proof of simplicity of Mathieu groups

proof of simplicity of Mathieu groups

We give a uniform proof of the

simplicity

of the

Mathieu groups

M

22

,

M

23

, and

M

24

, and the

alternating groups

A

n

(for

n

>

5

), assuming the simplicity of

M

21

≅

P

⁢

S

⁢

L

⁢

(

3

,

𝔽

4

)

and

A

5

≅

P

⁢

S

⁢

L

⁢

(

2

,

𝔽

4

)

. (Essentially, we are assuming that the simplicity of the

projective special linear groups

is known.)

Lemma 1

.

Let

G

act transitively on a set

S

. If

H

is a

normal subgroup

of

G

, then the

transitivity classes

of the action,
restricted to

H

, form a set of blocks for the action of

G

.

Proof.

If

T

,

U

are any transitivity classes for the restricted action, let

t

∈

T

,

u

∈

U

, and

g

∈

G

such
that

g

⁢

t

=

u

. Then

x

↦

g

⁢

x

is a

bijective

map from

T

onto

U

(here we use normality). Hence any element of

G

maps transitivity classes to transitivity classes.
∎

Hence it follows:

Corollary 2

.

Let

G

act primitively (

http://planetmath.org/PrimativeTransitivePermutationGroupOnASet

) on a set

S

. If

H

is a normal subgroup of

G

, then either

H

acts transitively on

S

, or

H

lies in the kernel of the action. If the action is

faithful

, then either

H

=

{

1

}

or

H

is

transitive

.

Theorem 3

.

Let

G

be a group acting primitively and faithfully on a set

S

. Let

K

be the

stabilizer

of some point

s

0

∈

S

, and assume that

K

is simple. Then if

H

is a nontrivial proper normal subgroup of

G

, then

G

is

isomorphic

to the

semidirect product

of

H

by

K

.

H

can be identified with

S

in such a way that

1

∈

H

is identified with

s

0

, the action of

H

becomes left

multiplication

, and the action of

K

becomes

conjugation

.

Proof.

Since

H

∩

K

is a normal subgroup of

K

, it is either

{

1

}

or

K

.

If

H

∩

K

=

K

, then

K

⊂

H

, and since

K

is

maximal

and

H

is proper, we have

K

=

H

. Since

H

is normal and

H

stabilizes

s

0

, then

H

stabilizes every point (since the action is transitive). Since the action is faithful,

K

=

H

=

{

1

}

, a

contradiction

. (This contradiction can also be reached by applying the corollary.)

Therefore,

H

∩

K

=

{

1

}

. So no element of

H

, other than 1, fixes

s

0

. Thus

H

acts freely and transitively on

S

. For any

g

∈

G

, if

g

⁢

s

0

=

s

and

h

⁢

s

=

s

0

, then

h

⁢

g

⁢

s

0

=

s

0

, hence

h

⁢

g

is in

K

. Thus

G

is generated by

H

and

K

. Since

H

is normal and

H

∩

K

=

{

1

}

,

G

is the (internal) semidirect product of

H

by

K

.
∎

Now we come to the main

theorem

from which we will deduce the simplicity results.

Theorem 4

.

Let

G

be a group acting faithfully on a set

S

. Let

s

0

∈

S

and let

K

be the stabilizer of

s

0

. Assume

K

is simple.

1. Assume the action of

G

is

doubly transitive

, and let

H

be a nontrivial proper normal subgroup of

G

. Then

H

is an

elementary abelian

p

-group for some prime

p

. Furthermore,

K

is isomorphic to a

subgroup

of

G

⁢

L

⁢

(

n

,

F

p

)

, and

G

is isomorphic to a subgroup of

A

⁢

G

⁢

L

⁢

(

n

,

F

p

)

, the group of

affine transformations

of

H

.

2. If the action of

G

is triply transitive and

|

S

|

>

3

, then any nontrivial proper normal subgroup of

G

is an elementary abelian 2-group.

3. If the action of

G

is quadruply transitive and

|

S

|

>

4

, then

G

is simple.

Proof.

For part 1, use the identification of

H

with

S

given by the previous theorem. Since the action is doubly transitive, the action by conjugation of

K

is transitive on

H

-

{

1

}

. Therefore, all non-identity elements of

H

have the same order, which must therefore be some prime

p

. Hence

H

is a

p

-group. The center

Z

⁢

(

H

)

is nontrivial, and is preserved by all

automorphisms

. By double

transitivity

again, there is an automorphism taking any

nontrivial element

to any other; hence

H

is

abelian

. Therefore

H

is an elementary abelian

p

-group.

For part 2, we know from part 1 that

H

is isomorphic to an elementary abelian

p

-group and

K

acts as

linear transformations

of

H

. Since the action of

G

is triply transitive, the action of

K

on the nonzero

elements

is doubly transitive. However, if

p

>

2

, then the linearity of the action disallows double transitivity (if

x

↦

y

, then

2

⁢

x

↦

2

⁢

y

so we do not have

complete

freedom for any two elements since

H

some element besides 0,

y

and

2

⁢

y

.)

(We note that when

|

S

|

=

3

, we have the example

G

=

S

3

,

H

=

A

3

,

K

=

S

2

.)

Here is an example illustrating part 2. The group

A

⁢

G

⁢

L

⁢

(

n

,

𝔽

2

)

acts triply transitively on

𝔽

2

n

, and the stabilizer of a point is

G

⁢

L

⁢

(

n

,

𝔽

2

)

, which is simple if

n

>

2

.

A

⁢

G

⁢

L

⁢

(

n

,

𝔽

2

)

contains the normal subgroup of

translations

, an elementary abelian 2-group.

For part 3, note that the action of

G

⁢

L

⁢

(

n

,

𝔽

2

)

on

𝔽

2

n

,

n

>

2

, is not triply transitive on nonzero elements, so the only

conclusion

left is that

G

is simple.
∎

Corollary 5

.

The Mathieu groups

M

21

,

M

22

,

M

23

, and

M

24

are simple.

Proof.

We take it as known that

M

21

≅

P

⁢

S

⁢

L

⁢

(

3

,

𝔽

4

)

is simple. Since

M

n

has

M

n

-

1

as point stabilizer, and has a triply transitive action on a set of

n

elements, we may work our way inductively up to

M

24

, using the previous theorem. The index (

http://planetmath.org/Coset

) of

M

n

-

1

in

M

n

is

n

, which is not a power of 2. Hence in all cases,

M

n

is simple.
∎

Corollary 6

.

The alternating groups

A

n

are simple for

n

≥

5

.

Proof.

Since the natural action of

A

n

on

n

letters is quadruply transitive for

n

≥

6

, and the point stabilizer of

A

n

is

A

n

-

1

, we may apply the theorem to deduce the simplicity of the alternating groups

A

n

,

n

≥

5

, from the simplicity of

A

5

≅

P

⁢

S

⁢

L

⁢

(

2

,

𝔽

4

)

≅

P

⁢

S

⁢

L

⁢

(

2

,

𝔽

5

)

.
∎

Title

proof of simplicity of Mathieu groups

Canonical name

ProofOfSimplicityOfMathieuGroups

Date of creation

2013-03-22 18:44:08

Last modified on

2013-03-22 18:44:08

Owner

monster (22721)

Last modified by

monster (22721)

Numerical id

8

Author

monster (22721)

Entry type

Proof

Classification

msc 20D08

Classification

msc 20B20

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