aarjaneiro/planetmath huggingface dataset:
Schur’s lemma

Schur’s lemma

Schur’s lemma is a fundamental result in

representation theory

,
an elementary observation about

irreducible modules

, which is nonetheless
noteworthy because of its profound applications.

Lemma

(Schur’s lemma).

Let

G

be a

finite group

and let

V

and

W

be

irreducible

G

-modules. Then, every

G

-

module homomorphism

f

:

V

→

W

is
either

invertible

or the trivial

zero map

.

Proof.

Note that both the kernel,

ker

⁡

f

, and the image,

im

⁡

f

, are

G

-

submodules

of

V

and

W

, respectively. Since

V

is irreducible,

ker

⁡

f

is either
trivial or all of

V

. In the former case,

im

⁡

f

is all of

W

— also because

W

is irreducible — and hence

f

is invertible. In
the latter case,

f

is the zero map.
∎

One of the most important consequences of Schur’s lemma is the following.

Corollary

.

Let

V

be a

finite-dimensional

, irreducible

G

-module taken over
an algebraically closed field. Then, every

G

-module homomorphism

f

:

V

→

V

is equal to a

scalar multiplication

.

Proof.

Since the ground field is

algebraically closed

, the

linear
transformation

f

:

V

→

V

has an

eigenvalue

; call it

λ

.
By definition,

f

-

λ

⁢

1

is not invertible, and hence equal to
zero by Schur’s lemma. In other words,

f

=

λ

, a scalar.
∎

Title

Schur’s lemma

Canonical name

SchursLemma

Date of creation

2013-03-22 13:08:01

Last modified on

2013-03-22 13:08:01

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

22

Author

rmilson (146)

Entry type

Theorem

Classification

msc 20C99

Classification

msc 20C15

Related topic

GroupRepresentation

Related topic

DenseRingOfLinearTransformations

Generated on Fri Feb 9 19:22:57 2018 by

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permutation representation

permutation representation

Let

G

be a group, and

S

any finite set on which

G

acts.

That means that for any

g

,

h

∈

G

;

𝐯

,

𝐰

∈

S

•

g

⁢

𝐯

∈

V

,

•

(

g

⁢

h

)

⁢

𝐯

=

g

⁢

(

h

⁢

𝐯

)

,

•

e

⁢

𝐯

=

𝐯

.

Notice that we almost have what it takes to make

S

a

representation

of

G

, but

S

is no

vector space

. We can however obtain a

G

-module (a vector space carrying a representation of

G

) as follows.

Let

S

=

{

𝐬

1

,

𝐬

2

,

…

,

𝐬

n

}

. And let

ℂ

⁢

S

=

ℂ

⁢

[

𝐬

1

,

𝐬

2

,

…

,

𝐬

n

]

be the vector space generated by

S

over

ℂ

. in other words,

ℂ

⁢

S

is made of all formal

linear combinations

c

1

⁢

𝐬

1

+

c

2

⁢

𝐬

2

+

⋯

+

c

n

⁢

𝐬

n

with

c

j

∈

ℂ

.
The sum is defined coordinate-wise as is

scalar multiplication

.

Then the action of

G

in

S

can be extended linearly to

ℂ

⁢

S

as

g

⁢

(

c

1

⁢

𝐬

1

+

c

2

⁢

𝐬

2

+

⋯

+

c

n

⁢

𝐬

n

)

=

c

1

⁢

(

g

⁢

𝐬

1

)

+

c

2

⁢

(

g

⁢

𝐬

2

)

+

⋯

+

c

n

⁢

(

g

⁢

𝐬

n

)

and then the map

ρ

:

G

→

G

⁢

L

⁢

(

ℂ

⁢

S

)

where

ρ

is such that

ρ

⁢

(

g

)

⁢

(

𝐯

)

=

g

⁢

𝐯

makes

ℂ

⁢

S

into a

G

-module. The

G

-module

ℂ

⁢

S

is known as the

permutation representation

associated with

S

.

Example.

If

G

=

S

n

acts on

S

=

{

𝟏

,

𝟐

,

…

,

𝐧

}

, then

ℂ

⁢

S

=

{

c

1

⁢

𝟏

+

c

2

⁢

𝟐

+

⋯

+

c

n

⁢

𝐧

}

.

If

σ

∈

S

n

, the action becomes

σ

⁢

(

c

1

⁢

𝟏

+

c

2

⁢

𝟐

+

⋯

+

c

n

⁢

𝐧

)

=

c

1

⁢

σ

⁢

(

𝟏

)

+

c

2

⁢

σ

⁢

(

𝟐

)

+

⋯

+

c

n

⁢

σ

⁢

(

𝐧

)

.

Since

S

forms a basis for this space, we can compute the matrices corresponding to the defining

permutation

and we will see that the corresponding

permutation matrices

are obtained.

References.

Bruce E. Sagan.

The

Symmetric Group

: Representations, Combinatorial Algorithms and Symmetric Functions.

2a Ed. 2000. Graduate Texts in Mathematics. Springer.

Title

permutation representation

Canonical name

PermutationRepresentation

Date of creation

2013-03-22 14:53:59

Last modified on

2013-03-22 14:53:59

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 20Cxx

Related topic

MatrixRepresentation

Generated on Fri Feb 9 19:22:59 2018 by

LaTeXML
Burnside p-q theorem

Burnside

p

-

q

theorem

Any group whose order is

divisible

by only two distinct primes is

solvable

.
(These two distinct primes are the

p

and

q

of the title.)

It follows that if

G

is a non-abelian finite simple group, then

|

G

|

must have at least three distinct

prime divisors

.

Title

Burnside

p

-

q

theorem

Canonical name

BurnsidePqTheorem

Date of creation

2013-03-22 13:15:58

Last modified on

2013-03-22 13:15:58

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D05

Generated on Fri Feb 9 19:23:01 2018 by

LaTeXML
Burnside’s Theorem

Burnside’s Theorem

Theorem 1

(Burnside’s Theorem).

Let

G

be a

simple group

,

σ

∈

G

. Then the number of

conjugates

of

σ

is not a prime power (unless

σ

is its own

conjugacy class

).

Proofs of this

theorem

are quite difficult and rely on

representation theory

.

From this we immediately get

Corollary 1

.

A group

G

of order

p

a

⁢

q

b

, where

p

,

q

are prime, cannot be a

nonabelian

simple group.

Proof.

Suppose it is. Then the center of

G

is

trivial

,

{

e

}

, since the center is a

normal subgroup

and

G

is simple nonabelian. So if

C

i

are the nontrivial conjugacy classes, we have from the

class equation

that

|

G

|

=

1

+

∑

|

C

i

|

Now, each

|

C

i

|

divides

|

G

|

, but cannot be

1

since the center is trivial. It cannot be a power of either

p

or

q

by Burnside’s theorem. Thus

p

q

∣

|

C

i

|

for each

i

and thus

|

G

|

≡

1

(

mod

p

⁢

q

)

, which is a

contradiction

.
∎

Finally, a corollary of the above is known as the Burnside

p

-

q

Theorem (

http://planetmath.org/BurnsidePQTheorem

).

Corollary 2

.

A group of order

p

a

⁢

q

b

is

solvable

.

Title

Burnside’s Theorem

Canonical name

BurnsidesTheorem

Date of creation

2013-03-22 16:38:14

Last modified on

2013-03-22 16:38:14

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20D05

Generated on Fri Feb 9 19:23:03 2018 by

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classification of semisimple groups

classification of semisimple groups

For every

semisimple group

G

there is a

normal subgroup

H

of

G

, (called the centerless
competely reducible

radical

) which

isomorphic

to a

direct product

of

nonabelian

simple groups

such that

conjugation

on

H

gives an

injection

into

Aut

⁢

(

H

)

. Thus

G

is isomorphic to a

subgroup

of

Aut

⁢

(

H

)

containing the

inner automorphisms

, and for every group

H

isomorphic
to a direct product of

non-abelian

simple groups, every such subgroup is

semisimple

.

Title

classification of semisimple groups

Canonical

name

ClassificationOfSemisimpleGroups

Date of creation

2013-03-22 13:17:10

Last modified on

2013-03-22 13:17:10

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 20D05

Generated on Fri Feb 9 19:23:05 2018 by

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semisimple group

semisimple group

In

group theory

the use of the phrase

semi-simple group

is used sparingly.
Standard texts on group theory including

[

1

,

2

]

avoid
the term altogether. Other texts provide precise definitions which are nevertheless
not

equivalent

[

3

,

4

]

. In general it is preferable to use
other terms to describe the class of groups being considered as there is
no uniform convention. However, below is a list of possible uses of for
the phrase

semi-simple group

.

1.

A group is

semi-simple

if it has no non-trivial normal

abelian

subgroups

[

3

, p. 89]

.

2.

A group

G

is

semi-simple

if

G

′

=

G

and

G

/

Z

⁢

(

G

)

is a

direct
product

of

non-abelian

simple groups

[

4

, Def. 6.1]

.

3.

A

product

of simple groups may be called

semi-simple

. Depending
on application, the simple groups may be further restricted to finite simple groups
and may also exclude the abelian simple groups.

4.

A Lie group whose associated Lie algebra is a semi-simple Lie algebra may
be called a

semi-simple

group and more specifically, a

semi-simple Lie group

.

Connections

with

algebra

The use of

semi-simple

in the study of algebras,

representation theory

, and modules
is far more precise owing to the fact that the various possible definitions are generally
equivalent.

For example. In a finite dimensional associative algebra

A

, if

A

it is a product of

simple algebras

then the

Jacobson radical

is

trivial

. In contrast, if

A

has trivial
Jacobson radical then it is a direct product of simple algebras. Thus

A

may be
called

semi-simple

if either:

A

is a direct product of simple algebras or

A

has trivial Jacobson radical.

The analogue fails for groups. For

instance

. If a group is defined as semi-simple
by virtue of having no non-trivial normal abelian subgroups then

S

n

is semi-simple
for all

n

>

5

. However,

S

n

is not a product of simple groups.

References

1

Aschbacher, M.

Finite group

theory

Cambridge studies in advanced mathematics 10,
Cambridge University Press, Cambridge, (1986).

2

Gorenstein, D.

Finite groups

Chelsea Publishing Company, New York, (1980).

3

Robinson, D. J.S.

A course in the theory of groups

Ed. 2, GTM 80, Springer, New York, (1996).

4

Suzuki, M.

Group Theory I,II

, (English) Springer-verlag, Berlin (1982, 1986).

Title

semisimple group

Canonical name

SemisimpleGroup

Date of creation

2013-03-22 13:17:07

Last modified on

2013-03-22 13:17:07

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 20D05

Related topic

socle

Defines

semi-simple group

Defines

semisimple group

Generated on Fri Feb 9 19:23:08 2018 by

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Janko groups

Janko groups

The

Janko groups

denoted by

J

1

,

J

2

,

J

3

, and

J

4

are four of the 26

sporadic groups

. They were discovered by . Janko in 1966 and published in the article ”A new finite simple group with

abelian

Sylow

2

-

subgroups

and its

characterization

.” (Journal of Algebra,

3

, 1966, 32: 147-186).

Each of these groups have very intricate

matrix representations

as maps into large

general linear groups

. For example, the matrix

K

corresponding to

J

4

gives a

representation

of

J

4

in

G

⁢

L

112

⁢

(

2

)

.

Title

Janko groups

Canonical

name

JankoGroups

Date of creation

2013-03-22 13:59:17

Last modified on

2013-03-22 13:59:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20D08

Related topic

ExamplesOfFiniteSimpleGroups

Related topic

Solvable

Generated on Fri Feb 9 19:23:15 2018 by

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Wielandt-Kegel theorem

Wielandt-Kegel theorem

Theorem

.

If a

finite group

is the

product

of two

nilpotent

subgroups

,
then it is

solvable

.

That is, if

H

and

K

are nilpotent subgroups of a finite group

G

,
and

G

=

H

⁢

K

, then

G

is solvable.

This result can be considered as
a

generalization

of Burnside’s

p

-

q

Theorem

(

http://planetmath.org/BurnsidePQTheorem

),
because if a group

G

is of order

p

m

⁢

q

n

, where

p

and

q

are distinct primes, then

G

is the product of a Sylow

p

-subgroup (

http://planetmath.org/SylowPSubgroup

) and Sylow

q

-subgroup, both of which are nilpotent.

Title

Wielandt-Kegel theorem

Canonical name

WielandtKegelTheorem

Date of creation

2013-03-22 16:17:37

Last modified on

2013-03-22 16:17:37

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D10

Synonym

Kegel-Wielandt theorem

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

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π-separable group

π

-separable group

Let

π

be a set of

prime numbers

.
A

finite group

G

is called

π

-

separable

if there exists a

composition series

{

1

}

=

G

0

⊲

⋯

⊲

G

n

=

G

such that each

G

i

+

1

/

G

i

is either
a

π

-group (

http://planetmath.org/PiGroupsAndPiGroups

)
or a

π

′

-group (

http://planetmath.org/PiGroupsAndPiGroups

).

A

{

p

}

-separable group, where

p

is a prime number,
is usually called a

p

-separable

group.

π

-separability can be thought of as
a

generalization

of solvability for finite groups;
a finite group is

π

-separable for all sets of primes
if and only it is

solvable

.

Title

π

-separable group

Canonical

name

piseparableGroup

Date of creation

2013-03-22 13:17:48

Last modified on

2013-03-22 13:17:48

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20D10

Defines

π

-separable

Defines

p

-separable

Defines

π

-separability

Defines

p

-separability

Defines

p

-separable group

Generated on Fri Feb 9 19:23:25 2018 by

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Čunihin’s theorem

Čunihin’s theorem

Theorem 1

(Čunihin).

Let

G

be a finite,

π

-

separable

group (

http://planetmath.org/Seperable

), for some set

π

of primes. Then

•

any

π

-

subgroup

(

http://planetmath.org/PiGroupsAndPiGroups

) is

contained

in a Hall

π

-subgroup (

http://planetmath.org/HallPiSubgroup

), and

•

any two Hall

π

-subgroups are

conjugate

of one another

Remarks

1.

For

π

=

{

p

}

, this essentially reduces to the

Sylow theorems

(with unnecessary hypotheses).

2.

If

G

is

solvable

, it is

π

-separable for all

π

, so such subgroups exist for all

π

. This result is often called Hall’s

theorem

. There is another Hall’s theorem, which is

similar

to this one, can be be found here (

http://planetmath.org/HallsTheorem2

).

References

1

Derek J.S. Robinson.

A Course in the Theory of Groups

, second edition. Springer (1995)

Title

Čunihin’s theorem

Canonical name

vCunihinsTheorem

Date of creation

2013-03-22 13:17:54

Last modified on

2013-03-22 13:17:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 20D10

Related topic

SylowPSubgroups

Defines

Hall’s theorem

Generated on Fri Feb 9 19:23:27 2018 by

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Burnside basis theorem

Burnside basis theorem

Theorem 1

If

G

is a finite

p

-group, then

Frat

⁢

G

=

G

′

⁢

G

p

, where

Frat

⁢

G

is the

Frattini subgroup

,

G

′

the

commutator subgroup

, and

G

p

is the

subgroup

generated by

p

-th powers.

The theorem implies that

G

/

Frat

⁢

G

is

elementary
abelian

, and thus has a minimal

generating set

of
exactly

n

elements, where

|

G

:

Frat

G

|

=

p

n

. Since any lift of such a
generating set also generates

G

(by the non-generating property of the
Frattini subgroup), the smallest generating set of

G

also
has

n

elements.

The theorem also holds for

profinite

p

-groups (

inverse limit

of finite

p

-groups).

Title

Burnside basis theorem

Canonical name

BurnsideBasisTheorem

Date of creation

2013-03-22 13:16:08

Last modified on

2013-03-22 13:16:08

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20D15

Generated on Fri Feb 9 19:23:29 2018 by

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characterization of finite nilpotent groups

characterization of finite nilpotent groups

Let

G

be a

finite group

. The following are

equivalent

:

1.

G

is

nilpotent

.

2.

Every

subgroup

(

http://planetmath.org/Subgroup

) of

G

is

subnormal

.

3.

Every

proper subgroup

of

G

is properly

contained

in its

normalizer

.

4.

Every

maximal subgroup

of

G

is normal.

5.

Every

Sylow subgroup

of

G

is normal.

6.

G

is a

direct product

(

http://planetmath.org/DirectProductAndRestrictedDirectProductOfGroups

) of

p

-groups (

http://planetmath.org/PGroup4

).

For proofs, see the article on

finite nilpotent groups

.

Condition 3 above is the

normalizer condition

.

Title

characterization of finite nilpotent groups

Canonical name

CharacterizationOfFiniteNilpotentGroups

Date of creation

2013-03-22 13:16:24

Last modified on

2013-03-22 13:16:24

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Theorem

Classification

msc 20D15

Classification

msc 20F18

Related topic

FiniteNilpotentGroups

Related topic

NilpotentGroup

Related topic

NormalizerCondition

Related topic

SubnormalSubgroup

Related topic

LocallyNilpotentGroup

Generated on Fri Feb 9 19:23:31 2018 by

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proof of the Burnside basis theorem

proof of the Burnside basis theorem

Let

P

be a

p

-group and

Φ

⁢

(

P

)

its

Frattini subgroup

.

Every

maximal subgroup

Q

of

P

is of index

p

in

P

and is therefore
normal in

P

. Thus

P

/

Q

≅

ℤ

p

. So given

g

∈

P

,

g

p

∈

Q

which proves

P

p

≤

Q

. Likewise,

ℤ

p

is

abelian

so

[

P

,

P

]

≤

Q

. As

Q

is any maximal subgroup, it follows

[

P

,

P

]

and

P

p

lie in

Φ

⁢

(

P

)

.

Now both

[

P

,

P

]

and

P

p

are

characteristic subgroups

of

P

so in particular

F

=

[

P

,

P

]

⁢

P

p

is normal in

P

. If we pass to

V

=

P

/

F

we find that

V

is abelian and every

element

has order

p

– that is,

V

is a

vector space

over

ℤ

p

. So the maximal subgroups of

P

are in a

1-1 correspondence

with the

hyperplanes

of

V

. As the

intersection

of all hyperplanes of a vector space is the origin, it follows the intersection of all maximal subgroups of

P

is

F

. That is,

[

P

,

P

]

⁢

P

p

=

Φ

⁢

(

P

)

.

Title

proof of the

Burnside basis theorem

Canonical name

ProofOfTheBurnsideBasisTheorem

Date of creation

2013-03-22 15:46:25

Last modified on

2013-03-22 15:46:25

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

12

Author

Algeboy (12884)

Entry type

Proof

Classification

msc 20D15

Generated on Fri Feb 9 19:23:38 2018 by

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

a nontrivial normal subgroup of a finite

p

-group

G

and the center of

G

have nontrivial intersection

Let

G

be a finite

p

-group, and let

H

be a nontrivial

normal subgroup

of

G

. Then

H

∩

Z

⁢

(

G

)

≠

{

1

}

.

Title

a nontrivial normal subgroup of a finite

p

-group

G

and the center of

G

have nontrivial

intersection

Canonical name

ANontrivialNormalSubgroupOfAFinitePgroupGAndTheCenterOfGHaveNontrivialIntersection

Date of creation

2013-03-22 14:21:05

Last modified on

2013-03-22 14:21:05

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

4

Author

gumau (3545)

Entry type

Theorem

Classification

msc 20D20

Generated on Fri Feb 9 19:23:40 2018 by

LaTeXML
Burnside normal complement theorem

Burnside normal complement theorem

Let

G

be a

finite group

, and

S

a

Sylow subgroup

such that

C

G

⁢

(

S

)

=

N

G

⁢

(

S

)

. Then

S

has a normal complement. That is, there exists a

normal subgroup

N

⊲

G

such that

S

∩

N

=

{

1

}

and

S

⁢

N

=

G

.

Title

Burnside normal complement theorem

Canonical name

BurnsideNormalComplementTheorem

Date of creation

2013-03-22 13:16:22

Last modified on

2013-03-22 13:16:22

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20D20

Generated on Fri Feb 9 19:23:42 2018 by

LaTeXML
class equation theorem

class equation theorem

Let

G

be a group acting on a

finite set

X

. Define the set of

invariants

in

X

by the action of

G

as

G

X

=

{

x

∈

X

|

g

x

=

x

∀

g

∈

G

}

. Then there are

H

1

,

…

,

H

r

subgroups

of

G

with

H

i

≠

G

∀

i

such that

#

X

=

#

G

X

+

∑

i

=

1

r

[

G

:

H

i

]

Title

class equation theorem

Canonical name

ClassEquationTheorem

Date of creation

2013-03-22 14:20:50

Last modified on

2013-03-22 14:20:50

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

5

Author

gumau (3545)

Entry type

Theorem

Classification

msc 20D20

Synonym

class equation

Related topic

Centralizer

Generated on Fri Feb 9 19:23:44 2018 by

LaTeXML
example of groups of order pq

example of groups of order pq

As a specific example, let us classify groups of order 21. Let

G

be a group of order 21. There is only one Sylow 7-subgroup

K

so it is normal. The possibility of there being

conjugate

Sylow 3-subgroups is not ruled out. Let

x

denote a

generator

for

K

, and

y

a generator for one of the Sylow 3-subgroups

H

. Then

x

7

=

y

3

=

1

, and

y

⁢

x

⁢

y

-

1

=

x

i

for some

i

<

7

since

K

is normal. Now

x

=

y

3

⁢

x

⁢

y

-

3

=

y

2

⁢

x

i

⁢

y

-

2

=

y

⁢

x

i

2

⁢

y

-

1

=

x

i

3

, or

i

3

=

1

mod

7

. This implies

i

=

1

,

2

, or 4.

Case 1:

y

⁢

x

⁢

y

-

1

=

x

, so

G

is

abelian

and

isomorphic

to

C

21

=

C

3

×

C

7

.

Case 2:

y

⁢

x

⁢

y

-

1

=

x

2

, then every

product

of the elements

x

,

y

can be reduced to one in the form

x

i

⁢

y

j

,

0

≤

i

<

7

,

0

≤

j

<

3

. These 21 elements are clearly distinct, so

G

=

⟨

x

,

y

∣

x

7

=

y

3

=

1

,

y

⁢

x

=

x

2

⁢

y

⟩

.

Case 3:

y

⁢

x

⁢

y

-

1

=

x

4

, then since

y

2

is also a generator of

H

and

y

2

⁢

x

⁢

y

-

2

=

y

⁢

x

4

⁢

y

-

1

=

x

16

=

x

2

, we have recovered
case 2 above.

Title

example of

groups of order pq

Canonical name

ExampleOfGroupsOfOrderPq

Date of creation

2013-03-22 14:51:15

Last modified on

2013-03-22 14:51:15

Owner

jh (7326)

Last modified by

jh (7326)

Numerical id

4

Author

jh (7326)

Entry type

Example

Classification

msc 20D20

Generated on Fri Feb 9 19:23:46 2018 by

LaTeXML
Frattini argument

Frattini argument

If

H

is a finite

normal subgroup

of a group

G

, and

S

is a

Sylow subgroup

of

H

, then

G

=

H

⁢

N

G

⁢

(

S

)

, where

N

G

⁢

(

S

)

is the

normalizer

of

S

in

G

.

Title

Frattini argument

Canonical name

FrattiniArgument

Date of creation

2013-03-22 13:16:02

Last modified on

2013-03-22 13:16:02

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20D20

Generated on Fri Feb 9 19:23:48 2018 by

LaTeXML