Mathieu groups

Mathieu groups

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automorphism group

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automorphism groups

The

Mathieu groups

are a of 5 sporadic simple groups discovered by the French mathematician Émile Léonard Mathieu. They are usually denoted by

M

11

,

M

12

,

M

22

,

M

23

,

M

24

. They are defined as automorphism groups of certain

Steiner systems

, and the

subscript

denotes the size of the underlying set of the Steiner system.

If

Ω

is a set of

n

elements

, then a

(

t

,

k

,

n

)

-Steiner system on

Ω

is a set

S

of subsets of

Ω

, each of size

k

, such that every subset of

Ω

of size

t

is

contained

in a unique element of

S

. The automorphism group of the Steiner system is defined as the

permutations

of

Ω

which map

S

to itself.

There exists a (5,8,24)-Steiner system, and it is unique up to permutation of the elements of

Ω

. It can be constructed as the set of octads of the

extended binary Golay Code

𝒢

24

. We denote it by

S

⁢

(

5

,

8

,

24

)

below.

There exists a (5,6,12)-Steiner system, and it is unique up to permutation of the elements. It can be constructed as follows. Take

Ω

′

to be a

dodecad

(element of weight 12) of

𝒢

24

. Then the subsets of size 6 in

Ω

′

which are contained in an octad of

𝒢

24

form a (5,6,12)-Steiner system. We denote it by

S

⁢

(

5

,

6

,

12

)

below.

1

Definition of the Mathieu groups

The group

M

24

is the automorphism group of

S

⁢

(

5

,

8

,

24

)

. It has order

244 823 040

=

2

10

⋅

3

3

⋅

5

⋅

7

⋅

11

⋅

23

.

The group

M

23

is the

subgroup

of

M

24

fixing a given of

Ω

. It is the automorphism group of a (4,7,23)-Steiner system). It has order

|

M

24

|

/

24

=

10 200 960

=

2

7

⋅

3

2

⋅

5

⋅

7

⋅

11

⋅

23

.

The group

M

22

is the subgroup of

M

24

fixing two given of

Ω

. It is a subgroup of index (

http://planetmath.org/Coset

) 2 in the automorphism group of a (3,6,22)-Steiner system. It has order

|

M

23

|

/

23

=

443 250

=

2

7

⋅

3

2

⋅

5

⋅

7

⋅

11

.

The group

M

12

is the automorphism group of

S

⁢

(

5

,

6

,

12

)

. It has order

95 040

=

2

6

⋅

3

3

⋅

5

⋅

11

.

The group

M

11

is the subgroup of

M

12

fixing a of

Ω

′

. It is the automorphism group of a (4,5,11)-Steiner system. It has order

|

M

12

/

12

|

=

7 920

=

2

4

⋅

3

2

⋅

5

⋅

11

.

For further on the Mathieu groups, consult the (

[

1

]

, Chapters 10 and 11) and (

[

2

]

, Chapters 5-7).

Note.

It is possible to continue the

pattern

above and define groups

M

21

,

M

20

,

M

10

,

M

9

,

M

8

. However, they are no longer sporadic simple groups. The group

M

21

is a subgroup of 3! = 6 in the automorphism group of a (2,5,21)-Steiner system, which are the points and lines of the

projective plane

over the field of 4 elements. In fact,

M

21

≅

P

⁢

S

⁢

L

⁢

(

3

,

𝔽

4

)

, a

simple group

of Lie type (

http://planetmath.org/ProjectiveSpecialLinearGroup

). The group

M

20

is a

solvable group

. The group

M

10

is not simple; it contains, with 2, the

alternating group

A

6

(sometimes denoted

M

10

′

in this

context

since it is the

derived subgroup

of

M

10

). The groups

M

9

and

M

8

are solvable.

References

1

J. H. Conway and N. J. A. Sloane.

Sphere Packings

, Lattices, and Groups. Springer-Verlag, 1999.

2

Robert L. Griess, Jr. Twelve

Sporadic Groups

. Springer-Verlag, 1998.

Title

Mathieu groups

Canonical name

MathieuGroups

Date of creation

2013-03-22 18:43:33

Last modified on

2013-03-22 18:43:33

Owner

monster (22721)

Last modified by

monster (22721)

Numerical id

7

Author

monster (22721)

Entry type

Definition

Classification

msc 20B20

Classification

msc 20D08

Defines

M

24

Defines

M

23

Defines

M

22

Defines

M

12

Defines

M

11

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