernikovGroup

Generated on Fri Feb 9 19:24:30 2018 by

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subnormal series

subnormal series

Let

G

be a

group with

a

subgroup

H

, and let

G

=

G

0

⊳

G

1

⊳

⋯

⊳

G

n

=

H

(1)

be a series of subgroups with each

G

i

a

normal subgroup

of

G

i

-

1

.
Such a series is called a

subnormal series

or a

subinvariant series

.

If in addition, each

G

i

is a normal subgroup of

G

,
then the series is called a

normal series

.

A subnormal series in which each

G

i

is a

maximal normal subgroup

of

G

i

-

1

is called a

composition series

.

A normal series in which

G

i

is a maximal normal subgroup of

G

contained in

G

i

-

1

is called a

principal series

or a

chief series

.

Note that a composition series need not end in the

trivial group

1

.
One speaks of a composition series (1) as a

composition series from

G

to

H

.
But the term

composition series for

G

generally means a composition series from

G

to

1

.

Similar remarks apply to principal series.

Some authors use normal series as a synonym for subnormal series. This usage is, of course, not compatible with the stronger definition of normal series given above.

Title

subnormal series

Canonical name

SubnormalSeries

Date of creation

2013-03-22 13:58:42

Last modified on

2013-03-22 13:58:42

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Definition

Classification

msc 20D30

Synonym

subinvariant series

Related topic

SubnormalSubgroup

Related topic

JordanHolderDecompositionTheorem

Related topic

Solvable

Related topic

DescendingSeries

Related topic

AscendingSeries

Defines

composition series

Defines

normal series

Defines

principal series

Defines

chief series

Generated on Fri Feb 9 19:24:32 2018 by

LaTeXML
subnormal subgroup

subnormal subgroup

Let

G

be a group, and

H

a

subgroup

of

G

.
Then

H

is a

subnormal subgroup

of

G

if there is a

natural number

n

and subgroups

H

0

,

…

,

H

n

of

G

such that

H

=

H

0

◁

H

1

◁

⋯

◁

H

n

=

G

,

where

H

i

is a

normal subgroup

of

H

i

+

1

for

i

=

0

,

…

,

n

-

1

.

Subnormality is a ,
as

normality of subgroups is not transitive

.

We may write

H

⁢

sn

⁡

G

or

H

◁

◁

G

or

H

⁢

⊴

⁢

⊴

⁢

G

to indicate that

H

is a subnormal subgroup of

G

.

In a

nilpotent group

, all subgroups are subnormal.

Subnormal subgroups are

ascendant

and

descendant

.

Title

subnormal subgroup

Canonical name

SubnormalSubgroup

Date of creation

2013-03-22 13:16:27

Last modified on

2013-03-22 13:16:27

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Definition

Classification

msc 20D35

Classification

msc 20E15

Synonym

subinvariant subgroup

Synonym

attainable subgroup

Related topic

SubnormalSeries

Related topic

ClassificationOfFiniteNilpotentGroups

Related topic

NormalSubgroup

Related topic

CharacteristicSubgroup

Related topic

FullyInvariantSubgroup

Defines

subnormal

Defines

subnormality

Generated on Fri Feb 9 19:24:34 2018 by

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