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GroupTheory

  

SymplecticGroup

  

construct a permutation group isomorphic to a symplectic group

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SymplecticGroup(n, q)

Parameters

n

-

an even positive integer

q

-

power of a prime number

Description

• 

The symplectic group Spn,q  is the group of all n×n matrices over the field with q elements that respect a fixed nondegenerate symplectic form. The integer n must be even.

• 

The SymplecticGroup( n, q ) command returns a permutation group isomorphic to the symplectic group Spn,q  .

• 

Note that for n=2 the groups Spn,q  and SLn,q  are isomorphic, so that a special linear group is returned in this case.

• 

The implemented ranges for n and q are as follows:

n=2

q100

n=4

q10

n=6

q4

n=8,10

q=2

• 

If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.

• 

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

Examples

withGroupTheory:

GSymplecticGroup4,5

GSp4,5

(1)

ifactorGroupOrderG

27325413

(2)

GroupOrderSylowSubgroup2,G

128

(3)

S3SylowSubgroup3,G

S3<a permutation group on 624 letters>

(4)

GroupOrderS3

9

(5)

IsCyclicS3

false

(6)

IdentifySmallGroupS3

9,2

(7)

GroupOrderSylowSubgroup5&comma;G

625

(8)

IsTrivialPCore5&comma;G

true

(9)

GroupOrderSylowSubgroup13&comma;G

13

(10)

GSymplecticGroup4&comma;3

GSp4&comma;3

(11)

DegreeG

80

(12)

IsSimpleG

false

(13)

GroupOrderCentreG

2

(14)

For n=2 the corresponding special linear group is returned.

SymplecticGroup2&comma;5

SL2&comma;5

(15)

Note the exceptional isomorphism:

AreIsomorphicSymplecticGroup4&comma;2&comma;Symm6

true

(16)

GSymplecticGroup6&comma;q

GSp6&comma;q

(17)

GroupOrderG

q9q21q41q61

(18)

ClassNumberSymplecticGroup8&comma;q

5q+q+1q+4q2+q2+q+3q+q4+q3+7q::even25q+51+q+4q+11q2+q2+4q+10q+q4+4q3otherwise

(19)

ClassNumberSymplecticGroup4&comma;11kassumingk::&apos;posint&apos;

511k+10+11k2

(20)

Compatibility

• 

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

• 

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

• 

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

See Also

GroupTheory[AreIsomorphic]

GroupTheory[ClassNumber]

GroupTheory[Degree]

GroupTheory[Generators]

GroupTheory[GroupOrder]

GroupTheory[ProjectiveSymplecticGroup]

GroupTheory[SpecialLinearGroup]

GroupTheory[SymmetricGroup]