finite nilpotent groups

finite nilpotent groups

The study of

finite nilpotent groups

mostly centers around the study of

p

-groups. This is because of the following two

theorems

.

Theorem 1

.

Finite (

http://planetmath.org/Finite

)

p

-groups are

nilpotent

.

Proof.

From the

class equation

we know the center of a finite

p

-group is non-trivial.
Thus by

induction

the upper central series of a

p

-group

P

terminates at

P

.
So

P

is nilpotent.
∎

Example.

Infinite

p

-groups may not always be nilpotent. In the extreme there are counterexamples like the

Tarski monsters

T

p

. These are infinite

p

-groups in which every

proper subgroup

has order

p

. Therefore given any two

non-trivial elements

x

,

y

in which

y

∉

⟨

x

⟩

generate

T

p

. In particular, the only central element is 1 so that the upper central series is

trivial

and therefore

T

p

is not nilpotent.

Indeed, Tarski monsters are not in fact

solvable groups

which is a weaker property than nilpotent.

□

Example.

Some infinite

p

-groups are nilpotent. Indeed, some infinite

p

-groups are even

abelian

such as

ℤ

p

∞

– the

countable

dimension

vector space

over the field

ℤ

p

– and the Prüfer group

ℤ

p

∞

– the inductive limit of

ℤ

p

n

.

□

Theorem 2

.

Let

G

be a

finite group

. Then all the following are

equivalent

.

1.

G

is nilpotent.

2.

Every

Sylow subgroup

of

G

is normal.

3.

For every prime

p

|

|

G

|

, there exists a unique Sylow

p

-

subgroup

of

G

.

4.

G

is the

direct product

of its Sylow subgroups.

For the proof recal the following

consequence

of the

Sylow theorems

:

Proposition 3

.

If

G

is a finite group and

P

a Sylow

p

-subgroup of

G

then

N

G

⁢

(

N

G

⁢

(

P

)

)

=

N

G

⁢

(

P

)

.

(See Subgroups Containing The

Normalizers

Of Sylow Subgroups

Normalize

Themselves)

Now we prove Theorem

2

Proof.

(

1

) implies (

2

). Suppose that

G

is nilpotent and that

P

is a Sylow

p

-subgroup of

G

. Then as

G

is nilpotent, every subgroup of

G

is

subnormal

in

G

, meaning, if

H

is properly

contained

in

G

then

N

G

⁢

(

H

)

properly contains

H

. Thus

N

G

⁢

(

N

G

⁢

(

P

)

)

is larger than

N

G

⁢

(

P

)

or

N

G

⁢

(

P

)

=

G

. However because

P

is a Sylow

p

-subgroup we know

N

G

⁢

(

P

)

=

N

G

⁢

(

N

G

⁢

(

P

)

)

so we conclude

N

G

⁢

(

P

)

=

G

. Therefore every Sylow

p

-subgroup of

G

is normal in

G

.

(

2

) implies (

3

). Suppose every Sylow subgroup of

G

is normal in

G

. Then by the Sylow theorems we know that for every prime

p

dividing

|

G

|

there is exactly one Sylow

p

-subgroup of

G

– as all Sylow

p

-subgroups are

conjugate

and here by

assumption

all are also normal.

(

3

) implies (

4

). Suppose that there is a unique Sylow

p

-subgroup of

G

for every

p

|

|

G

|

. Then by the Sylow theorems every Sylow subgroup of

G

is normal in

G

. Furthermore, if

P

and

Q

are two distinct Sylow subgroups then they they are Sylow subgroups for different primes so that by Lagrange’s theorem their

intersection

is trivial. Let

P

1

,

…

,

P

k

the Sylow subgroups of

G

. Then as each

P

i

is normal in

G

we have

G

=

P

1

⁢

⋯

⁢

P

k

and we have also demonstrated

P

1

⁢

⋯

⁢

P

i

∩

P

i

+

1

=

1

for

2

≤

i

≤

k

therefore

G

is the direct product of

P

1

,

…

,

P

k

.

(

4

) implies (

1

). Suppose that

G

is a

product

of its Sylow subgroups. Then as every Sylow subgroup is a

p

-group,

G

is a product of

nilpotent groups

so

G

itself is nilpotent.
∎

Title

finite nilpotent groups

Canonical name

FiniteNilpotentGroups

Date of creation

2013-03-22 15:46:40

Last modified on

2013-03-22 15:46:40

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

18

Author

Algeboy (12884)

Entry type

Topic

Classification

msc 20D15

Classification

msc 20E15

Classification

msc 20E34

Related topic

NilpotentGroup

Related topic

DirectProductAndRestrictedDirectProductOfGroups

Related topic

ClassificationOfFiniteNilpotentGroups

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