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MathieuGroup

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

MathieuGroup( n, formopt )

Parameters

n

-

an integer in { 9, 10, 11, 12, 21, 22, 23, 24 }

formopt

-

(optional) equation of the form form = F, where F is either "permgroup" (the default) or "fpgroup"

Description

• 

The Mathieu groups Mn, for n in 9,10,11,12,21,22,23,24 are a family of transitive permutation groups studied by Émile Mathieu in the late nineteenth century.  The simple groups in the family are examples of highly transitive groups. The Mathieu group Mn is simple for n in 11,12,21,22,23,24.

• 

Note that while the Mathieu group M21 of order 20160 is simple, it is not sporadic, being isomorphic to the group PSL3,4 .

• 

The MathieuGroup( n ) command returns a permutation group isomorphic to the Mathieu group of degree n, where the degree n must be in { 9, 10, 11, 12, 21, 22, 23, 24 }. This is a sporadic finite simple group for n=11, 12, 22, 23, 24.

• 

The Mathieu group M9 is, in fact a soluble group.

• 

The form = F option controls the form of the group returned. By default, a permutation group is returned; this is equivalent to passing the option form = "permgroup". A finitely presented group can be obtained by passing the option form = "fpgroup".

• 

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

withGroupTheory:

MathieuGroup11

GroupTheory:-MathieuGroup11

(1)

GMathieuGroup23

GroupTheory:-MathieuGroup23

(2)

typeG,PermutationGroup

true

(3)

GroupOrderG

10200960

(4)

TransitivityG

4

(5)

GMathieuGroup12

GroupTheory:-MathieuGroup12

(6)

DegreeG

12

(7)

TransitivityG

5

(8)

GMathieuGroup9:

IsSolubleG

true

(9)

DerivedSeriesG

GroupTheory:-DerivedSeriesmodule...end module

(10)

DisplayCharacterTableMathieuGroup10

C

1a

2a

3a

4a

4b

5a

8a

8b

|C|

1

45

80

90

180

144

90

90

 

 

 

 

 

 

 

 

 

χ__1

1

1

1

1

1

1

1

1

χ__2

1

1

1

1

−1

1

−1

−1

χ__3

9

1

0

1

−1

−1

1

1

χ__4

9

1

0

1

1

−1

−1

−1

χ__5

10

−2

1

0

0

0

−I2

I2

χ__6

10

−2

1

0

0

0

I2

−I2

χ__7

10

2

1

−2

0

0

0

0

χ__8

16

0

−2

0

0

1

0

0

GMathieuGroup11,form=fpgroup

GroupTheory:-FPGroupa,b,a,a,b,b,b,b,a,b,b,a,b,b,a,b,b,a,b,b,a,b,b,a,b,b,a,b,a,b,a,1b,a,b,a,b,b,a,1b,a,b,a,1b,a,1b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b

(11)

Compatibility

• 

The GroupTheory[MathieuGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

• 

The GroupTheory[MathieuGroup] command was updated in Maple 2020.

See Also

GroupTheory[CharacterTable]

GroupTheory[Degree]

GroupTheory[DerivedSeries]

GroupTheory[GroupOrder]

GroupTheory[IsSoluble]

GroupTheory[IsTransitive]

GroupTheory[Transitivity]