Klein 4-group

Klein 4-group

The

Klein 4-group

is the

subgroup

V

(

Vierergruppe

) of

S

4

(see

symmetric group

) consisting of the following
4

permutations

:

(

)

,

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

.

(see

cycle notation

). This is an

abelian group

,

isomorphic

to the

product

ℤ

2

⊕

ℤ

2

.
The group is named after

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Klein.html

Felix Klein,
a pioneering figure in the field of geometric group theory.

1

Klein 4-group as a symmetry group

The group

V

is isomorphic to the

automorphism group

of various

planar
graphs

, including graphs of 4 vertices. Yet we have

Proposition 1

.

V

is not the automorphism group of a

simple graph

.

Proof.

Suppose

V

is the automorphism group of a simple graph

G

.
Because

V

contains the permutations

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

and

(

14

)

⁢

(

23

)

it follows the degree of every vertex is the same – we can map
every vertex to every other. So

G

is a

regular graph

on 4 vertices.
This makes

G

isomorphic to one of the following 4 graphs:

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

1

=

"

1

"

;

(

1

,

0

)

*

+

2

=

"

2

"

;

(

1

,

1

)

*

+

3

=

"

3

"

;

(

0

,

1

)

*

+

4

=

"

4

"

;

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

1

=

"

1

"

;

(

1

,

0

)

*

+

2

=

"

2

"

;

(

1

,

1

)

*

+

3

=

"

3

"

;

(

0

,

1

)

*

+

4

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

1

=

"

1

"

;

(

1

,

0

)

*

+

2

=

"

2

"

;

(

1

,

1

)

*

+

3

=

"

3

"

;

(

0

,

1

)

*

+

4

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

"

4

"

;

"

1

"

*

*

@

-

;

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

1

=

"

1

"

;

(

1

,

0

)

*

+

2

=

"

2

"

;

(

-

0.5

,

0.86

)

*

+

3

=

"

3

"

;

(

-

0.5

,

-

0.86

)

*

+

4

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

1

"

;

"

3

"

*

*

@

-

;

"

1

"

;

"

4

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

2

"

;

"

4

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

.

In order the automorphism groups of these graphs are

S

4

,

⟨

(

12

)

,

(

34

)

⟩

,

⟨

(

12

)

,

(

1234

)

⟩

and

S

4

.
None of these are

V

, though the second is isomorphic to

V

.
∎

Though

V

cannot be realized as an automorphism group of a planar graph
it can be realized as the set of

symmetries

of a

polygon

, in particular,
a non-square

rectangle

.

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

1

=

"

1

"

;

(

2

,

0

)

*

+

2

=

"

2

"

;

(

2

,

1

)

*

+

3

=

"

3

"

;

(

0

,

1

)

*

+

4

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

"

4

"

;

"

1

"

*

*

@

-

;

We can rotate by

180

∘

which corresponds to the permutation

(

13

)

⁢

(

24

)

. We can also flip the rectangle over the horizontal

diagonal

which gives the permutation

(

14

)

⁢

(

23

)

, and finally also over the vertical
diagonal which gives the permutation

(

12

)

⁢

(

34

)

.

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

3

=

"

1

"

;

(

2

,

0

)

*

+

4

=

"

2

"

;

(

2

,

1

)

*

+

1

=

"

3

"

;

(

0

,

1

)

*

+

2

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

"

4

"

;

"

1

"

*

*

@

-

;

,

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

4

=

"

1

"

;

(

2

,

0

)

*

+

3

=

"

2

"

;

(

2

,

1

)

*

+

2

=

"

3

"

;

(

0

,

1

)

*

+

1

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

"

4

"

;

"

1

"

*

*

@

-

;

,

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

2

=

"

1

"

;

(

2

,

0

)

*

+

1

=

"

2

"

;

(

2

,

1

)

*

+

4

=

"

3

"

;

(

0

,

1

)

*

+

3

=

"

4

"

;

"

1

"

;

"

2

"

*

*

@

-

;

"

2

"

;

"

3

"

*

*

@

-

;

"

3

"

;

"

4

"

*

*

@

-

;

"

4

"

;

"

1

"

*

*

@

-

;

.

An important corollary to this realization is

Proposition 2

.

Given a square with vertices labeled in any way by

{

1

,

2

,

3

,

4

}

, then the
full

symmetry group

(the

dihedral group

of order 8,

D

8

) contains

V

.

2

Klein 4-group as a vector space

As

V

is isomorphic to

ℤ

2

⊕

ℤ

2

it is a 2-dimensional

vector space

over the Galois field

ℤ

2

. The

projective geometry

of

V

– equivalently, the

lattice of subgroups

– is given in the following
Hasse diagam:

{xy}

<

10

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

0

)

*

+

⟨

(

)

⟩

=

"

1.1

"

;

(

-

2

,

1

)

*

+

⟨

(

12

)

(

34

)

⟩

=

"

2.1

"

;

(

0

,

1

)

*

+

⟨

(

13

)

(

24

)

⟩

=

"

3.1

"

;

(

2

,

1

)

*

+

⟨

(

14

)

(

23

)

⟩

=

"

4.1

"

;

(

0

,

2

)

*

+

V

=

"

5.1

"

;

"

2.1

"

;

"

1.1

"

*

*

@

-

;

"

3.1

"

;

"

1.1

"

*

*

@

-

;

"

4.1

"

;

"

1.1

"

*

*

@

-

;

"

5.1

"

;

"

4.1

"

*

*

@

-

;

"

5.1

"

;

"

3.1

"

*

*

@

-

;

"

5.1

"

;

"

2.1

"

*

*

@

-

;

The automorphism group of a vector space is called the

general linear
group

and so in our

context

Aut

⁡

V

≅

G

⁢

L

⁢

(

2

,

2

)

. As we can interchange
any basis of a vector space we can label the

elements

e

1

=

(

12

)

⁢

(

34

)

,

e

2

=

(

13

)

⁢

(

24

)

and

e

3

=

(

14

)

⁢

(

23

)

so that we have the permutations

(

e

1

,

e

2

)

and

(

e

2

,

e

3

)

and so we generate all permutations on

{

e

1

,

e

2

,

e

3

}

. This proves:

Proposition 3

.

Aut

⁡

V

≅

G

⁢

L

⁢

(

2

,

2

)

≅

S

3

. Furthermore, the affine linear group
of

V

is

A

⁢

G

⁢

L

⁢

(

2

,

2

)

=

V

⋊

S

3

.

3

Klein 4-group as a normal subgroup

Because

V

is a subgroup of

S

4

we can consider its

conjugates

. Because

conjugation

in

S

4

respects the cycle

structure

. From this we see that
the

conjugacy class

in

S

4

of every element of

V

lies again in

V

. Thus

V

is normal. This now allows us to combine both of the previous

sections

to outline the exceptional nature (amongst

S

n

families) of

S

4

. We
collect these into

Theorem 4

.

1.

V

is a

normal subgroup

of

S

4

.

2.

V

is

contained

in

A

4

and so it is a normal subgroup of

A

4

.

3.

V

is the Sylow 2-subgroup of

A

4

.

4.

V

is the

intersection

of all Sylow 2-subgroups of

S

4

, that is,
the

2

-core of

S

4

.

5.

S

4

/

V

≅

S

3

.

6.

S

4

≅

A

⁢

G

⁢

L

⁢

(

2

,

2

)

≅

V

⋊

S

3

.

Proof.

We have already argued that

V

is normal in

S

4

. Upon inspecting the
elements of

V

we see

V

contains only

even permutations

so

V

≤

A

4

and consequently

V

is normal in

A

4

as well. As

|

A

4

|

=

12

and

|

V

|

=

4

we establish

V

is a Sylow 2-subgroup of

A

4

. But

V

is normal so it
the Sylow 2-subgroup of

A

4

(

Sylow subgroups

are conjugate.)

Now notice that the dihedral group

D

8

acts on a square and so it is
represented as a

permutation group

on 4 vertices, so

D

8

embeds in

S

4

.
As

|

D

8

|

=

8

and

|

S

4

|

=

24

,

D

8

is a Sylow 2-subgroup of

S

4

and so
all Sylow 2-subgroups of

S

4

are

embeddings

of

D

8

(in particular various
relabellings of the vertices of the square.) But by

Proposition

2

we know that each embedding contains

V

. As there are 3 non-equal
embeddings of

D

8

(think of the 3 non-equal

labellings

of a square) we
know that the intersection of these

D

8

is a

proper subgroup

of

D

8

.
As

V

is a

maximal subgroup

of each

D

8

and contained in each,

V

is
the intersection of all these embeddings.

Now the action of

S

4

by conjugation on the Sylow 2-subgroups

D

8

permutes all 3 (again Sylow subgroups are conjugate) so

S

4

↦

S

3

.
Indeed,

V

is in the kernel of this action as

V

is in each

D

8

.
Indeed a three cycle

(

123

)

permutes the

D

8

’s with no

fixed point

(consider the relabellings) and

(

12

)

fixes only one. So

S

4

maps
onto

S

3

and so the kernel is precisely

V

. Thus

S

4

/

V

=

S

3

.

Now we can embed

S

3

into

S

4

as

⟨

(

123

)

,

(

12

)

⟩

so

V

∩

S

3

=

1

,

V

⁢

S

3

=

S

4

so

S

4

=

V

⋉

S

3

. Finally,

A

⁢

G

⁢

L

⁢

(

2

,

2

)

acts transitively on the four points of the vector space

V

so

A

⁢

G

⁢

L

⁢

(

2

,

2

)

embeds in

S

4

. And by Proposition

3

we conclude

S

4

≅

A

⁢

G

⁢

L

⁢

(

2

,

2

)

.
∎

We can make

similar

arguments

about subgroups of symmetries for
larger

regular polygons

. Likewise for other 2-dimensional vector spaces
we can establish similar structural properties. However it is only
when we study we involve

V

that we find these two methods intersect
in a this exceptionally

parallel

fashion. Thus we establish the exceptional
structure of

S

4

. For all other

S

n

’s,

A

n

is the only proper normal
subgroup.

We can view the properties of our

theorem

in a geometric way as follows:

S

4

is the group of symmetries of a

tetrahedron

. There is an

induced

action
of

S

4

on the six edges of the tetrahedron. Observing that this action preserves

incidence relations

one gets an action of

S

4

on the three pairs
of opposite edges.

4

Other properties

V

is non-cyclic and of smallest possible order with this property.

V

is

transitive

and

regular

. Indeed

V

is the (unique)

regular representation

of

ℤ

2

⊕

ℤ

2

. The other 3 subgroups of

S

4

which are isomorphic to

ℤ

2

⊕

ℤ

2

are not transitive.

V

is the symmetry group of the Riemannian curvature tensor.

Title

Klein 4-group

Canonical name

Klein4group

Date of creation

2013-03-22 12:49:02

Last modified on

2013-03-22 12:49:02

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

26

Author

Algeboy (12884)

Entry type

Topic

Classification

msc 20K99

Synonym

Klein four-group

Synonym

Viergruppe

Related topic

GroupsInField

Related topic

Klein4Ring

Related topic

PrimeResidueClass

Related topic

AbelianGroup2

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