subgroups of S_4

subgroups of

S

4

The

symmetric group

on

4

letters,

S

4

, has

24

elements

. Listed by

cycle type

, they are:

Cycle type

Number of elements

elements

1

,

1

,

1

,

1

1

(

)

2

,

1

,

1

6

(

12

)

,

(

13

)

,

(

14

)

,

(

23

)

,

(

24

)

,

(

34

)

3

,

1

8

(

123

)

,

(

132

)

,

(

124

)

,

(

142

)

,

(

134

)

,

(

143

)

,

(

234

)

,

(

243

)

2

,

2

3

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

4

6

(

1234

)

,

(

1243

)

,

(

1324

)

,

(

1342

)

,

(

1423

)

,

(

1432

)

Any

subgroup

of

S

4

must be generated by some subset of these elements, and must have order (

http://planetmath.org/OrderGroup

) dividing

24

, so must be one of

1

,

2

,

3

,

4

,

6

,

8

, or

12

.

Think of

S

4

as acting on the set of “letters”

Ω

=

{

1

,

2

,

3

,

4

}

by permuting them. Then each subgroup

G

of

S

4

acts either transitively or intransitively. If

G

is

transitive

, then by the

orbit-stabilizer theorem

, since there is only one orbit we have that the order of

G

is a

multiple

of

|

Ω

|

=

4

. Thus all the transitive subgroups are of orders

4

,

8

, or

12

. If

G

is intransitive, then

G

has at least two orbits on

Ω

. If one orbit is of size

k

for

1

≤

k

<

4

, then

G

can naturally be thought of as (

isomorphic

to) a subgroup of

S

k

×

S

n

-

k

. Thus all intransitive subgroups of

S

4

are isomorphic to subgroups of

S

2

×

S

2

≅

ℤ

/

2

⁢

ℤ

×

ℤ

/

2

⁢

ℤ

≅

V

4

S

1

×

S

3

≅

S

3

Looking first at subgroups of order

12

, we note that

A

4

is one such subgroup (and must be transitive, by the above

analysis

):

A

4

=

{

e

,

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

,

(

123

)

,

(

132

)

,

(

124

)

,

(

142

)

,

(

134

)

,

(

143

)

,

(

234

)

,

(

243

)

}

Any other subgroup

G

of order

12

must contain at least one element of order

3

, and must also contain an element of order

2

. It is easy to see that if

G

contains two elements of order three that are not

inverses

, then

G

=

A

4

, while if

G

contains exactly two elements of order three which are inverses, then it contains at least one element with cycle type

2

,

2

. But any such element together with a

3

-cycle generates

A

4

. Thus

A

4

is the only subgroup of

S

4

of order

12

.

We look next at order

8

subgroups. These subgroups are

2

-

Sylow subgroups

of

S

4

, so they are all

conjugate

and thus isomorphic. The number of them is odd and divides

24

/

8

=

3

, so is either

1

or

3

. But

S

4

has three

conjugate subgroups

of order

8

that are all isomorphic to

D

8

, the

dihedral group

with

8

elements:

{

e

,

(

1324

)

,

(

1423

)

,

(

12

)

(

34

)

,

(

14

)

(

23

)

,

(

13

)

(

24

)

,

(

12

)

,

(

34

)

}

{

e

,

(

1234

)

,

(

1432

)

,

(

13

)

(

24

)

,

(

12

)

(

34

)

,

(

14

)

(

23

)

,

(

13

)

,

(

24

)

}

{

e

,

(

1342

)

,

(

1243

)

,

(

14

)

(

23

)

,

(

13

)

(

24

)

,

(

12

)

(

34

)

,

(

14

)

,

(

23

)

}

and so these are the only subgroups of order

8

(which must also be transitive).

All subgroups of order

6

must be intransitive by the above analysis since

4

∤

6

, so by the above, a subgroup of order

6

must be isomorphic to

S

3

and thus must be the image of an

embedding

of

S

3

into

S

4

.

S

3

is generated by

transpositions

(as is

S

n

for any

n

), so we can determine embeddings of

S

3

into

S

4

by looking at the image of transpositions. But the images of the three transpositions in

S

3

are determined by the images of

(

12

)

and

(

13

)

since

(

23

)

=

(

12

)

⁢

(

13

)

⁢

(

12

)

. So we may send

(

12

)

and

(

13

)

to any pair of transpositions in

S

4

with a common element; there are four such pairs and thus four embeddings. These correspond to four distinct subgroups of

S

4

, all conjugate, and all isomorphic to

S

3

:

{

e

,

(

12

)

,

(

13

)

,

(

23

)

,

(

123

)

,

(

132

)

}

{

e

,

(

13

)

,

(

14

)

,

(

34

)

,

(

134

)

,

(

143

)

}

{

e

,

(

23

)

,

(

24

)

,

(

34

)

,

(

234

)

,

(

243

)

}

{

e

,

(

12

)

,

(

14

)

,

(

24

)

,

(

124

)

,

(

142

)

}

(The fact that transpositions in

S

3

must be mapped to transpositions in

S

4

rather than elements of cycle type

2

,

2

is left to the reader).

We shall see that some subgroups of order

4

are transitive while others are intransitive. A subgroup of order four is clearly isomorphic to either

ℤ

/

4

⁢

ℤ

or to

ℤ

/

2

⁢

ℤ

×

ℤ

/

2

⁢

ℤ

. The only elements of order

4

are the

4

-cycles, so each

4

-cycle generates a subgroup isomorphic to

ℤ

/

4

⁢

ℤ

, which also contains the inverse of the

4

-cycle. Since there are six

4

-cycles,

S

4

has three

cyclic subgroups

of order

4

, and each is obviously transitive:

{

e

,

(

1234

)

,

(

13

)

(

24

)

,

(

1432

)

}

{

e

,

(

1243

)

,

(

14

)

(

23

)

,

(

1342

)

}

{

e

,

(

1324

)

,

(

12

)

(

34

)

,

(

1423

)

}

A subgroup isomorphic to

ℤ

/

2

⁢

ℤ

×

ℤ

/

2

⁢

ℤ

has, in

addition

to the

identity

, three elements

σ

1

,

σ

2

,

σ

3

of order

2

, and thus of cycle types

2

,

1

,

1

or

2

,

2

. There are several possibilities:

•

All three of the

σ

i

are of cycle type

2

,

1

,

1

. Then the

product

of any two of those is a

3

-cycle or of cycle type

2

,

2

, which is a

contradiction

.

•

Two of the

σ

i

are of cycle type

2

,

2

. Then the third is as well, since the product of any pair of the elements of

S

4

of cycle type

2

,

2

is the third such. In this case, the group is

{

e

,

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

}

This group acts transitively.

•

One of the

σ

i

is of cycle type

2

,

2

and the other two are of cycle type

2

,

1

,

1

. In this case, the two

2

-cycles must be

disjoint

, since otherwise their product is a

3

-cycle, so the group looks like

{

e

,

(

12

)

,

(

34

)

,

(

12

)

⁢

(

34

)

}

or one of its conjugates (of which there are two). These groups are intransitive, each having two orbits of size

2

.

Finally, we have a number of subgroups of order

2

and

3

generated by elements of those orders; all of these are intransitive.

Summing up,

S

4

has the following subgroups up to

isomorphism

and

conjugation

:

Order

Conjugates

Group

12

1

A

4

(transitive)

8

3

{

e

,

(

1324

)

,

(

1423

)

,

(

12

)

⁢

(

34

)

,

(

14

)

⁢

(

23

)

,

(

13

)

⁢

(

24

)

,

(

12

)

,

(

34

)

}

≅

D

8

(transitive)

6

4

{

e

,

(

12

)

,

(

13

)

,

(

23

)

,

(

123

)

,

(

132

)

}

≅

S

3

(intransitive)

4

3

{

e

,

(

1234

)

,

(

13

)

⁢

(

24

)

,

(

1432

)

}

≅

ℤ

/

4

⁢

ℤ

(transitive)

4

1

{

e

,

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

}

≅

V

4

(transitive)

4

3

{

e

,

(

12

)

,

(

34

)

,

(

12

)

⁢

(

34

)

}

≅

V

4

(intransitive)

3

4

{

e

,

(

123

)

,

(

132

)

}

(intransitive)

2

6

{

e

,

(

12

)

}

(intransitive)

2

3

{

e

,

(

12

)

⁢

(

34

)

}

(intransitive)

1

1

{

e

}

(intransitive)

Of these, the only proper nontrivial

normal subgroups

of

S

4

are

A

4

and the group

{

e

,

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

,

(

14

)

⁢

(

23

)

}

≅

V

4

(see the article on normal subgroups of the symmetric groups).

The

subgroup lattice

of

S

4

is thus (listing only one group in each

conjugacy class

, and taking liberties identifying isomorphic images as subgroups):

\xymatrix

⁢

@

⁢

R

⁢

1

⁢

p

⁢

c

⁢

@

⁢

C

⁢

1

⁢

p

⁢

c

⁢

&

⁢

&

⁢

S

4

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

⁢

d

⁢

d

]

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

(

24

)

⁢

&

⁢

&

⁢

A

4

⁢

\ar

⁢

@

-

[

l

⁢

d

⁢

d

⁢

d

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

(

12

)

⁢

&

⁢

&

⁢

&

⁢

&

⁢

D

8

⁢

\ar

⁢

@

-

[

l

⁢

d

⁢

d

]

⁢

\ar

⁢

@

-

[

d

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

(

8

)

⁢

S

3

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

d

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

&

⁢

&

⁢

(

6

)

⁢

&

⁢

&

⁢

&

⁢

⟨

(

12

)

⁢

(

34

)

,

(

13

)

⁢

(

24

)

⟩

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

⟨

(

12

)

,

(

34

)

⟩

⁢

\ar

⁢

@

-

[

d

⁢

d

]

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

⁢

d

]

⁢

&

⁢

⟨

(

1234

)

⟩

⁢

\ar

⁢

@

-

[

l

⁢

d

⁢

d

]

⁢

&

⁢

(

4

)

⁢

&

⁢

⟨

(

123

)

⟩

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

&

⁢

&

⁢

(

3

)

⁢

&

⁢

&

⁢

⟨

(

12

)

⟩

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

⟨

(

12

)

⁢

(

34

)

⟩

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

d

]

⁢

&

⁢

&

⁢

(

2

)

⁢

&

⁢

&

⁢

{

e

}

⁢

&

⁢

&

⁢

&

⁢

&

⁢

(

1

)

Title

subgroups of

S

4

Canonical name

SubgroupsOfS4

Date of creation

2013-03-22 17:40:54

Last modified on

2013-03-22 17:40:54

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

13

Author

rm50 (10146)

Entry type

Topic

Classification

msc 20B30

Classification

msc 20B35

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