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GroupTheory

  

ClassifyFiniteSimpleGroup

  

classify a finite simple group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

ClassifyFiniteSimpleGroup( G )

ClassifyFiniteSimpleGroup( n )

Parameters

G

-

a finite simple group

n

-

a positive integer; the order of a finite simple group

Description

• 

The ClassifyFiniteSimpleGroup( G ) command returns an object of type CFSG describing the classification of the finite simple group G as belonging to one of several families of finite simple groups as described below.

• 

Since much of the classification depends solely on the order of G, you can also use a positive integer n that is the order of some finite simple group. In this case, if there are two possibilities (and there are at most two for any given order), an expression sequence of CFSG objects is returned describing the two possibilities.

• 

The returned CFSG object c supports several methods that you can use to query the object for information about how the given group fits into the classification of finite simple groups. See CFSG for details on these methods, and for more information about the classification itself.

• 

If the group G is not simple, or if the positive integer n is not the order of a finite simple group, then an exception is raised.

• 

By default, the ClassifyFiniteSimpleGroup checks that the input group is simple. Since this can be expensive for a large group, if you know that your group is simple, you can avoid this check by passing the check = false option. This has no effect when the first argument is an integer n.

Examples

withGroupTheory:

ClassifyFiniteSimpleGroupAlt5

CFSG: Alternating Group A5

(1)

ClassifyFiniteSimpleGroupAlt8

CFSG: Alternating Group A8

(2)

ClassifyFiniteSimpleGroupAlt500

CFSG: Alternating Group A500

(3)

ClassifyFiniteSimpleGroupPSL3,4

CFSG: Chevalley Group A24=PSL3,4

(4)

ClassifyFiniteSimpleGroupCyclicGroup17

CFSG: Cyclic Group C17

(5)

ClassifyFiniteSimpleGroupHeldGroup

CFSG: Sporadic Group He

(6)

ClassifyFiniteSimpleGroupOrthogonalGroupO10+(2)

CFSG: Chevalley Group D52=PΩ+10,2

(7)

ClassifyFiniteSimpleGroupithprime100

CFSG: Cyclic Group C541

(8)

ClassifyFiniteSimpleGroup241313567211131719233147

CFSG: Sporadic Group

(9)

The alternating group of degree 4 is not simple, so the following command raises an exception.

ClassifyFiniteSimpleGroupAlt4

Error, (in GroupTheory:-ClassifyFiniteSimpleGroup) group is not simple

The check = false option can be useful in a situation like the following.

Ga,b|a2,b3,a·b5=1

Ga,ba2,b3,ababababab

(10)

ClassifyFiniteSimpleGroupG

Error, (in GroupTheory:-IsSimple) cannot determine whether a general finitely presented group is simple; try converting to a permutation group

ClassifyFiniteSimpleGroupG,check=false

CFSG: Alternating Group A5

(11)

Alternatively since, in this case, the group is small, you could convert it to a permutation group, as follows.

ClassifyFiniteSimpleGroupPermutationGroupG

CFSG: Alternating Group A5

(12)

Compatibility

• 

The GroupTheory[ClassifyFiniteSimpleGroup] command was introduced in Maple 2020.

• 

For more information on Maple 2020 changes, see Updates in Maple 2020.

See Also

GroupTheory[AlternatingGroup]

GroupTheory[CFSG]

GroupTheory[CyclicGroup]

GroupTheory[HeldGroup]

GroupTheory[IsSimple]

GroupTheory[PSU]

ithprime